Summarize three readings on critical issues of social justice and equity (total of 4 pages)

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Summarize three readings on critical issues of social justice and equity that are relevant to teaching elementary mathematics with technology. You will be given three articles. Write a one page summary for each article, then write a one page synthesis combining all three articles for the fourth page.

DISCUSSION PAPER What are we Afraid of? Arguments against Teaching Mathematics with Technology in the Professional Publications of Organisations for US Mathematicians By Hilary Smith Risser Department of Mathematical Sciences Montana Tech, 1300 W. Park, Butte, MT 59701 More than twenty years after the introduction of the first handheld graphing calculator the mathematics community appears to still be struggling with the use of technology in the teaching and learning of mathematics. One major venue for arguments against technology use in the teaching and learning of mathematics is the news magazines of professional organizations for mathematicians. These magazines are widely read in the mathematics community. An examination of the articles, opinions, and letters written for two such magazines between 2001 and 2009 reveals why some mathematicians are concerned with the use of technology in the learning and teaching of mathematics. The arguments against technology use centre on three main issues: whether technology should change the focus of mathematics curriculum, whether technology use changes how students conceptualize mathematics, and whether the benefits of technology outweigh the costs. These arguments provide a revealing look at what some mathematicians fear are the negative effects of technology use on the learning of mathematics. 1 MIND THE GAP At the beginning of 2001, Madison (2001) described a gulf between United States high school and college mathematics classrooms. Despite the fifteen years that had passed since the introduction of the first handheld graphing calculator, Madison asserted that the difference in the amount of technology use, specifically scientific and graphing calculators, in the teaching and learning of mathematics was one of the major causes of difficulty for students transitioning from K-12 mathematics to college mathematics. While Madison cited anecdotal evidence collected during his work as the chair of the Mathematics Association of America (MAA) Task Force on Articulation in his article, his assertions are supported by research. The National Assessment of Education Progress (NAEP) is conducted to measure K-12 achievement and to survey K-12 administrators, teachers, and students about school characteristics and classroom practices. The survey results from the 2005 administration of NAEP reveal that 84% of United States 12th graders reported using calculators in mathematics class at least once a week and at least 61% reported using them daily (U.S. Department of Education, 2006). When the results are compared to results from surveys of teaching methods at US colleges the gap in the use of technology is clear. Since 1965, the Conference Board of Mathematical Sciences (CBMS) has been compiling data on the state of undergraduate mathematics in the United States. The survey of postsecondary mathematics departments includes questions on the use of technology in the teaching of undergraduate mathematics. While the NAEP data demonstrates a nearly ubiquitous use of calculators in the teaching of secondary mathematics, the CBMS report (Lutzer, Rodi, Kirkman and Maxwell, 2007) indicates that less than half of the sections of introductory mathematics at four year colleges utilised graphing calculators during the teaching of the course. From the CBMS data, it is clear that the use of computers in teaching these courses is even less common. Fewer than 20% of the sections of introductory mathematics at four year colleges utilised computers in instruction. The purpose of this article is to identify some of the fears that could be the potential cause of the gap between K12 and postsecondary technology use through an examination of negative attitudes towards technology use in two news magazines for US mathematicians. Both the MAA and the American Mathematical Society (AMS) publish news magazines which contain the opinions of mathematicians on a variety of issues including technology use. From 2001-2009, both the Notices of the AMS published eighty-eight issues and MAA Focus published seventy-eight issues. The Notices publishes approximately two to four feature articles, one short opinion article, and three to seven other shorter news articles per issue. In addition, some issues include between one and seven letters to the editor. Focus publishes approximately eight to twelve articles per issue. Some of these articles report news of interest to the larger mathematics community, while others are published by regular features departments. As with the Notices, some issues of Focus also contain letters to the editor. In order to locate articles and letters that discussed technology use in teaching, all issues of both magazines from 2001 to 2009 were searched using the keywords “technology”, “computer”, “software”, “online”, and “calculator”. Once the articles and letters that contained these keywords were identified, they were examined to determine whether perceived drawbacks of technology use in International Journal for Technology in Mathematics Education, Vol 18 No 2 [98 H Smith Risser the teaching and/or learning of mathematics were presented by the author. Two opinions discussing perceived negative consequences of technology use in the teaching and/or learning of mathematics were found in the Notices (Quinn, 2005; Quinn 2009) and five were found in Focus (Bressoud, 2009; Dillon, 2006; Jardine, 2001; Vestal, 2008; Wildstrom, 2006). Three letters to the editor were written in response to articles published in the Notices that advocated a specific use of technology in the learning of mathematics (Baker, 2004; Escobales, 2004; Norwood, 2004). In Focus, six letters to the editor were written that asserted negative effects of technology use (Ding, 2007; Kenefic, 2006; Myerson, 2007; Norfolk, 2008; Norwood, 2006; Starr, 2007). Five of these six letters were written in response to articles or opinions advocating a particular use of technology. It is important to note that not all of the mathematicians that expressed concern about the use of technology viewed technology use in general as negative. Indeed many of the authors indicated that they used technology in their research and classified themselves as both early adopters of new technologies and strong proponents of technology use in teaching. Their personal experiences with technology use in teaching and learning gave them a view of both the benefits and drawbacks of technology use. Since this article deals primarily with mathematicians’ fears concerning technology use, arguments against technology use in each article or letter to the editor were identified and grouped. From these groups, three main concerns with the adoption of technology in the teaching and learning of mathematics were identified: concerns about potential changes in curriculum as a result of technology use, concerns about perceived negative effects of technology use on students’ mathematical cognition, and concerns as to whether the drawbacks of technology use outweighed the benefits. 2 SHOULD TECHNOLOGY CHANGE WHAT WE TEACH? The availability of low cost or free technology that is capable of solving many of the routine problems found in K12 or introductory mathematics textbooks has created a debate concerning the value of these types of routine tasks. Before the advent of these technologies, these routine problems could only be done using pencil-and-paper analysis. With the advent of computer algebra systems (CAS) and graphing software, many of the routine manipulation problems that fill high school and introductory college textbooks can be done quickly using technology. In this brave new mathematical world, the question as to whether some skills and/or topics in mathematics are no longer of value. In the analysed articles and letters, mathematicians in the Notices and Focus primarily debate the necessity of teaching computational algorithms for multi-digit operations which previously formed the backbone of the elementary mathematics curriculum (Verschaffel and Corte, 1996). In a 2001 interview (Jackson, 2001), incoming AMS president Hyman Bass summed up the fears of mathematicians as the © Research Information 2011. All rights reserved. “perceived neglect of basic skills” (p. 314) due to the early incorporation of technology into teaching. Anthony Ralston, a numerical mathematician and an outspoken opponent of the emphasis on computational algorithms in elementary school curriculum provided the catalyst for much of this debate. Over the eight year period, Ralston published five separate articles that contained statements advocating for the use of calculators in the teaching of elementary school mathematics (Ralston, 2003; Ralston, 2004; Ralston, 2006; Ralston, 2007; Ralston 2008). One such statement occurs in a review of the book “California Dreaming: Reforming Mathematics Education”, published in the Notices. In this review, Ralston (2003) asserts that the scientific calculator makes pencil-and-paper computational algorithms obsolete. In a letter to the editor Baker (2004) responded to Ralston’s supposition that teaching the algorithm for long division was unnecessary. Baker cited the fact that the algorithm for the long division of polynomials, a common technique in upper level mathematics, relies on an understanding of its similarity to the long division algorithm. Escobales (2004) responded to one of Ralston’s articles advocating calculator use with the criticism that the use of calculators deprives students of their first exposure to algorithms: an essential building block for solving mathematical problems. Both Baker and Escobales seem to fear that eliminating pencil-and-paper techniques will deprive students of basic skills necessary for the study of higher level mathematics. Norwood (2006) articulates a concern that the computational algorithms will only be eliminated from schools with students of lower socioeconomic status. Thus creating a nation of mathematical haves and have nots. Norwood’s unspoken concern is that students that do not learn these computational algorithms will be unprepared to study higher level mathematics. Debates over pencil-and-paper skills are not confined to topics in lower level mathematics courses. In 2006, Bhatnagar argued that curve sketching in calculus was no longer a necessary skill for students to learn in Calculus courses since graphing calculators could produce graphs so quickly. In a letter to the editor, Ding (2007) responded that the ability of the calculator to sketch graphs does not decrease the value of teaching curve sketching. Two other letters in response to Bhatnagar’s article (Myerson, 2007; Starr, 2007) pointed out that despite the powerful numerical methods employed by the graphing calculator there are problems that will not yield to these methods. In all of the arguments against technology changing the content of mathematics curriculum, a fear that the elimination of penciland-paper work will lead to a poorer mathematical understanding in students is clear. 3 DOES TECHNOLOGY CHANGE HOW STUDENTS DEVELOP MATHEMATICALLY? Several of the articles focus on the question of how technology use affects student thinking. These fears are concerned not with the effects of changes in curriculum, but instead are concerned with changes in cognition. Quinn What are we Afraid of? Arguments Against Teaching Mathematics with Technology… (2009) asserts that it is how technology represents mathematics may lead to some misunderstandings. He speculated that how students are required to enter computations into a calculator may be somewhat to blame for creating some mathematical misunderstandings. Most basic four function calculators do not require parentheses. Consider the problem (1 + 2 − 3) / 4 which evaluates to 0. When written without parentheses, the expression 1 + 2 − 3 / 4 evaluates to 9/4. Yet, when the expression without parentheses is entered into a four function calculator, the calculator often finds the result of 0. Quinn fears that the use of the calculator in computations could create a misunderstanding concerning the necessity of parenthesis in the minds of students. In another article Quinn (2005) speculates that the numeric focus of calculators prevents students from developing an ability to think abstractly. In a report on plenary panels from the 11th International Congress on Mathematical Education (ICME), Selden (2008) asserted that “[t]here are questions about how students relate to new technologies” (p.11). These questions certainly appear in both magazines. Norwood (2004) speculated that calculator use created students without an ability to differentiate between mathematical tasks. He characterised students without underlying paper-and-pencil computation skills as “indifferent as to whether they push the plus button or the times button” (p. 608). Yopp (2008) wrote of his concern that students in his courses had used a computer algebra system as a way to avoid rather than explore mathematics. Dillon (2006) stated her belief that students that rely upon a calculator to compute lack the ability to “interpret calculations done by machine” (p. 25). Wildstrom (2006) described situations in which students don’t understand the relationship between mathematical content and capabilities of technology and Norfolk (2008) articulates his concern that calculator use harms student development of basic number sense. Kenefic (2006) wrote a letter to the editor speculating on the effects of calculator use on the mathematical development of engineering students. Kenefic noticed that students who had performed well while using a calculator did not perform as well without a calculator. He noted that these students “misused the equal sign, … tried to do too much work in their heads, and failed to organise their work so that they could follow it” (p. 36). Reliance on the calculator to perform the steps prevented students from learning how to communicate the mathematical information and ideas in a meaningful way. Evidence from other disciplines indicates that relying on technology does change how individuals think and process information. For example in a study of automated navigation systems use, results indicated that a reliance on technology to navigate in an environment causes individuals to develop less spatial knowledge about their environment (Parush, Ahuvia and Erev, 2007). Individuals in the study that solely relied on the technology to navigate did not form their own spatial understanding of their environment. It is clear that some members of the mathematical community fear that technology has the potential of having a similar 99] negative effect on the development of mathematical understanding. 4 AT WHAT COST? The integration of a new technology into mathematics courses often does not go smoothly. Students may encounter difficulties working with the technology (Jardine, 2001) or the instructor may discover that the technology is not capable of accomplishing some of the desired tasks (Vestal, 2008). In 2001, Jardine pointed out that time spent teaching students how to use technology to solve mathematics is time not spent teaching students mathematics. Time, after all, is a zero sum game and the sophisticated computer programs typically used in college teaching can have a steep learning curve for students. While these programs are capable of helping students to explore rich mathematics content, there is concern that including complex technologies in college mathematics instruction can be detrimental to the amount of mathematics contained in a course (Bressoud, 2009; Dillon, 2006). There is also concern expressed in both magazines that technology rich environments themselves are a distraction. In 2006, Dillon expressed concern that students in technology rich classrooms were being supplied with “electronic dope” (p. 25). In the era of computer classrooms, instructors wonder whether students are using the computer to explore mathematics or to check e-mail and update their Facebook status. Similarly, graphing calculators are powerful tools for the exploration of mathematical concepts, but they are also capable of being programmed with noneducational games. In these articles, mathematicians express a fear that the technology is really a distraction from the actual task at hand: mathematics. 5 WHERE DO WE GO FROM HERE? First, while analysis of these publications does give a glimpse of reasons that some in the mathematics community fear the use of technology in the teaching of mathematics, it is only a glimpse. There is no evidence as to how widespread these fears are in the mathematical community. The articles and letters examined in this publication represent an extremely small percentage of the total articles and letters published over the time period. There were also many other articles and letters which extolled the benefits of technology use. Perhaps the gap is not as wide as it appears in the examination of national data sets. There is also no way to determine which fears are most widespread or whether there are other fears which this analysis did not reveal. As such, one of the first steps should be to conduct a large-scale study of US mathematicians and mathematics educators to determine what the mathematics community fears concerning the use of technology. Once the specific fears concerning technology have been determined, it will be possible to address those fears. Ralston (2004) hypothesised that it is the use of anecdotal evidence that allows calculator use to become the scapegoat upon which all poor mathematics preparation can be blamed. Certainly many of the claims made by authors of International Journal for Technology in Mathematics Education, Vol 18 No 2 [100 H Smith Risser articles were not supported by citations of research. For instance, Cuoco (2003) states that “A small percentage of teachers uses dynamic geometry environments, spreadsheets, statistical packages, and even computer algebra systems” (p.780) without specifying what evidence can be used to support this claim. Interestingly, between 2001 and 2009 only one article was published in either magazine reporting on results of research concerning technology use. The article, published in the May/June 2007 issue of Focus reported on the results of a department of education study on technology use. The report indicated that “no measurable differences in learning outcomes for students who used various kinds of technology in their classrooms” (Gouvêa, 2007, p. 11). While the summary did include a link to both a longer article published in Education Week and the report from the Institute for Education Sciences, the short two paragraph summary did not address the methodology in the studies or what types of technology were used in the learning of mathematics. In particular, the article did not address that the study in question only examined the effectiveness of three computer software programs designed to do automated mathematics instruction and tutoring. The study did not examine the use of technology products like graphing calculators, computer algebra systems, or dynamic geometry software all of which previous research indicates are related to positive effects on student outcomes (Almeqdadi, 2000; Funkhouser, 2002; Harskamp, Suhre and Van Streun, 2000). Publicising the results of research that demonstrates the benefits of technology use in professional magazines could allay some of the general concerns of mathematicians concerning technology use. Reports of findings concerning the benefits of technology use (Ellington, 2006) and the variables that may contribute to these benefits (Li and Ma, 2008) should be presented in these news magazines. The presentation should be balanced and contain detailed information concerning methods, conclusions, and implications for the mathematics community. In addition to the results on the general effectiveness of technology, results could be used to address the specific fears of the mathematics community. For instance, research indicates that regular graphing calculator use can actually strengthen the relationship between students’ understanding of graphical and symbolic forms of equations (Ruthven, 1990) and that visualisation software can help students develop spatial understanding (Sorby, Drummer, Hungwe, Parolini and Molzan, 2006). If there are specific fears that have not already been addressed by research, then research studies aimed at answering these concerns would need to be conducted. For instance, long-term studies could be conducted to determine the relationship between an individual’s computational fluency and their long-term understanding of algebraic computations in order to determine if changes in curriculum prompted by technology have the negative long-term effects mathematicians fear. Unfortunately, not all fears can be addressed empirically. The fear that time spent teaching with © Research Information 2011. All rights reserved. technology is time not spent teaching mathematics must be addressed philosophically. Mathematicians and K-12 mathematics teachers must perform cost-benefit analyses of various technologies to determine if the benefits of a particular technology outweigh its drawbacks. It may also be possible to find a technology with similar benefits, but fewer drawbacks. Free resources like Geogebra and Wolfram Alpha that are easy to learn or do not require the learning of a specific syntax make the use of technology more time efficient for instructors and more cost efficient for schools and universities. The mathematics community must strive to continuously educate one another concerning new developments in technology and to evaluate the costs of these new technologies. Finally, we must be willing to admit that the cause of the technology gap may not be fear at all. While K-12 teachers appreciate the low cost and portability of calculators, mathematicians often prefer computer software programs with more computational range and power. K-12 and post-secondary mathematics instructors face very different barriers to incorporating technology in their instruction. These different barriers may result in the adoption of or resistance to certain technologies. If it is the preference for the quality rather than the quantity of technology use the drives the divide between K-12 and postsecondary mathematics, all members of the mathematical community must explore other ways to help students to bridge the gap. REFERENCES Almeqdadi, F. (2000) The effect of using the Geometer’s Sketchpad (GSP) on Jordanian students’ understanding of geometrical concepts, Paper presented at the International Conference on Technology in Mathematics Education, Beruit, Lebanon, Retrieved from Baker, P. (2004) Long division by hand, Notices of the AMS, 51(3), 310. Bhatnagar, S. C. (2006) Curve sketching: a disappearing delight, MAA Focus, 26(6), 31. Bressoud, D. (2009) Technology in support of the classroom, MAA Focus, 29(3), 9. Cuoco, A. (2003) Teaching mathematics in the United States, Notices of the AMS, 50(7), 777-787. Dillon, M. (2006) What I learned from teaching with technology, MAA Focus, 26(6), 25. Ding, P. (2007) Curve-sketching: an eternal delight, MAA Focus, 27(1), 20-21. Escobales, R. (2004) Paper-and-pencil math, Notices of the AMS, 51(7), 734-735. Ellington, A. (2006) The effects of non-CAS graphing calculators on student achievement and attitude levels in What are we Afraid of? Arguments Against Teaching Mathematics with Technology… mathematics: a meta-analysis, Mathematics, 106(1), 16-26. School Science and 101] Ralston, A. (2004) Research mathematicians and mathematics education: a critique, Notices of the AMS, 51(4), 403-411. Funkhouser, C. (2002) The effects of computer-augmented geometry instruction on student performance and attitudes, Journal of Research on Technology in Education, 35(2), 163175. Ralston, A. (2006) K-12 mathematics education: how much common ground is there? MAA Focus, 26(1), 14-15. Gouvêa, F. (2007) Short takes, MAA Focus, 27(5),11. Ralston, A. (2007) Focus on focal points: a commentary on the NCTM curriculum focal points for prekindergarten through grade 8 mathematics, MAA Focus, 27(2), 29-31. Harskamp, E., Suhre, C. and Van Streun, A. (2000) The graphics calculator and students’ solution strategies. Mathematics Education Research Journal, 12(1), 37-52. Jackson, A. (2001) Presidential views: interview with Hyman Bass, Notices of the AMS, 48(3), 312-315. Jardine, D. (2001) A different pencil: use technology wisely, MAA Focus, 21(6), 20-21. Kenefic, R. (2006) Calculators and cyborgs, MAA Focus, 26(8), 36. Li, Q. and Ma, X. (2008) Technology in mathematics classrooms: a meta-analysis of the recent literature, Journal on School Educational Technology, 3(4), 34-54. Lutzer, D., Rodi, S., Kirkman, E. and Maxwell, J. (2005) Statistical abstract of undergraduate programs in the mathematical sciences, Available from: Madison, B. (2001) Bumps in the road from school to college mathematics, MAA Focus, 21(1), 10-11. Myerson, G. (2007) Fooling the calculator II, MAA Focus, 27(3), 22. Ralston, A. (2008) A nation still at risk, MAA Focus, 28(6),18-19. Ruthven, K. (1990) The influence of graphic calculator use on translation from graphic to symbolic forms, Educational Studies in Mathematics, 21(5), 431-450. Selden, A. (2008) ICME-11: Mexico, mathematics, and mariachis, MAA Focus, 28(9), 10-11. Sorby, S., Drummer, T., Hungwe, K., Parolini, L. and Molzan, R. (2006) Preparing for engineering studies: improving the 3-D spatial skills of K-12 students, Proceedings of the 2006 International Conference on Engineering Education, Puerto Rico, T3E-6 – T3E-10. Starr, N. (2007) Fooling the calculator I, MAA Focus, 27(3), 22. U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics (2006) NAEP 2005 Data. Accessed from: Verschaffel, L. and De Corte, E. (1996) Number and Arithmetic, in Bishop, A., Clements, K., Keitel, C., Kilpatrick, J. and Laborde, C. (eds) International Handbook of Mathematics Education, Netherlands: Kluwer Academic Publishers, 99-138. Norfolk, T. (2008) Is Ralston right? MAA Focus, 28(7), 9. Norwood, R. (2004) Teach more math K-12, Notices of the AMS, 51(6), 608-609. Norwood, R. (2006) Should we teach less? Focus, 26(3), 29. Parush, A., Ahuvia, S. and Erev, I. (2007) Degradation in spatial knowledge acquisition when using automatic navigation systems, Proceedings of the 8th International Conference on Spatial Information Theory, Australia, 238254. Quinn, F. (2005) The K-12 math test conundrum, Notices of the AMS, 52(4), 399. Vestal, S. (2008)What I learned…by using an online homework system in Calculus I, MAA Focus, 28(9), 22-23. Wildstrom, S. (2006) AP Calculus: friend or foe? MAA Focus, 26(6), 30-31. Yopp, D. (2008) Portfolios and scoring rubrics in technology-aided instruction, MAA Focus, 28(5), 14-16. BIOGRAPHICAL NOTES Hilary Smith Risser is a former K-12 schoolteacher and is currently an associate professor of mathematics. Dr. Smith Risser’s main area of research is technology in mathematics education. Quinn, F. (2009) K-12 calculator woes, Notices of the AMS, 56(5), 559. Ralston, A. (2003) Book review: California dreaming, Notices of the AMS, 50(10), 1245-1249. 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Journal of Urban Mathematics Education December 2012, Vol. 5, No. 2, pp. 44–52 ©JUME. PUBLIC STORIES OF MATHEMATICS EDUCATORS Critical Transformation of Mathematics Educators Maggie Lee McHugh University of Wisconsin La Crosse Jennifer Kosiak University of Wisconsin La Crosse hat can be done to enhance pre-service educators’ knowledge of social justice and its role in the elementary mathematics classroom? I (Maggie) pondered this question as a graduate student and mathematics educator. Following the advice of Cochran-Smith (1991) who advocated for connecting reform-minded teachers and pre-service teachers, I approached my colleague Dr. Jennifer (Jenn) Kosiak, an associate professor of mathematics education, about assisting in my research. Although Jenn had little background knowledge in social justice, her strong passion to continually learn and grow made her an ideal traveling companion as we set off on a journey to enhance per-service mathematics teachers’ awareness of social justice issues and their relevance in the mathematics classroom. This public story focuses not only on the pre-service educators we worked with but also, and more so, on our own professional and personal growth through the journey. We found that we could not speak of the transformation that we witnessed in these future teachers without addressing our personal transformations. Throughout this story, I will share our journey from my perspective. As part of the research involved written reflections from Jenn, I will incorporate her voice throughout the article in Arial font to address her growth in her own words. Before embarking on this journey, I read literature about how to best implement a social justice curriculum. I found that, in education, social justice themes are enacted to enhance students’ learning and their life chances by challenging the inequities of school and society (Michelli & Keiser, 2005). Yet, social justice education cannot be implemented without the dedication of critically aware teacher educators who have examined their own identity and privilege so W MAGGIE LEE MCHUGH is an associate lecturer and Learning Center Director at the University of Wisconsin-La Crosse, 1007 Cowley Hall, 1725 State Street, La Crosse, WI, 54601; email: Her research interests include critical pedagogy, social justice mathematics, and equitable uses of technology in education. JENNIFER KOSIAK is an associate professor of mathematics education at the University of Wisconsin-La Crosse, 1004 Cowley Hall, 1725 State Street, La Crosse, WI 54601; e-mail: Her research interests include the mathematics content knowledge for teaching, the integration of technologies to support the teaching and learning of mathematics, and social justice in mathematics. McHugh & Kosiak Public Stories that they are able to guide pre-service teachers in the process of developing critical consciousness. Quin (2009) explicitly states the goal of social justice educators is “to empower [both] educators and learners to act in anti-oppressive ways for social justice” (p. 110, emphasis in original). To implement such pedagogy, social justice educators must create what Freire (1970) calls horizontal relationships, relationships where teacher and learner work together toward a common goal. This is the type of relationship that developed between Jenn and me. No longer were Jenn and I limited to our respective roles of professor and graduate student; rather, we were both mathematics educators with the desire to create a dynamic classroom where pre-service educators could learn about social justice through critical dialogue. Analyzing Identity Through my initial readings in critical pedagogy, I learned that our first step as educators was to reflect upon our own identities and privileges. As two White females from small towns with little diversity of race, ethnicity, or language, we had a lot of examining to do. Reflecting upon our own personal and educational upbringing, we asked the following questions:      As a teacher, what is my role in the classroom? As a White person, what innate biases do I hold regarding student learning? As a female, what privileges may be affecting my teaching practices? As a White female, how do the tasks that I select for my students either include or exclude students in my classroom? Overall, how do these lenses affect my view of social justice in a mathematics classroom? Many of my biases come from my upbringing and education. Most of my peers were from two-parent, middle-class homes. Having experienced little diversity, I believed that all students had the ambition to go to college, to become educated, and to find a fulfilling career. It was not until I started working in public schools while in college that I came to understand that many students, whether urban, rural, or suburban, come from backgrounds much different from mine. Desiring to engage students through relevancy in learning, I began to search for resources, conferences, and like-minded colleagues who could help me to examine my privilege in order to become a critical educator who can embrace all students. Although I summarize my desire to change in a few sentences, this transformative process occurred over multiple years (and continues). In the midst of learning more about critical pedagogy, I approached Jenn about my research idea. Journal of Urban Mathematics Education Vol. 5, No. 2 45 McHugh & Kosiak Public Stories When Maggie first approached me about incorporating social justice into my elementary mathematics methods course, I was apprehensive because I thought that social justice merely meant making my classroom equitable to diverse student populations. With the National Council of Teachers of Mathematics (NCTM, 2000) Equity Principle in mind, my beliefs about social justice suggested that I should create a classroom environment that was both challenging and supportive of student learning regardless of their background. “I already do this, right?” I thought to myself. However, listening to Maggie’s discussion about social justice and critical pedagogy caused me to reformulate my understanding of social justice as a mechanism to investigate injustice in the world and create change. Immediately, I thought of issues such as distribution of world wealth and poverty rates and how I could use statistics and number facts to model math concepts to my pre-service students. I believed that purposefully modeling pedagogical practices would help these future teachers understand the role that social justice could play in the classroom, especially the mathematics classroom. Defining Social Justice Mathematics Much of the planning for this project occurred over the summer preceding the mathematics methods course. When the first class assembled, I had some jitters. Were the students willing and ready to embark on this journey with Jenn and me? Would they believe in the value of transforming the mathematics classroom into a place where all students can feel empowered to enact social change? We were about to find out. We administered a pre-survey to all of the pre-service educators in three sections of a mathematics methods course in order to ascertain their beliefs and abilities regarding social justice and mathematics. Like Jenn, the students began the semester with little knowledge of what social justice meant, specifically what it meant in the mathematics classroom. Similarly, these students were also influenced by the NCTM Equity Principle, which was introduced the week before, as much of their pre-survey definitions focused on supporting all students in the classroom. This finding made me realize that these students needed a firmer grounding in general social justice literature before I could move them toward integrating social justice and mathematics. However, given the limitations of a onesemester course, Jenn and I decided to focus on the concept of social justice within the mathematics classroom, all the while maintaining that the principles of social justice can be utilized in every classroom regardless of content. When introducing social justice mathematics to the pre-service teachers, I was apprehensive about how they would perceive this different course expecta- Journal of Urban Mathematics Education Vol. 5, No. 2 46 McHugh & Kosiak Public Stories tion. I tried to find concrete examples of what social justice pedagogy would look like in the elementary mathematics classroom. I believe these examples were critical based upon the students’ initial difficulty in defining social justice education both inside and outside of mathematics. Overall, I was relieved that no one questioned why I was including social justice in the mathematics classroom. In fact, many of the students expressed their satisfaction with the practicality of including social justice in a content area given that no other content methods course addressed this mode of teaching. To engage the pre-service educators in creating a more formal definition of social justice mathematics, Jenn and I asked them to critically reflect upon the article “A Social Justice Data Fair: Questioning the World Through Math” by Alexander and Munk (2010). This article showcases how a Canadian elementary school introduced social justice through an activity called The Welfare Diet. As part of their reflections, we asked these students to develop a working definition of social justice as well as an idea of how they might use this kind of activity with their future students. I was happy that these pre-service teachers took the definition of social justice beyond equity to include connecting mathematics to real-world problems as well as to the students’ lives and communities. Many of my students found that the article helped them to see a new approach to teaching mathematics. Additionally, this article allowed the pre-service teachers to see that mathematics is not a singular course; it can be effectively integrated across the curriculum into areas such as social studies and language arts. Their course reflections continually shaped my own vision of social justice in the mathematics classroom. Perhaps, not having a preset notion of what social justice is and is not allowed all of us to explore, investigate, and conjecture about social justice through the lens of mathematics. With our building concept of social justice mathematics, I introduced critical mathematics pedagogues such as Gutstein (2006) and Gutiérrez (2007). Using my research to spur a more formalized definition, we decided that teaching mathematics for social justice hinges on a teaching and learning environment where    students are introduced to the various issues of equity, diversity, and social injustices; students increase and strengthen their mathematical content knowledge; and students learn to use mathematics to identify and examine social issues with the intent to enact change. Journal of Urban Mathematics Education Vol. 5, No. 2 47 McHugh & Kosiak Public Stories Modeling Social Justice Mathematics The Association of Teacher Educators (2003) states: “In order for teacher educators to impact the profession, they must successfully model appropriate behaviors in order for those behaviors to be observed, adjusted, replicated, internalized, and applied appropriately to learners of all levels and styles” (p. 1). With this charge in mind, our first task was to slowly but intentionally model social justice pedagogy in the math classroom. Embracing our love of children’s literature, Jenn and I chose to read the book If the World Were a Village (Smith & Armstrong, 2006) that scales the world population to a village of 100. This book provides the reader with demographics and statistics such as “20 villagers earn less than one dollar a day” or “60 are always hungry.” As I listened to Jenn read this book aloud to the pre-service teachers and heard all of the descriptions of the world’s population in terms of percents and fractions out of 100, I wondered if they, too, were reacting to the reality behind those numbers. After reading If the World Were a Village, we asked students to pick a page from the book related to language, ethnicity, age, or wealth and to make mathematical models to represent the data presented. I wondered if these pre-service teachers would concentrate solely on the numbers and ignore the real-world inequities that were so vividly portrayed. Jenn and I were delighted when they discussed not only the mathematics but also the inequities across the world’s population. For example, they were surprised that roughly one-third of the population is illiterate and almost 40% of the population does not have access to running water. This activity led to a richer discussion of how percents and fractions can be used in the mathematics classroom to illustrate larger social issues. We discussed ways in which they could extend such an activity into their future classrooms, including creating a real-life version of the book entitled If Wisconsin Schools Were a Village. This project would entail investigating the make-up of the classroom and comparing that to statistics in Wisconsin including the number of minority students, ELL students, and students with special needs. Experiencing Social Justice Mathematics In addition to modeling social justice mathematics, we wanted to develop a purposeful learning experience where these pre-service teachers could apply their definitions of social justice to a mathematical concept. This desire led to the implementation of The Accessible Playground Project, adapted from a social justice lesson plan developed by the Centre for Urban Schooling. I walked into the classroom one day and declared, “All children have the right to relax and play!” After some initial wonderment about my strong declaration, Jenn and I outlined The Accessible Playground Project. Groups of students Journal of Urban Mathematics Education Vol. 5, No. 2 48 McHugh & Kosiak Public Stories were given the task to research both playground equipment and accessibility requirements. They then completed a scale 2D blueprint and a 3D model using mathematical concepts such as geometric shapes and solids, area and perimeter, scale and proportional reasoning. Additionally, they calculated an estimated cost for the playground. These pre-service teachers summarized their findings and outlined the criteria they established in a formal report. At the end of this 3-week activity, the groups presented their models in the classroom linking their presentation to not only why their model was safe and accessible but also to the UNICEF (1990) Convention on the Rights of a Child, Article 31, which prompted the activity. Through this activity, groups of pre-service teachers determined what accessibility meant not only in a playground situation but also their own classroom. Our conversations began with the appropriate angle of an accessible slide and the cost function to model this task, but quickly shifted. We found ourselves discussing accessibility is a classroom-based issue focused on things like location of white boards, heights of desks and lab stations as well as instructional materials that would meet the needs of all students regardless of ability. Jenn and I facilitated discussion amongst the pre-service teachers about including bilingual signs in schools, conservation of the environment, and recycling both inside and outside of the classroom. The idea of going out to play at recess was linked to childhood obesity and school lunch programs. All of these discussions spurned from the initial concept of playground equity. Additionally, part of our established criteria for a social justice mathematics lesson is to use the mathematics to critically examine an inequity, therefore, we asked the pre-service educators to analyze cost differentials between an accessible and traditional playground. Almost every group found that the cost of accessible playgrounds was fairly equal to the cost of traditional school playgrounds. They also found themselves examining the playgrounds at their clinical field sites to give them an accessibility ranking. Creating Social Justice Mathematics Jenn and I used discussion, article reflections, purposeful modeling, and learning experiences to teach our students about social justice pedagogy and its relevance to the mathematics classroom. Finally, we implemented a pedagogical project that allowed the pre-service educators to explore their understandings of social justice. The culminating activity was the creation of a mathematical concept plan that embedded a social justice theme. Initially, students were apprehensive about the type of activity that they would use in order to satisfy this course assignment and Jenn and I did not know what to expect. They continually asked, Journal of Urban Mathematics Education Vol. 5, No. 2 49 McHugh & Kosiak Public Stories “Is this a social justice idea?” To which we replied, “You need to decide that for yourself.” As the semester progressed, students would reflect on different ways they could include social justice in their concept plans and the idea of linking mathematics through socially relevant activities emerged. Some students discussed how they could link measurement to the inequities of fresh water supply in the United States versus India. Other discussions evolved into how they could use percents and fractions to investigate how school lunch programs are related to childhood obesity. They used numbers and operations to model poverty in the United Sates and its impact on student achievement. Even at the earliest grade levels, these preservice teachers were able to not only have students use the mathematics but also engage in the discussion of the social justice themes. Jenn and I were pleasantly surprised at the depth of mathematical connections to social justice these pre-service teachers created. Mathematical concepts of fractions and percents were tied to child labor; coordinate planes and graphing helped locate the Native American tribes of Wisconsin; data regarding cyber bullying was graphed to find the increasing rate of change. Tasks began with critical questions such as: How many steps do you take to get a glass of fresh water? Do you think all people on Earth can measure the distance to fresh water in steps? These pre-service teachers found ways to critically embed social justice into mathematics lessons. Their growth throughout the semester was reflected in their post-survey where all students could define social justice in the mathematics classroom, but more so in their final reflections about the value they found in not just “doing” mathematics but “using” mathematics to explore local, regional, national, and global concerns. Yet, just as important to the research was the unintended outcome of our transformation as critical educators. Overall, the experience of incorporating social justice into the mathematics class was rewarding as I saw not only growth in my pre-service teachers’’ understandings of social justice but also in my own. The main challenge was that I did not fully grasp the concept of social justice in education at the onset of the course. Indeed, I often felt that I learned more about social justice as I processed article reflections with the students and read their concept plans. I also wonder if my own identity might have influenced how I modeled social justice to my students. As a White female educator teaching mainly White female students, I think it is important for all of us to examine how our beliefs shape our teaching and the types of activities we give to our students. If you asked me a year ago if I would incorporate themes of societal inequalities into my mathematics methods classroom, I would have probably said, “Don’t they get that in Social Studies?” I have always been a believer of purposeful modeling about math concepts including integrat- Journal of Urban Mathematics Education Vol. 5, No. 2 50 McHugh & Kosiak Public Stories ing reading and writing into the mathematics classroom. This project has helped me understand how all curricula can be integrated in a manner that will enhance students’ academic and social learning. Though I had studied social justice practices and had begun my transformation as a critical educator, I had worked with others in their exploration of these difficult concepts. While moving with these pre-service educators towards a deeper mathematics pedagogy guided by social justice principles, I realized just how difficult it is to maintain an atmosphere appropriate for critical pedagogy. I would constantly critique and reflect upon everything that occurred in class from the tasks we chose to the language we used. Sometimes, I could tell the preservice educators were frustrated with Jenn and me because we never “gave them the answer.” The perception of mathematics is that all problems have a direct solution. However, one of the most difficult tasks was attempting to open the preservice teachers’ perception from a purely quantitative view of mathematics to a qualitative, narrative understanding that highlights problem solving and process more than the solution. Indeed, this view of mathematics as a dynamic set of knowledge still plagues me. I fall into the trap of teaching to skill building without providing a critical context for the learning. Nevertheless, in small and large ways, social justice pedagogy has transformed my teaching. My students engage in an activity that examines the wage gap between White men and White women, Black men, Black women, Latinos, and Latinas. I ask my students not only to engage in the mathematics of finding models of equations and when White men and afore mentioned groups would earn equal pay, but I also encourage them to reflect upon their future career. I recognize that the experience of leading others to an awareness of social justice truly transformed me into an emerging critical educator. I realize I will continue to explore my identity and reflect upon my teaching practices; however, with the lens of social justice, I know that my awareness of social justice will lead to praxis in my teaching and daily living. One year after engaging in the research project, Jenn and I have come to realize that we are critically transforming as mathematics educators. It is hard to believe that prior to this research project, we viewed mathematics and social justice as separate entities. The idea of enacting change through examining societal inequalities in mathematics continues to transform both our classes and the preservice teachers with whom we engage. Today, we share a formulaic definition of social justice in the mathematics classroom: Critical Mathematics + Societal Inequity Investigations = Transformative Change for All. References Alexander, B., & Munk, M. (2010). A social justice data fair: Questioning the world through math. Rethinking Schools, 25(1), 51–54. Journal of Urban Mathematics Education Vol. 5, No. 2 51 McHugh & Kosiak Public Stories Association of Teacher Educators. (2003). Standards for teacher educators. Retrieved from Cochran-Smith, M. (1991). Learning to teach against the grain. Harvard Educational Review, 61, 279–309. Freire, P. (1970). Pedagogy of the oppressed. New York, NY: Continuum. Gutiérrez, R. (2007). (Re)defining equity: The importance of a critical perspective. In N. S. Nasir & P. Cobb. (Eds.), Improving access to mathematics: Diversity and equity in the classroom. (pp. 37–50). New York, NY: Teachers College Press. Gutstein, E. (2006). Reading and writing the world with mathematics: Toward a pedagogy for social justice. New York, NY: Routledge. Michelli, N. M., & Keiser, D. L. (Eds.) (2005). Teacher education for democracy and social justice. New York, NY: Routledge. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. Quin, J. (2009). Growing social justice educators: A pedagogical framework for social justice education. Intercultural Education, 20(2), 109–125. Smith, D. J., & Armstrong, S. (2006). If the world were a village: A book about the world’s people. Toronto, Canada: Kids Can Press. UNICEF. (1990). The conventions on the rights of the child. 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Journal of Urban Mathematics Education July 2017, Vol. 10, No.1, pp. 16–31 ©JUME. PUBLIC STORIES OF MATHEMATICS EDUCATORS Responding to Inequities in Mathematics Education: Opening Spaces for Dialogue Megan H. Wickstrom Montana State University Susan A. Gregson University of Cincinnati R ecently, mathematics educators have discussed the challenges of preparing teachers to effectively teach all students. When examining these challenges, researchers have acknowledged problems that often arise because “teachers–– largely white, female, monolingual, and middle class––are not effectively prepared to teach mathematics to an increasingly racially, ethnically, linguistically, and socioeconomically diverse student population with which they often have had limited previous interaction” (Bartell, 2012, p. 113). This lack of preparation can be attributed, at least in part, to the fact that many teachers have limited personal experience with the types of inequities that exist across the educational system (Stinson, 2004). When we consider teachers’ readiness to teach mathematics equitably, it is also important to step back and consider our roles as mathematics teacher educators. For many of us, our lived experiences in the educational system are not far from those of the teachers we instruct. Therefore, we must ask how we can adequately advise and support preservice and inservice teachers to teach mathematics in equitable ways when many of us are still learning to navigate this terrain? There has been a recent call to offer cases and stories to support mathematics teacher educators in discussing inequities in the classroom (White, Crespo, & Civil, 2016). Moreover, equity scholars have urged mathematics educators to “engage colleagues and friends in explicitly talking about race, class, gender, and other systems of privilege and oppression” (Aguirre et al., 2017, p. 140). This engagement requires a willingness to enter a “brave space in which some of our assumptions are questioned” (p. 128). We hope to add to these conversations by narrating our experiences with confronting inequities in K–12 classrooms, including our personal efforts toward finding the courage to present these accounts. Supporting teachers to MEGAN H. WICKSTROM is an assistant professor in the Department of Mathematical Sciences – Montana State University, 2-235 Wilson Hall, Bozeman, MT, 59717-2400; email: Her research interests include the teaching and learning of mathematical modeling at the elementary level, creating mathematical tasks that promote equitable learning opportunities for all, and investigating and supporting teachers’ applications of research into practice. SUSAN A. GREGSON is an assistant professor in the Curriculum and Instruction and Middle Childhood Education programs – University of Cincinnati, 511D Teachers-Dyer Complex, 2610 McMicken Circle, Cincinnati, OH, 45221-0022; email: Her research interests include equitable classroom practice, political knowledge for teaching mathematics, and the preparation of mathematics teachers for effective teaching of marginalized students. Wickstrom & Gregson Public Stories navigate inequities in mathematics classrooms involves continued uncertainty because no single solution works across all cases. It involves coming to understand the history and people in the communities in which we work, making a deliberate choice to address inequity, creating space for reflection and dialogue, and revising our strategies based on insights gained from working with others. Moreover, it means expecting that we will make mistakes along the way and figuring out how to be simultaneously comfortable and uncomfortable with that fact. While a certain level of comfort is necessary to move forward, it can make us complacent in situations where our expertise is limited. A theme that connects our stories is our focus on creating spaces to “listen well” (Powell, 2012, p. 26) and to learn as a critical first step toward confronting inequity. In this public story, we account our respective strategies for creating these spaces with teachers and with each other as colleagues. By listening to teachers and engaging with them in practice, we are working to become more adept at creating opportunities for productive changes toward equity. We also recognize that improving our practice requires creating opportunities to connect with other mathematics teacher educators to discuss challenges in our practice and how we might address them. We hope that our stories may provide other teacher educators with examples of what the process of opening spaces for addressing inequity with teachers and with other mathematics teacher educators might look like. We begin by describing our respective stances and a case involving inequitable mathematics teaching that we have encountered in our practice. We discuss our attempts to create spaces for productive change along with the teachers we mentor. Each account includes acknowledgment of our remaining questions and tensions. We then connect our cases by describing how and why we came together to write this article. We discuss our collaboration process and provide examples of how this process opened spaces for our own learning. We conclude with remarks about both the nature of efforts to address inequity in our practices as mathematics teacher educators and the continued challenges we see for this effort. Megan’s Story I worked with Mrs. Cate,1 a fourth grade teacher with 6 years of experience, as part of a 2-year professional development (PD) project focused on integrating learning trajectories as a formative assessment tool in elementary classrooms. For the study, I interviewed her and observed her class throughout the project. Mrs. Cate taught at Terrace Elementary School located in the Midwestern United States. Terrace is an urban school where the majority of students identify as Black or Latina/o and 80% of students qualify for free or reduced-price school meals. At the 1 All proper names are pseudonyms. Journal of Urban Mathematics Education Vol. 10, No. 1 17 Wickstrom & Gregson Public Stories time, I was a 28-year-old White female mathematics education doctoral student and PD provider. As a former middle school teacher, I respect the intensity of classroom teachers’ work. I was raised in a city that was similar to, and in close proximity to, the one in which my story took place. In addition, I student taught and volunteered in schools like Terrace Elementary, so I felt comfortable when working with students and teachers there. I believe that mathematics should be co-constructed by the teacher and the learners and that it becomes meaningful through shared experiences and interpretations. All students are capable of learning mathematics, but it is critical to find ways to relate mathematics to their lived experiences. I focus on Mrs. Cate here because I perceived components of her teaching as inequitable, and it was difficult for me, as a mathematics educator, to determine if and how I should respond. Equity Tensions Initially, Mrs. Cate’s classroom seemed like a well-structured environment for student learning. During her first interview, she highlighted several features of her instruction that she felt promoted student discussion, reflection, and growth. Mrs. Cate explained that she organized students in groups to promote discussion. She also highlighted a bulletin board on the back wall lined with clipboards. She said that this was where she kept track of things each student was doing well and something for them to improve. In addition, Mrs. Cate implemented a mathematics journal to encourage students to express their ideas in writing. Although Mrs. Cate articulated a solid rationale for the instructional strategies that she had put in place, I experienced them differently. Even though students were arranged in groups, Mrs. Cate rarely allowed them to talk or work together. I did not perceive the clipboards in the back of the room as a tool for accolades, as Mrs. Cate had intended, but rather as a way to compare and to demean students. In front of the class, she often said things to students like “I wish I could find something good to put up here, but I haven’t seen any good work from you in 3 weeks.” Lastly, the mathematics journal was often used as a punishment activity for when students’ behavior was not appropriate during class. My foremost concern was how she framed students’ intelligences. Sternberg (2007) documented that although intelligence is often perceived as objective, it is very much subjective. Students’ perceived performance in class is often tied to how the teacher perceives what it means to be “smart” and how well students’ behaviors align with the teacher’s expectations (Hatt, 2012; Wickstrom, 2015). Mrs. Cate evaluated and rewarded students based more on their behavior than their efforts toward mathematical learning goals. In this classroom, being good at mathematics was associated with listening, being quiet, not fidgeting or making faces, and speaking when called on. Hence, Mrs. Cate favored students who worked on tasks quietly, answered questions quickly and correctly, and “behaved” during instruction. Journal of Urban Mathematics Education Vol. 10, No. 1 18 Wickstrom & Gregson Public Stories When students behaved appropriately, she rewarded them with candy or prizes. Students only discussed ideas or questions when directly asked, and students who acted out or misbehaved were sent outside or to the principal’s office. Mrs. Cate had two students––Hadley, a White girl, and Kirby, a White boy––on whom she called frequently and often used as exemplars for the rest of the class. She told me that these two students were her top students because they were able to answer mathematics questions quickly and often correctly. Through observations, I noticed both Hadley and Kirby sat quietly in class. Mrs. Cate said, “Look how nicely Hadley is sitting and listening” to the rest of the classroom. In contrast, several students did not behave according to Mrs. Cate’s expectations. From the first day, I took note of her relationship with Timothy. Timothy was a Black boy, and his desk was positioned next to Mrs. Cate’s desk and removed from the other students. Timothy often sat with his arms crossed and head down. One day during instruction, Timothy became bored and made faces at his friends across the room. When Mrs. Cate caught him, a confrontation erupted, and she sent Timothy out of the room. Mrs. Cate had a special desk for Timothy in the hallway where he would sit until she thought it was time for him to return. Timothy missed most of the mathematics classes and was often forgotten out in the hallway for long periods of time, sometimes over an hour. When I asked Mrs. Cate about Timothy she said, [Timothy] is one of my highest testing math students, like the computer lab testing, which is kind of like standardized testing. But, those tests obviously don’t tell us everything because in class, he doesn’t get it [math]. Mrs. Cate’s statement surprised me, because, I had a different perspective on why Timothy wasn’t learning; he was not allowed to participate in class. As an observer, I knew what was happening was not equitable. I often left observations feeling uncomfortable and concerned for students like Timothy. Whether knowingly or unknowingly, I felt Mrs. Cate’s approach was creating racial divides in her classroom. As reflected in the school statistics, most of the students in Mrs. Cate’s class were not White. In fact, there were only three White students in her classroom. Mrs. Cate positioned Hadley and Kirby as top students because they aligned with her expectations of what a well-behaved student should be. She allowed White students special opportunities such as explaining concepts and going to the chalkboard while often limiting opportunities or completely taking them away from Black students like Timothy. As Mrs. Cate continually equated “good” behavior with mathematical proficiency, both White and Black students missed opportunities to grow mathematically. Simultaneously, this approach unfairly marked Black students as mathematically inferior to White students. As part of the PD, I encouraged Mrs. Cate to engage in mathematical discussions and activities with students. In the first week of observation, she would begin Journal of Urban Mathematics Education Vol. 10, No. 1 19 Wickstrom & Gregson Public Stories with an inquiry-based task but quickly resorted to quiet work time after she felt students were becoming out of control. When debriefing these lessons, Mrs. Cate made statements like “that’s difficult with these kids” or “when working with these kids, I have to….” The language of “these kids” sparked my attention and I wondered if it referred to race, socioeconomic status, ability, or a combination of these. In getting to know Mrs. Cate, I found out that she commuted a half hour to Terrace from a rural, primarily White community, which was not uncommon for teachers in the district. She discussed that her mother was a teacher and her mother’s passion for teaching motivated her to follow suit. She also revealed that even though her mother was an enthusiastic and hands-on teacher, she did not feel like she could use similar teaching strategies with “these kids.” When discussing the job at Terrace, she said, “But coming into the city was a whole different ballgame for me as far as what I saw growing up as a student and I guess I have to be strict and more firm just because of the city.” She also asserted that she had never been around a “minority” child until college. In the first few weeks working with Mrs. Cate, I had the sense that she wanted to be an engaging teacher but felt she had to teach in a certain way because of her students’ backgrounds. Mrs. Cate’s story is not new in educational research. There is often a racial and cultural mismatch between teachers and their students (Goldenberg, 2014), and instead of recognizing and engaging racial and cultural differences, many teachers take the stance that learning means working harder and behaving (Haberman, 1991). In addition, I knew Mrs. Cate’s teaching style was not the only approach being used in her school; other teachers in her building had learned to navigate these tensions and to engage in inquiry- and equity-based mathematics. Creating Space with Mrs. Cate It was difficult for me to know what to do in this situation, and I considered several possibilities. From the beginning, I knew I could not confront Mrs. Cate directly about her teaching practices for several reasons. First, Mrs. Cate perceived me as an outsider who did not understand the day-to-day realities of her teaching that made her teaching practices necessary and effective with her students. In addition, I was conducting research for my doctoral dissertation, so addressing my concerns with Mrs. Cate meant risking the study and my ability to work with her or other teachers in the district. I began to address what I observed by listening to Mrs. Cate in the interviews and asking her questions related to some of her comments and strategies. I hoped some reflection might allow her to consider her actions and to gain insight into her practices. For example, after she sent Timothy out of the room, I asked her to talk about Timothy informally after class and then more specifically during interviews. Journal of Urban Mathematics Education Vol. 10, No. 1 20 Wickstrom & Gregson Public Stories In these conversations, she seemed more comfortable telling me about her students and her teaching. Consequently, I realized Mrs. Cate’s perceptions were deeply engrained and reaffirmed daily. It seemed difficult to challenge these perceptions because as long as her students’ behaviors aligned with her preconceived expectations for them, my questions would not likely shift her beliefs. Instead, I chose to demonstrate possibilities for more equitable instruction. I thought that if Mrs. Cate could see her students differently then she might begin to change how she viewed them. Moreover, given that my frequent interviews with Mrs. Cate took up her time, I wanted to reciprocate by assisting her with her work in some way. I offered to teach for her, to free up time for her to reflect on her students’ thinking. Glaser (1982) and Lather (1986) describe this approach as reciprocity or “the exchange of favors and commitments, the building of a sense of mutual identification and feeling of community” (Glaser, 1982, p. 50). My purpose was twofold. I wanted to give back to Mrs. Cate, but I also hoped that demonstrating other ways of interacting with her students might provide openings for talking about equity. When I proposed the arrangement to Mrs. Cate, she hesitantly agreed that I could teach one or two lessons a week. In the first few weeks, I stuck to Mrs. Cate’s lesson plans to gain her trust. Although the lessons were not student-centered, I made a point to elicit multiple students’ perspectives, check with students to see how they were doing, provide scaffolding, and hold high expectations for everyone. As Mrs. Cate became more comfortable with me teaching, she asked if I would be interested in teaching mathematics intervention. Intervention occurred several times per week and consisted of students practicing facts on computers or by playing games. She directed that students should practice math facts and concepts but gave me the freedom to choose what I wanted to teach and how. Instead of having students independently practice facts, I designed mini-lessons for them to work cooperatively. For example, when students were studying area and perimeter, I asked them to help me design a backyard fence for a pet. This project led to discussions on how to use the space, the size of the pet, and whether the house could be used as a border. Students were excited by the tasks and often discussed them with me days after the lesson. Teaching with and for Mrs. Cate created an opportunity for dialogue. After the first few intervention classes, Mrs. Cate made comments like “I need to try that” or “I was really surprised how [a specific student] kept working on the task.” Eventually, she tried some of the tasks from intervention in her own classes and asked me for help in designing similar tasks. I believe Mrs. Cate wanted to be an engaging teacher, but she did not believe that her students could engage in rich mathematics. Providing students with challenging tasks gave us a glimpse of what they were capable of as well as concrete examples that highlighted particular students as creative and competent. It was difficult to discuss beliefs with Mrs. Cate directly, but I wit- Journal of Urban Mathematics Education Vol. 10, No. 1 21 Wickstrom & Gregson Public Stories nessed several incidents that challenged her beliefs about students’ mathematical abilities. Remaining Questions and Tensions This case highlights the reality that opening much-needed spaces for conversations about teachers’ practices and beliefs about students is difficult. My main concern in working with Mrs. Cate was that students were not receiving equal opportunities for quality mathematics education. I attempted to address this by modeling approaches that allowed students in an intervention class to demonstrate mathematical competence beyond good behavior. I saw evidence that teaching for and working with Mrs. Cate opened space to discuss students’ mathematical thinking and provided concrete evidence that countered Mrs. Cate’s perceptions of struggling students. And yet, as our collaboration ended, I was left wondering, “Did I do enough?” While I witnessed some positive change in Mrs. Cate’s instruction, my lingering tension is that by focusing my efforts on developing rich tasks accessible to all students, I skirted issues of race and equity. I continue to wonder if I could have done more to help Mrs. Cate challenge her assumptions about Black students and to productively address the racial bias that I observed in her practice. More generally, I continue to grapple with how to broach topics like racism, classism, and ableism without fracturing relationships with my teacher partners. Susan’s Account As a mathematics teacher educator who has been a classroom teacher, schoolbased coach, and researcher in urban and rural schools, I respect the work of classroom teachers and believe that it is not possible to transform education to meet the needs of marginalized students without teacher knowledge, collaboration, and agency. I work in a teacher education program whose mission includes preparing teachers to work in urban schools. A significant tension of my practice is my desire to help early-career teachers challenge inequitable practices while avoiding the pitfall of portraying urban educators––especially those whose racial and economic backgrounds differ from their students––as the primary obstacle to equity. However, as a White middle class teacher, I routinely encounter and participate in situations where deficit notions of students of color––an intrinsic ideology of inequitable teaching––go unchallenged. I position myself as an equity researcher, yet I am troubled in situations like these in which I lack the tools or the courage to disrupt the status quo. Moreover, I find that when I operate alone, outside of a community of educators regularly committed to equity issues, the tools I have acquired to resist deficit perspectives become dull. Therefore, participating in communities where Journal of Urban Mathematics Education Vol. 10, No. 1 22 Wickstrom & Gregson Public Stories educators with diverse experiences address tensions of equitable practice is essential for my own professional development. The context of my account is a voluntary professional development group for preservice and early-career educators, the Mathematics Equity Group (MEG). I both facilitate and participate in the MEG. The MEG’s goals include supporting teachers as they “identify and challenge discourses that further ingrain inequalities,” develop “political knowledge and experiences necessary to negotiate the system,” and develop “working networks of educators who share their emancipatory visions” (Gutiérrez, 2013, p. 62). MEG participants follow a modified version of Gutiérrez’s (2012) In My Shoes discussion protocol in which a teacher describes a problematic scenario from personal practice. After clarifying questions are addressed, the group discusses strategies for addressing the situation with follow-up questions such as: What would that strategy look like? Is that something you can see yourself doing? Teachers are encouraged to consider the scenario with respect to their own practice as educators working toward equity. This account focuses on Mr. David’s In My Shoes experience in the spring of 2014. Mr. David, an African American, was a preservice teacher in a multi-level (grades 4–6) urban field placement. Mr. David had significant pre-certification urban teaching experience as a full-time substitute in local schools with high poverty rates (over 98%) and large numbers of African American students (more than 95%). Six other MEG teachers, all White, and myself, participated in the discussion. Equity Tensions in Working with the MEG Mr. David and I had previously discussed challenges in his field placement prior to this In My Shoes experience, so I thought I knew what to expect. He was concerned with both the exclusively procedural nature of the enacted curriculum, and a potential personality conflict with his cooperating teacher, Ms. Marcus. I knew that he had worked through these issues to some extent, so I encouraged him to share his story in MEG. However, as Mr. David presented, it became clear that his concerns were more complicated than I thought. He described a learning environment where students largely worked in silence; where norms for behavior were rigid and enforced punitively; and where seating, participation, and discipline were highly racialized. Mr. David told the group about Jasmine, a Black child who he saw as being frequently and unfairly disciplined. For example, when another child walked by Jasmine’s desk and inadvertently knocked a piece of paper on the ground, Ms. Marcus noticed and pulled Jasmine aside. Mr. David began shouting to imitate Ms. Marcus’ tone: “I can’t believe you had a piece of paper under your desk. I told you last semester. I told you this semester. Clean up under your desk.” According to Mr. David’s account, the teacher “reams her for like two to three minutes. And then the Journal of Urban Mathematics Education Vol. 10, No. 1 23 Wickstrom & Gregson Public Stories little girl has to take the [math] test.” In a voice that was both incredulous and outraged, Mr. David went on to describe patterns he noticed in student seating: “All the little White girls sit in the front row in the middle. The Black girls sit behind them. The Black boys sit in the back right corner and the White boys sit in the front right corner.” Another participant interjected, “It’s that clear cut and obvious?” Mr. David acknowledged his own concern that perhaps he was simply imagining a bias. And so, he “started to keep track. [Ms. Marcus] only calls on the little White girls. I’ve seen the whole row [of Black girls] raise their hand. And she calls on the one little White girl, Ashley.” Mr. David recounted another incident where he approached a Black student to talk about a fraction worksheet. As soon as Ms. Marcus saw them talking, she reportedly said, “What did you say Darnell? Do this reflection!” Reflections were a form of punishment in the class. Mr. David described trying to respectfully defend Darnell: Mr. David: I talked to him. He wasn’t talking. I talked to him. Ms. Marcus: No! He knew what he was doing. He came in and he sat down next to you and he talked because he knew you would talk to him. Mr. David: No ma’am. I asked him. Ms. Marcus: If you are a student in this class and you think you can come in and talk to another teacher about anything that is going on, you are going to get a reflection! As Mr. David told his story, I wrestled with multiple emotions. First I was horrified. I remember saying, “This teacher does not belong in the classroom!” and it was hard for me to move beyond this immediate thought. The situation felt like an extreme case of a racialized, authoritarian, and repressive environment where “the achievement gap is a mirror image to the punishment gap” (Yang, 2009, p. 51). But the example, though extreme, was not inconsistent with other situations I have encountered in schools that serve high numbers of marginalized students. I have witnessed both effective and ineffective colleagues and administrators forcefully reprimanding students. I have done some yelling myself over the years. Without firsthand experience in the setting, and without knowing more about a teacher and her practice, it could be possible to mistake, for example, warm demander approaches that involve “mean-talk” (Ware, 2006, p. 438) as oppressive. I trusted Mr. David’s perspective of the situation, but I was uncertain how others in the group might interpret Ms. Marcus’ behavior. Teachers with limited experience of the range of effective discipline practices may misread classroom situations. They may view even warm demander pedagogy as harsh and inappropriate. Conversely, they may believe that discipline styles they would never choose for their own children are requirements for teaching marginalized students. I struggled with whether I should jump into the conversation to provide nuance. I also won- Journal of Urban Mathematics Education Vol. 10, No. 1 24 Wickstrom & Gregson Public Stories dered whether without my intervening, the participants might dismiss this example as so extreme that they would never likely face a similar situation in their own practice? My response that “this teacher does not belong in the classroom,” was genuine, but could also be used as an excuse to dismiss efforts to learn to confront similar behavior that teachers might encounter with peers in the future. I also worried that giving this example too much attention and treating it as typical might make participants hesitant to see themselves acting because they feel too inexperienced to tackle such a pervasive problem. Despite these tensions, I fought the urge to share every concern that popped into my head. When you have experience, you want to share it––especially as a trained teacher educator. Yet, I have learned from multiple In My Shoes discussions, that when I limit my contributions, other participants’ questions and comments take the group in insightful directions. The MEG protocol is designed to probe further, to connect participants with each other’s experiences, and to develop each person’s capacity for acting in related situations in ways that align with the kind of teachers we want to be. So, I decided to trust the protocol. I encouraged Mr. David to phrase his concerns about this inequitable situation as a question. He responded with, “What is it that I can do in the limited time I have in this practicum to give those kids a different experience?” Creating Spaces through MEG The participants began with clarifying questions such as, “Who is Ms. Marcus?” and “What is her teaching background?” A generative moment came when Ms. Shelby, a first-year teacher, revealed that she was in the same teacher’s classroom for her first practicum, 2 years earlier. Ms. Shelby was eager to discuss her frustration in that placement where the way students were treated made her “feel terrible.” Ms. Shelby shared her earliest attempts at teaching in the placement describing how Ms. Marcus both discouraged her from trying student-centered approaches and appeared vindicated when Ms. Shelby tried them, and they did not go well. Ms. Shelby reported “feeling like a failure” in that experience. Discussion turned to the suggestion phase as participants applied possible strategies developed in previous MEG discussions to Mr. David’s situation. For example, Ms. Cass, suggested “playing dumb.” Mr. David might use his position as a novice to question Ms. Marcus’s methods, bringing attention to both her problematic behavior and potentially opening space for discussion through questions like, “What is the purpose of the reflections?” Another approach was “claiming a requirement.” Mr. David might claim that making sure all students’ voices are heard is an official component of his practicum, and therefore, he would have an excuse to use methods that insure students are called on randomly. A third suggestion was Journal of Urban Mathematics Education Vol. 10, No. 1 25 Wickstrom & Gregson Public Stories that Mr. David might “highlight the competence”2 of Black students by amplifying their thinking with the cooperating teacher with comments like, “So and so has a really good idea.” Mr. David asked the group to consider whether any of these ideas might backfire. Following Gutiérrez’s (2012) protocol and hoping to broaden participants thinking about this question, I posed an additional question to the group: “Pretend you are in your first year of teaching and Ms. Marcus is on your team. What would you do?” The discussion opened to more carefully consider Ms. Marcus’s background, and to an extent, the sociocultural and institutional practices that may have shaped her views. Ms. Cass offered this speculation: Maybe she really is racist, but maybe she has been trained that way? I am not saying it is good training, but what if having all the girls sit in front was her training? Knowing this would put things into perspective to me about how you would handle things if this was your team teacher. This comment helped the group to step back and consider the institutional conditions under which teachers’ perspectives develop. Discussion shifted slightly from how to “fix this colleague” to the circumstances under which teaching behaviors like those Mr. David observed may have developed and could continue unchecked. As an example of institutional conditions that normalize inequity, I noted “Culture of Poverty” trainings that have been required in many districts and which promote racialized stereotypes of low-income children and their educational needs (Gorski, 2008). Other participants asked questions about the school climate and if other adults were aware of the atmosphere in Ms. Marcus’s classroom. We speculated about whether the lack of African American male teachers in both the building and as student teachers coming from our program might have affected this teacher’s perceptions of the appropriate mathematical goals and roles for Black students and helped to make it seem acceptable to discipline them more harshly than White students. We talked about some of the reasons that a new teacher might be afraid to speak up against injustice. To close the session, Mr. David explained that he had already implemented a “killing with kindness” approach with Ms. Marcus in which he strategically and deliberately praised her for everything she did in the classroom––even actions he disagreed with. He offered to help with everything from making copies to cleaning the board to tutoring challenging students. Mr. David reported that this strategy allowed him to achieve key short-term goals despite the limits of this placement. Ms. Marcus became more approachable to him; she offered advice and mentorship, although that advice was sometimes questionable relative to equity. Because she allowed him to take over the mathematics teaching when he was present, Mr. Marcus 2 MEG participants revised this term from “assigned competence” Cohen (1998). Journal of Urban Mathematics Education Vol. 10, No. 1 26 Wickstrom & Gregson Public Stories gained experience teaching conceptually rich mathematics in a setting that was more racially and economically diverse than his pre-cohort experiences. While teaching, he highlighted the strengths of the range of students in the class and provided an example for all students of a Black male mathematics teacher. He did not expect that his approach would change Ms. Marcus’s future behavior, but it allowed him space to provide some support for Black and White students and also to hone his skills for future equity battles. Remaining Questions and Tensions Mr. David’s story provided an opportunity for preservice and early-career mathematics teachers to consider, question, and analyze one inequitable classroom environment from multiple perspectives. Participants used the example to engage more nuanced questions about the nature of the institutional climate under which such inequitable conditions exist. The MEG participants had the opportunity to think specifically about how they would handle similar situations in their own practice. For me, questions and tensions remain. Ms. Shelby’s revelation about her experience with Ms. Marcus surprised me and was a reminder that MEG participants, like all learners, advance from their current understandings. At some point, I asked Ms. Shelby why she did not raise her concerns with this teacher in the group while they were happening. She reported that she had not thought about the problems as equity issues until she heard Mr. David’s story. While I was pleased that the discussion opened space for Ms. Shelby to see her own field placement differently, I was disheartened to think that neither our preservice urban education program nor previous MEG sessions had prepared Ms. Shelby to speak up about a classroom environment where she felt this uncomfortable. Issues of power were likely at play. Unlike Mr. David, whose prior experiences in urban schools gave him perspective, without that experience, Ms. Shelby, may have been less willing to question her placement––trusting that our program would not have put her into a situation where she would experience inappropriate teaching. And, like it or not, I cannot separate my role as a professor in my program from my role as a participant in MEG. I represent the institution that made Ms. Shelby’s placement. Thus, it makes sense that she might not have felt comfortable questioning her placement in my presence. It took Mr. David’s participation in MEG to bring problems to light. Likewise, many of my lingering questions involve discussing educational racism in an environment––mathematics teacher education––with low numbers of African American teachers. There is no doubt that Mr. David’s story was powerful because his experience with racism in American life and schools provides him with authority about what is and is not normal. Yet, he cannot speak for all Black people. How can I, as a White facilitator of MEG, do a better job of challenging participants to consider inequity beyond the superficial when they do not share Mr. Da- Journal of Urban Mathematics Education Vol. 10, No. 1 27 Wickstrom & Gregson Public Stories vid’s intimate knowledge of its effects? How can I provide direction for novice teachers without claiming to have all the answers? Where should my models come from? For example, the fact that Mr. David, a Black male graduate student with advanced mathematics skills, had to go to such lengths simply to be accepted as a legitimate teacher in the eyes of his cooperating teacher was a lesson for all of us. The group had the possibility to directly consider how subject position, level of experience with racism, and amount of urban teaching experience, can influence how one views a similar situation. Yet, I did little to explicitly help the group make these connections or to consider what such subjectivity means for promoting equity. Moreover, I believe I had a responsibility as a facilitator to frame this reality with hope and possibility for change; but in this case, our discussions never got to that level. Connecting Our Cases We (Susan and Megan) met at a conference for mathematics teacher educators. Megan was inspired by a presentation about the MEG because it was the first time she had heard open, detailed discussion of the challenges of learning to address inequitable teaching practices, especially racialized practices, in a professional setting. Megan saw many similarities between Mrs. Cate and Ms. Marcus, but, until that point, had been nervous about discussing the situation with Mrs. Cate openly. Megan was not sure what others could learn from her case and questioned the appropriateness of how she addressed the situation. Susan admitted that she, too, was nervous, especially about revealing such details of teaching without having the example either seen as business as usual and therefore demonizing to urban teachers, or having the example be dismissed as an abnormal outlier and therefore not important to address. Susan was also nervous about opening details of MEG for scrutiny when she questioned her own expertise at addressing racism. But a central goal of MEG is to learn to talk about and act against injustice even when it makes participants nervous. Thus, we decided we should further discuss our stories together to see what we might be able to learn and to share with others. In discussing our work, we asked “Is it appropriate to share these stories?” and “What is our purpose in sharing our stories?” We both felt inherent tensions within these cases. First, both of our accounts involved practicing teachers and the question of inequitable behaviors in their classrooms. It was challenging to retell these accounts without feeling like we were portraying the teachers as ineffective, uncaring, and racist while portraying ourselves as just and equitable researchers. We knew that both cases were complex and that publishing these accounts meant risking that both teachers’ practices and our own responses could be painted in binary “right or wrong” terms. In short, we did not want to paint classroom teachers from a deficit perspective, but working with teachers toward equity often creates Journal of Urban Mathematics Education Vol. 10, No. 1 28 Wickstrom & Gregson Public Stories these awkward situations, and it is important to be honest about our struggles and to share our strategies, no matter how imperfect. We also had to face the fact that writing about inequitable teaching practices and our own ability to confront such practice would not be as clean cut or straightforward as reporting on traditional mathematics education research. This complexity made us both uncomfortable and forced us to confront our own uncertainty about what counts as important knowledge for the field and where in the literature stories like ours belong. Lastly, as White, middle class teacher educators, we had concerns about whether our voices warranted being heard (Megan) and whether we have enough perspective to do justice to the complexity of the issues (Susan). We decided the best way to proceed would be to write down our cases and continually talk to work through our tensions as we wrote. As we worked, our discussion points repeatedly centered on the following questions:     Why did we respond to these scenarios the way we did, and why did we feel, at the time, that our responses were appropriate? What was and was not helpful in how we each responded to our situation? How were our responses too safe, and how could we have pushed our boundaries further? How would we respond now if faced with a similar situation? Why? In writing our accounts and discussing these questions, we learned about each other and ourselves. For example, in discussing how and why we responded the way we did, Megan wondered if she could have done more to explicitly discuss issues of equity with Mrs. Cate. Susan helped Megan to realize that the ways we address inequities are situated in our positionality and context. They can be improved over time. If faced with a similar situation now, Megan would feel more comfortable approaching Mrs. Cate differently. For example, when Megan witnessed Mrs. Cate favoring certain students, she might now ask, “How can we involve all students in the lesson and help them feel confident in their abilities?” She would consider saying directly, “In my view, it is not okay that all students are not working on the task.” She might also challenge Mrs. Cate to look for other students in the classroom who demonstrated competency. Likewise, Susan learned much from repeated discussions with Megan. Unlike Megan, Susan’s main research focus is equitable mathematics practice. Thus, our experiences with the equity research base were quite different. There were times, especially at the beginning of our partnership, when Susan’s experience with the literature got in the way of her own growth. For example, Megan suggested early on that the Pedagogy of Poverty framework (Haberman, 1991) might situate both of our experiences. At first, thinking that those ideas were dated and did not focus enough on racism to meet our needs, Susan questioned Megan’s idea. However, Journal of Urban Mathematics Education Vol. 10, No. 1 29 Wickstrom & Gregson Public Stories Megan’s thoughts about how the frame put her situation into perspective pushed Susan to take a second look at what she thought she knew. After reflection, Susan decided that Haberman’s (1991) findings were still relevant and useful in many ways, especially for the novice teachers in her MEG group. In future MEG discussions, Susan now plans to discuss the Pedagogy of Poverty frame by encouraging the participants to consider how the tenets of the original frame relate to racial injustice. As we participated in these conversations across time, our discussions opened spaces to process the equity challenges each of our stories raised. We found time to consider and talk through other approaches for acting the next time we are faced with such situations. And we developed a stronger understanding of each other’s perspective, which has allowed us to see each other as allies in in this work. A crucial element to having productive discussions was the fact that we both acknowledge that racial inequities exist and that addressing racism in the field is an essential component of our job as mathematics teacher educators. Both of us have worked with colleagues who accept inequitable teaching practice as the status quo or believe it is a wasted effort to address inequities because teachers’ beliefs cannot be changed. Moreover, we both have experienced instances where addressing inequity was treated as something to check off a list of expectations rather than an ongoing process. Addressing inequities involves developing, refining, and rehearsing potential strategies. Our collaboration allowed space to reflect and be better prepared when faced with a similar situation. It was through these conversations that we realized why our voices warranted being heard. The purpose of sharing our stories is to highlight that we all have the capacity to enact change, but we need the support and courage to start somewhere as well as the understanding that addressing inequity is not an all or nothing endeavor. Advocating for equitable teaching practices takes continual reflection and dialogue. It involves reflecting on when and where to assert yourself and why and discussing the appropriateness of your actions with others. We realized that our conversations gave us space to feel like we were being heard and an outlet where someone else was acknowledging our tensions so that in the future we would have tools to assert ourselves in appropriate ways. We hope that our accounts motivate other mathematics educators to continue to discuss issues of inequity in their work as well as spark a larger conversation on the creation of mathematics education equity discussion groups at the post-secondary level. References Aquirre, J., Herbel-Eisenmann, B., Celedón-Pattichis, S., Civil, M., Wilkerson, T., Stephan, M., Pape, S., & Clements, D. H. (2017). Equity within mathematics education research as a political act: Moving from choice to intentional collective professional responsibility. Journal for Research in Mathematics Education, 48(2), 124–147. Journal of Urban Mathematics Education Vol. 10, No. 1 30 Wickstrom & Gregson Public Stories Bartell, T. (2012). Is this teaching mathematics for social justice? Teachers’ conceptions of mathematics classrooms for social justice. In A. A. Wager & D. W. Stinson (Eds.), Teaching mathematics for social justice conversations with educators (pp. 113–125). Reston, VA: National Council of Teachers of Mathematics. Cohen, E. G. (1998). Making cooperative learning equitable. Educational Leadership, 56(3), 18–21. Glaser, M. (1982). The threat of the stranger. Vulnerability, reciprocity, and fieldwork. In J. E. Sieber (Ed.), Ethics of social research: Fieldwork, regulation, and publication (pp. 49–70). New York, NY: Random House. Goldenberg, B. M. (2014). White teachers in urban classrooms: Embracing non-white students’ cultural capital for better teaching and learning. Urban Education, 49(1), 111–144. Gorski, P. C. (2008). Peddling poverty for profit: Elements of oppression in Ruby Payne’s framework. Equity & Excellence in Education, 41(1), 130–148. Gutiérrez, R. (2012, October 8). Developing political knowledge for teaching mathematics: One way of making classrooms more equitable for all students. A webinar presented to the Association of Mathematics Teacher Educators (AMTE). Retrieved from Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education, 44(1), 37–68. Haberman, M. (1991). The pedagogy of poverty versus good teaching. Phi Delta Kappan, 73(4), 290–294. Hatt, B. (2012). Smartness as a cultural practice in schools. American Educational Research Journal, 49(3), 438–460. Lather, P. (1986). Issues of validity in openly ideological research: Between a rock and a hard place. Interchange, 17(4), 63–84. Powell, A. (2012). The historical development of critical mathematics education. In A. A. Wager & D. W. Stinson (Eds.), Teaching mathematics for social justice conversations with educators (pp. 21–34). Reston, VA: National Council of Teachers of Mathematics. Sternberg, R. J. (2007). Who are the bright children? The cultural context of being and acting intelligent. Educational Researcher, 36(3), 148–155. Stinson, D. W. (2004). Mathematics as “gate-keeper” (?): Three theoretical perspectives that aim toward empowering all children with a key to the gate. The Mathematics Educator, 14(1), 8– 18. Ware, F. (2006). Warm demander pedagogy: Culturally responsive teaching that supports a culture of achievement for African-American students. Urban Education, 41(4), 427–456. White, D. Y., Crespo, S. & Civil, M. (2016). Facilitating conversations about inequities in mathematics classrooms. In D. Y. White, S. Crespo, & M. Civil (Eds.), Cases for mathematics teacher educators: Facilitating conversations about inequities in mathematics classrooms (pp. 1–6). Charlotte, NC: Information Age. Wickstrom, M. H. (2015). Challenging a teacher’s perceptions of mathematical smartness through reflections on student’s thinking. Equity & Excellence in Education, 48(4), 589–605. Yang, K. W. (2009). Discipline or punish: Some suggestions for school policy and teacher practice. Language Arts, 87(1), 49–51. Journal of Urban Mathematics Education Vol. 10, No. 1 31 Copyright of Journal of Urban Mathematics Education is the property of Georgia State University, College of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.

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