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The mathematical medium of data visualization and other data representations (DV) has served as a primary means of communicating about the COVID-19 crisis. DVs about the pandemic are highly visible across news journalism and include an increasingly innovative and diverse set of representational forms. These representational forms employ multimodal, interactive, and narrative elements, among others, that create new possibilities for data storytelling. Building on current efforts to expand the teaching and learning of data practices in K-12 mathematics education, we argue that innovative DVs create new opportunities for teaching and learning mathematics, particularly during times of crisis. We illustrate our argument using three examples of innovative DVs from news journalism. We discuss how these DVs could serve as complementary resources alongside conventional graphs to support students as they use mathematics and mathematical representations to make sense of crises such as the COVID-19 pandemic. Our commentary seeks to bring current trends in data representation to bear in mathematics education. Leveraging such trends offers artifacts useful for teaching and opens up space for elevating emotion and experience as important aspects of mathematics curricula.

From the Archives highlights articles from NCTM’s legacy journals, previously discussed by the MTLT Journal Club.

Data visualizations have proliferated throughout the COVID-19 pandemic to communicate information about the crisis and influence policy development and individual decision-making. In invoking exponential growth, mathematical modelling, statistical analysis, and the like, these data visualizations invite opportunities for mathematics teaching and learning. Yet data visualizations are social texts, authored from specific points of view, that narrate particular, and often consequential, stories. Their fundamental reliance on quantification and mathematics cements their social positioning as supposedly objective, reliable, and neutral. The reading of any data visualization demands unpacking the role of mathematics, including how data and variables have been formatted and how relationships are framed to narrate stories from particular points of view. We present an approach to a critical reading of data visualizations for the context of mathematics education that draws on three interrelated concepts: mathematical formatting (what gets quantified, measured, and how), framing (how variables are related and through what kind of data visualization), and narrating (which stories the data visualization tells, its potential impacts and limits). This approach to reading data visualisations includes a process of reimagining through reformatting, reframing and renarrating. We illustrate this approach and these three concepts using data visualizations published in the New York Times in 2020 about COVID-19. We offer a set of possible questions to guide a critical reading of data visualizations, beyond this set of examples.

Place-based education has traditionally had a limited focus on mathematics, and critical mathematics education often lacks explicit attention to place and space. We explore integrating theories of ...

In this article, we share our experiences in mathematics teacher education around professional development for teachers with a focus on student participation as an opportunity to learn. We describe a process through which teacher educators can support teachers in increasing and improving classroom participation opportunities for their students. We present Lesson Activity Bars and Difference in Participation Proportion, complementary tools that quantify and represent student participation in the mathematics classroom. We demonstrate the effectiveness of these tools in supporting teacher growth in the context of a professional growth project for teachers in urban secondary schools. In general, the teachers in this project increased the amounts of active participation they made available to their students. The cases of two teachers are analyzed in detail, using Clarke and Hollingsworth's (2002) Interconnected Model of Professional Growth, to add depth and nuance to our understanding of processes of teacher growth around increasing student participation opportunities in the mathematics classroom.

This article presents a set of spatial tools for classroom learning about spatial justice. As part of a larger team, we designed a curriculum that engaged 10 learners with 3 spatial tools: (a) an oversized floor map, (b) interactive geographic information systems (GIS) maps, and (c) participatory mapping. We analyze how these tools supported learning using notions of politicization. The floor map fed conceptual understandings of the map as a representational text and served as the terrain for an embodied activity to support proportional reasoning about inequitable distributions of resources. The data-rich GIS maps and their zoomability allowed for coordinating across multiple variables to connect patterns in inequities to other social processes. The participatory mapping enabled learners to make discoveries about, connect, and share beyond the individual classroom counterstories from people in the lived streets of their neighborhood. In aggregate, this set of spatial tools produced a complex, hybrid view ...

Borders – territorial, political, economic, and ideological – are processes of social division. They monitor and exclude and are typically regulated, patrolled, maintained, and defended by an array of power regimes, but borderlands are also sites of movement, agency, and resistance. We draw on Thomas Nail’s and Gloria Anzaldua’s theories about borders to elaborate on processes of social division around gender and sexuality in mathematics education. The goal in this chapter is to recognize and challenge salient borders around gender, sexuality, and other identity categories in mathematics education and to work toward opportunities for hybridity created by these borders and in the blurring or queering of them. We provide a review of literature documenting the extent to which sexist and heterosexist ideologies patrol, reinforce, and perpetuate borders in mathematics that often marginalize women and queer people. We conclude the chapter with recommendations around how to blur borders around gender and sexuality in mathematics pre-service teacher education.

In this reflective essay, Laurie H. Rubel, Maren Hall-Wieckert, and Vivian Y. Lim present a design heuristic for teaching mathematics for spatial justice (TMSpJ) based on their development of two curricular modules, one about the state lottery and the other about financial services in a city. Spatial tools, including data visualizations on maps and participatory mapping, were designed for youth to examine spatial injustices in these systems. The authors' findings report reflections about supporting students to “read and write the world with mathematics” (Freire & Macedo, 1987; Gutstein, 2003). These reflections inform an expanded design heuristic for TMSpJ.

Take a chance on introducing the mathematics of the lottery. It will further an understanding of not only probability, combinatorics, large and small numbers, and data analyses but also social justice.

Expand the focus on diversity and equity to discuss the importance of gender and sexual identity for mathematics education.

This article explores integrating place-based education with critical mathematics toward teaching mathematics for spatial justice. Local Lotto, a curricular module with associated digital tools, was designed to investigate the lottery as a critical spatial phenomenon and piloted in urban high schools. This article describes findings from the second iteration in a remedial class in a low-income neighborhood. The research questions consider how the spatial focus supported the learning of mathematics and provided opportunities for students to think critically about the lottery using that mathematics. Findings include student interest in and engagement with the theme of the lottery familiar from outside of school with associated social justice implications. Students used mathematics and spatial evidence, at various levels of spatial scale, to support arguments about the lottery with greater success at narrower levels of scale. Suggestions about further innovations to scaffold place in a “critical peda...

This material may not be copied or distributed electronically or in any other format without written permission from NCTM. 84 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 21, No. 2, September 2015 “l“Low threshold, high ceiling” tasks (McClure 2011) are accessible to diverse learners; invite a wide range of approaches; and hold the potential to further challenge, strengthen, and extend everyone’s mathematical reasoning. We present a family of Brick Pyramid problems (Wittmann 1995; Selter 1997; Muller 2003) as examples of “low threshold, high ceiling” tasks. By eliciting the practices of noticing and describing patterns and then symbolizing and generalizing those patterns, Brick Pyramid problems hold great potential for engaging students in “algebraizing” (Freudenthal 1991). The pyramid in fi gure 1 has fi ve bricks in the bottom row, and each subsequent higher row contains one fewer brick. Numbers are assigned to bricks according to the following rule “Just Anot her Brick in the Wall”

This chapter investigates student learning in the context of a module about a city’s two-tiered financial system of banks and alternative financial institutions (i.e., pawnshops), held in ten sessions in a high school advisory class led by a mathematics teacher. The module exemplifies teaching mathematics for spatial justice, by extending teaching mathematics for social justice (Gutstein, 2006) with the idea that place matters (Gruenewald, 2003) and that justice has a geography (Soja, 2010). In this module, students use the concepts of percent to mathematize the costs of loans, and they analyze intensive variables to investigate spatial data about the density of these categories of institutions across their city. Spatial data is presented with GIS maps, layerable with demographic data and locations of financial institutions. The spatial distribution of the financial institutions reflects the inequalities of the spatial patterns in the city’s social demographics. Contextualized in the disparity of interest rates across these financial institutions, the spatial pattern not only reflects but reinforces those social inequalities. This chapter presents findings from one round of piloting and pursues the question: How did the module’s spatial justice orientation support the development of conceptual understanding of percent and ratio? Analysis focuses on growth with respect to mathematical understanding of percent and learning in class sessions organized around the use of ratios to understand the distribution. Findings include student adoption of strategies indicative of conceptual understandings of percent and development of critical opinions about their city’s two-tiered personal lending system.

Gender-based achievement and participation gaps in mathematics among Palestinian/Arab Israelis favor girls and women, in the face of various intersecting challenges. We follow the stories of Palestinian/Arab Israeli women students in mathematics education to better understand the context of their success in school mathematics and early career trajectories. Our analysis shows emergent related master-narratives: of Israel as a meritocracy; diligence as key to success in mathematics for girls and women; and mathematics achievement holding precious exchange-value. Guided by critical race feminism and Scott’s notions of public/hidden transcripts, our intersectional analysis shows how the women’s stories confirm, challenge, and inflect these narratives. We propose directions for further research and additional implications.

Redistricting can provide a real-world application for use in a wide range of mathematics classrooms.

This article reports findings from a research and professional development project at two high schools located in low-income, urban communities of color. The project collaborates with teachers on improving their instructional practices, using a framework of culturally relevant mathematics pedagogy, which is described in detail here. We present results from a qualitative and quantitative analysis of mathematics instruction in 68 classroom observations of seven teachers. In particular, we use culturally relevant mathematics pedagogy as a lens through which to analyze instruction and the associated opportunities to learn mathematics provided to students.

Maps at four levels of scale—global, national, regional, and local—provide a context for mathematical investigations that help teachers learn about their students.

This article examines middle school and high school student thinking about the law of large numbers, using a coin toss task. The analysis identifies common justification categories for incorrect responses as well as correct responses. These results are then discussed through the lens of in-school mathematical thinking and out-of-school mathematical thinking.

The concept of revoicing has recently received a substantial amount of attention within the mathematics education community. One of the primary purposes of revoicing is to promote a deeper conceptual understanding of mathematics by positioning students in relation to one another, thereby facilitating student debate and mathematical argumentation. Our study reexamines revoicing in a multilingual high school algebra classroom; our findings challenge the assumption that revoicing is necessarily tightly connected with classroom argumentation. We demonstrate that a single discursive form, such as revoicing, can play a wide range of valuable functions within the classroom. More importantly, we investigate systematic differences in the ways that revoicing is used, by a particular teacher, across languages. Implications for policy and practice are discussed.

The National Council of Teachers of Mathematics (2000) recommends that students at all grade levels be provided with instructional programs that enable them to “create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; and use representations to model and interpret physical, social, and mathematical phenomena” (p. 67). This article describes a particular classroom activity used to highlight the significance of mathematical representations.

NCTM's Communication Principle states that instructional programs at all grade levels should enable stu dents to "organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teachers, and others; [and] analyze and evaluate the math ematical thinking and strategies of others" (NCTM 2000, p. 60). One way for teachers and schools to achieve these goals is through the implementation of cooperative learning, where groups of students explore mathematical ideas, form conjectures, discuss results, and compare mathematical strate gies together. Research has found that cooperative learning can facilitate greater student participation (Webb 1991), promote positive attitudes about learning mathematics (Leikin and Zaslavsky 1997), and thereby increase student achievement in math ematics (Artzt and Yaloz-Femia 1999; Webb 1991). Beginning teachers, however, often struggle with how to implement cooperative learning in their classes. Memories of their own school experiences, when the teacher would say, "Slide your desk so that you're next to a partner, and compare answers to last night's homework," do not seem to embody fully the true spirit of cooperative learning or NCTM's Com munication Principle (NCTM 2000). In my experi

Readers are likely to be familiar with the infamous Monty Hall problem, played on the Let's Make a Deal game show and later addressed in the “Ask Marilyn” column in a 1990 issue of Parade.

“EXCELLENCE IN MATHEMATICS education requires equity—high expectations and strong support for all students.” Thus begins the Equity Principle found in Principles and Standards for School Mathematics (NCTM 2000, p. 12). It goes on to detail some of the dimensions of equity: high expectations, strong support, reasonable accommodations, worthwhile opportunities, and access to technology. People are unlikely to disagree with the stance implied by the Equity Principle. “Mathematics for all” is a rallying cry akin to “liberty and justice for all.” To disagree with the Principle would be like speaking out against motherhood or apple pie. Yet the pathway to mathematics for all is not always clear. What does equity in mathematics look like in our classrooms and schools? How would individual teachers proceed if the Equity Principle was guiding their teaching?

Local Lotto is a 14-session curriculum designed for high school students to learn mathematics through an examination of the local lottery. The curriculum is organized around investigations of how local lottery games are won, who plays, how many people play, and where lottery revenues and prizes are distributed. A web-based application is integrated into the curriculum to allow students to explore the lottery in their school neighborhood, examine local lottery data, and assemble and justify their own arguments about the lottery. In this chapter, the authors describe technology's role in shaping a rich curriculum that engages students in investigating a local phenomenon while also addressing the content and practices of the Common Core State Standards of Mathematics. The chapter concludes with an outline of the challenges of integrating custom technologies into mathematics curricula and provides recommendations for future work.

The preparation and training of mathematics teachers to teach in “urban” contexts is one of the most pressing issues in mathematics teacher education. The term “urban,” however, takes on multiple meanings in American vernacular as well as educational research discourse. Sometimes it is used as a placeholder for race, so that students from underrepresented groups become “urban youth,” regardless of whether or not they live in a city. Other times, the term “urban” is used to connote a high-poverty or high-crime area. In this chapter, the term “urban” specifically indicates “of or relating to a city.” Defined in this way, urban-ness of context is significant for mathematics teaching and learning because of its inherent challenges and often overlooked benefits.

This article reports on a subset of results from a larger study which examined middle and high school students' probabilistic reasoning. Students in grades 5, 7, 9, and 11 at a boys' school (n=173)...