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In this literature synthesis, the three authors bring their embodied voices to collectively retrace the political-cultural-racialized-gendered topographies where key conceptualizations of language, mathematics, and the learner have emerged. Recognizing that they themselves have never been outside these landscapes as mere objective observers, the authors project their situated voices to critically consider--always with a sense of ownership and responsibility--what the field of language and mathematics research has done with/to these conceptualizations. These critical considerations are intended as a collective reminder that the field of language and mathematics education research can move forward by (re)engaging a dialogue between "what is" and "what could be" In this sense, the purpose of this literature synthesis is not to report yet another state of the art in the field, but rather to pierce through ongoing discourses to re-engage the field in reimagining possibilities for reopening, revising, and reimagining the concepts, methodologies, and philosophies that inform research in language and mathematics education. The synthesis concludes with a discussion of how and why our multivocal literature synthesis matters in the field of mathematics education research.

In many schools across the world, students experience mathematical concepts as ideas empty of wisdom and possibility. In this paper the authors analyze a philosophical conversation in which fifth-grade students were caught up in the animacy of the concept of space. Challenging the common view of mathematics as dealing with absolute truths and certainty, these students, the materials, and the concept itself formed dynamic assemblages that, through movement and the senses reanimated philosophical considerations regarding the concept of space. Over the course of an hour, students emerged as young philosophers whose ideas about space converged with those of philosophers from diverse fields, including physics, indigenous education, feminist theory, and dance. Using a philosophical analysis, the paper focuses on the vibrancy of a space filled with forces--images, movement, ideas, and materials--that shaped students' relationships with mathematics and language. Based on this analysis, the paper provides insights for reimagining the teaching and learning of mathematical concepts as philosophical topics that can support a renewed understanding of the relationship between learner, language, and mathematics.

This paper conceptualizes children's mathematical thinking from a materialist perspective on language and mathematics. This perspective considers human and non-human bodies as ontologically equivalent; that is, as both being agentive, vibrant, and animated, thus resisting static representations. This conceptualization is an alternative to the interactionist and language-based models that have dominated research in language and mathematics. This conceptual approach informs the paper's non-hierarchical analysis of a bilingual classroom studying the concept of equivalent fractions. The non-hierarchical nature of this analysis permits a dynamic shift of focus from learners to materials to mathematical concepts. This non-traditional analysis informs the paper's final discussion regarding the nature of children's mathematical thinking. The discussion highlights how working outside language-based interactionist models provides an alternative view of children and artefacts as equal partners working within and as part of the vibrant boundaries between matter and meaning.

This article elaborates a theoretical, methodological, and analytical approach intended to highlight the materiality and reciprocity of noticing in mathematics classrooms. Drawing from highly resonant concepts from materialism and Indigenous Knowledges--two perspectives that researchers rarely bring into dialogue--this alternative approach explores the reciprocal, material, and more-than-human nature of noticing. By focusing on the role of movement in noticing, the approach discusses the indivisibility of sensing and making sense as students and teachers mobilize mathematical concepts. Applying this approach in a non-traditional analysis of two classroom episodes, the paper provides a discussion centered on what noticing looks like when humans are decentered, concepts are mobilized, and sensing meets making sense.

The teaching and research communities in mathematics education agree that mathematical discussions pose challenges in elementary classrooms. These challenges continue to motivate research on mathematical discussions, with a focus on how students use talk in discussions. This study addresses the question, "What can teachers and researchers learn about mathematical discussions, when such discussions are led by bilingual students?" Evidence of bilingual mathematical discussions at the student-to-student level is analyzed, and implications for both Spanish-English bilingual and English monolingual mathematics instruction are offered.

In this paper, I theorize reciprocal noticing as a relational practice through which teachers and students exchange roles as knowers by reciprocating each other's noticing as they study mathematics concepts. Findings from a unit on measuring time implemented in two classrooms with non-dominant students illustrate how teachers and students--through their reciprocal noticing--mobilize concepts back to previous understandings and forward to possible new meanings. These findings serve to recalibrate the focus on professionalization of the teaching practice by considering how an inclusive understanding of noticing permits to appreciate the process of teachers and students sharing the process of learning mathematics in relationships.

In this essay, the authors provide a rationale for the need to break the silence of privilege and oppression in mathematics education. They begin by providing a brief rationale from their personal and professional perspectives, which includes background about planning and executing the Privilege and Oppression in the Mathematics Preparation of Mathematics Teachers Educators (PrOMPTE) conference, which motivated this special issue of the "Journal of Urban Mathematics Education." The authors then move into a (more typical) literature-based rationale for a focus on exploring and engaging with systems of privilege and oppression in relationship to themselves as mathematics teacher educators and in the preparation of new mathematics teacher educators.

In many classrooms, students solve problems posed by others--teachers, textbooks, and test materials. These problems typically describe a contrived situation followed by a question about an unknown that students are expected to resolve. Unsurprisingly, many students avoid reading these problems for meaning and instead engage in a suspension of sense making characterized by rule-following behavior and keyword searches. Problems in everyday situations, however, do not come preformulated. Instead, these problems and the reasoning that they instantiate develop simultaneously as problem solvers informally question the situation and begin to formulate conjectures and possible pathways for solving these problems. Word problem difficulties have been investigated primarily by looking at what students do or do not do, including looking at student errors or students' failure to use linguistic knowledge. These explanations suggest that students, not the problems need to change. In this article, the author suggests using a windows-and-mirrors framework for encouraging students to be problem posers. Like a window, a problem should be an opportunity for students and teachers to look out for what makes sense to solve the problem. Simultaneously, like a mirror, a problem should be an opportunity for students and teachers to look into what students notice as relevant for solving a problem. Readjusting the window and/or the mirror--as it is recognized in word problems--signifies reformulating a problem. Reformulating problems is important because asking students to solve preformulated problems has shown that students do not make sensible connections of mathematics to situations. When such experiences repeat every day, students develop self-perceptions that they are not good at math. The aim of this article is to explain how to help culturally diverse and socioeconomically disempowered students use problem posing to reflect their knowledge and gain insight into new knowledge. This article discusses four research-based strategies that can support all students in problem posing: (1) Let students specify some quantities; (2) Let students frame problem questions; (3) Promote problem posing at various points; and (4) Invite students to interpret representations. The author also provides questions that can elicit student noticing by restoring experiential mirrors and opening mathematical windows.

In this study, a temporal analysis and the analytical category of intersubjectivity are used to investigate how teachers and Latino/a bilingual students constructed shared communicative spaces in group mathematical discussions in an after school mathematics program in a culturally, linguistically, and economically diverse primary school. Intersubjectivity refers to the "common ground" or "shared understandings" that participants achieve through dialogue around joint activity, such as solving and discussing mathematical problems. Understanding how participants achieve intersubjectivity in mathematical discussion requires a temporal analysis, which includes attention to how particular instances of intersubjectivity have a history in previous interactions and a future in related activity. Using two illustrative cases, this study traced the histories and futures of the students' shared understandings, or intersubjetivities, across time. Findings highlight the importance of intersubjectivity as a catalyzer of a range of linguistic, cultural, and material resources, including Spanish and English, multiple representations, gestures, revoicing, and inscriptions, to help participants to attune to one another's understandings and further their intersubjectivity. These shared understandings were sometimes facilitated by divergent interpretations or confusions that teachers and students in ethical, emotional, respectful ways managed to address.

Textbooks, as well as classroom instruction, tend to present estimation as a stand-alone exercise, ignoring that "the process of estimation depends on the situation itself as well as the estimator." When taught this way, estimation is stripped of its power to help students judge the reasonableness of answers and make sense before, during, and after solving problems. Authors, Melissa Adams, a novice fourth-grade Latina bilingual teacher, and Higinio Dominguez, her bilingual Latino research partner, collaborated in a teaching unit on estimation. Early in the unit, they noticed that their students--all Latino and Latina bilinguals--were overapplying the rounding rule or not applying it at all. Noticing the difficulty that students were experiencing with the tasks, Adams commented, "I don't know why they're not getting something that seems so easy, you know?" Besides student difficulties, the authors began to notice a more fundamental problem: The teacher and students were attending to different aspects of the tasks. On the one hand, the teacher was attending to the computational aspect of estimation--learning how to estimate but not necessarily learning the purpose of estimating. Adams and Dominguez developed mathematical tasks that capitalized on the teacher and students' mutual noticing process. In the next interaction, Adams elicited from students multiple meanings related to the concept of estimation. Adams and Dominguez consistently privileged the informal words "más o menos," "casi," "about," and "almost," all common in students' everyday bilingual vocabularies--and deferred the more formal terms "aproximadamente" and "approximately" for the important point in the unit when students had had multiple opportunities to develop a strong understanding of the concept. In this teacher-researcher collaboration, the teacher learned to see estimation through her students' eyes.

This study investigated discursive positioning moves that facilitated Latino/a English learners' (ELs) opportunities to take on agentive problem-solving roles in group mathematical discussion. A focus on mechanisms that support students' agentive participation is consistent with the authors' view that recurrent experiences participating and being positioned in particular ways contribute to identity development. Findings suggest several ways that discursive positioning facilitated ELs' agentive participation. (Contains 3 tables, 3 figures, and 6 footnotes.)

In this study, the author represented what matters to bilingual students in their everyday lives--namely bilingualism and everyday experiences--in school-based mathematical problems. Solving problems in pairs, students demonstrated different patterns of organizing and coordinating talk across problem contexts and across languages. Because these patterns bear consequences for how bilinguals experience mathematics learning, the author takes these patterns as the basis to argue that mathematics education for bilingual students should capitalize on bilingualism and experiences as cognitive resources.

Although equal sharing problems appear to support the development of fractions as multiplicative structures, very little work has examined how children's informal solutions reflect this possibility. The primary goal of this study was to analyze children's coordination of two quantities (number of people sharing and number of things being shared) in their solutions to equal sharing problems and to see to what extent this coordination was multiplicative. A secondary goal was to document children's solutions for equal sharing problems in which the quantities had a common factor (other than 1). Data consisted of problem-solving interviews with students in 1st, 3rd, 4th, and 5th grades (n=112). We found two major categories of strategies: (a) Parts Quantities strategies and (b) Ratio Quantities strategies. Parts quantities involved children's partitions of continuous units expressed in terms of the number of pieces that would be created. Ratio quantities involved children's creation of associated sets of discrete quantities. Within these strategies, children drew upon a range of relationships among fractions, ratio, multiplication, and division to mentally or physically manipulate quantities of sharers and things to produce exhaustive and equal partitions of the items. Additionally, we observed that problems that included number combinations with common factors elicited a wider range of whole-number knowledge and operations in children's strategies and therefore appeared to support richer interconnections than problems with relatively prime or more basic number combinations.

This research focuses on eliciting bilingual students' problem-solving reasoning by providing mathematical tasks designed to involve the students in the action described in the problem by making connections to contexts familiar to them. Results showed these tasks allowed children to articulate their mathematical reasoning through gestures and speech. Two questions guided this investigation: (a) How do second-grade bilingual students communicate their mathematical reasoning? and (b) What is the role of the mathematical tasks in eliciting this mathematical reasoning? Analysis of seven second-grade Spanish-English bilingual students solving addition and subtraction problems showed that students simultaneously used words and gestures to communicate their mathematical reasoning to others and to regulate their own cognitive activity. In general, the students demonstrated that their developing bilingual proficiency did not constitute an impediment for guiding mathematical tasks to resolution; rather, students imparted mathematical meaning to each task by mutually supporting their verbal and nonverbal behaviors. The paper discusses how researchers and teachers can benefit from parallel attention to bilingual students' verbal and nonverbal communication. Both processes invite reflection on what it means to know and learn mathematics bilingually in early grades. (Contains 1 table.)

A research focusing on eliciting bilingual students' problem-solving reasoning is done by providing mathematical tasks designed to involve the students in the action described in the problem by making connections to context familiar to them. The results indicated that these tasks allowed children to articulate their mathematical reasoning through gestures and speech.

The article presents information on teaching estimation to bilingual fourth-grade students in the U.S. The authors look at their experiences using both English and Spanish languages to present a mathematical unit on estimation. The article discusses interpretations of estimation by students in the classroom, including exact calculation, double rounding, and selective rounding. The authors also explain how they created meanings related to the concept of estimation.