My aim in this article is to explore 3 perspectives on bilingual mathematics learners and to consider how a situated and sociocultural perspective can inform work in this area. The 1st perspective focuses on acquisition of vocabulary, the 2nd focuses on the construction of multiple meanings across registers, and the 3rd focuses on participation in mathematical practices. The 3rd perspective is based on sociocultural and situated views of both language and mathematics learning. In 2 mathematical discussions, I illustrate how a situated and sociocultural perspective can complicate our understanding of bilingual mathematics learners and expand our view of what counts as competence in mathematical communication.

This article reviews two sets of research studies from outside of mathematics education to consider how they may be relevant to the study of bilingual mathematics learners using two languages. The first set of studies is psycholinguistics experiments comparing monolinguals and bilinguals using two languages during arithmetic computation (language switching). The second set of studies is sociolinguistic research on young bilinguals using two languages during conversations (code switching). I use an example of a mathematical discussion between bilingual students to illustrate how sociolinguistics can inform analyses of bilingual mathematical conversations.

a b s t r a c tThis paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously.© 2015 Elsevier Inc. All rights reserved.This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. 1 The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion and questions adapted from Gee's (1999) questions for Discourse analysis, to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction, arguing that although these three components are important for all mathematics learners, it is essential that mathematics instruction for ELs include and maintain a simultaneous focus on all three components.The view of academic literacy in mathematics presented here is different than previous approaches to academic language in several ways. First, the definition includes not only cognitive aspects of mathematical activity-such as mathematical reasoning, thinking, concepts, and metacognition-but also sociocultural aspects-participation in mathematical practices-and discursive aspects-participation in mathematical discourse. A sociocultural perspective of academic literacy in mathematics provides a complex view of mathematical proficiency as participation in discipline-based practices that involve conceptual understanding and mathematical discourse. 2 Most importantly for ELs, this integrated view, rather than separating * Tel.: +1 8314592004.E-mail address: jmoschko@ucsc.edu 1 Although there are many labels used to refer to students who are learning English, I will use the term English Learners, abbreviated as ELs. Research on ELs may also be relevant to bilingual or multilingual mathematics lea...

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This article examines a classroom discussion of multiple interpretations of the scales on two distance versus time graphs. The analysis describes how two students and a teacher used multiple meanings for phrases of the form "I went by" and coordinated these meanings with different views of the scales. Students' ambiguous and shifting meanings did not prove to be obstacles to this discussion. Instead, this teacher used student interpretations as resources, built on them, and connected them to canonical mathematical concepts-in particular by highlighting (Goodwin, 1994) a "unitized" (Lamon, 1994(Lamon, , 1996(Lamon, , 2007 view of the scales. Research in mathematics education describes teaching that promotes conceptual development as having two central features: One is that teachers and students attend explicitly to concepts, and the other is that students wrestle with important mathematics (Hiebert & Grouws, 2007). Not only does this classroom discussion provide an example that it is possible to balance these two features, but the analysis provides the details of how instruction can simultaneously provide explicit attention to concepts while allowing students to wrestle with these concepts.Graphs, tables, and equations are ubiquitous in mathematics classrooms. Student interpretations of these inscriptions and the connections among these three representations are taken as evidence of conceptual understanding. We should not, however, assume that students and teachers interpret these inscriptions in the same ways. Researchers have described how students interpret algebraic symbols such

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