Education

To address issues of language in mathematics education, researchers need to use theoretical perspectives that integrate current views of mathematics learning and teaching with current views on language, discourse, bilingualism, and second language acquisition. This volume contributes to the development of such integrated approaches by describing theoretical perspectives and methodological issues in the study of language issues in mathematics education. The volume provides interdisciplinary reviews of the research literature from four different perspectives: mathematics education (Moschkovich), Cultural-Historical-Activity Theory (Gutiérrez, Sengupta-Irving, & Dieckmann), systemic functional linguistics (Schleppegrell), and assessment (Solano-Flores). This volume offers graduate students and researchers new to the study of language in mathematics education an introduction to resources for conceptualizing, framing, and designing research studies. For those already involved in examining language issues, the volume provides useful and critical reviews of the literature as well as recommendations for moving forward in designing research. Lastly, the volume provides a basis for dialogue across multiple research communities engaged in collaborative work to address these pressing issues.

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, 1996, 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.

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.

This case study uses a sociocultural perspective and the concept of appropriation (Newman, Griffin and Cole, 1989; Rogoff, 1990) to describe how a student learned to work with linear functions. The analysis describes in detail the impact that interaction with a tutor had on a learner, how the learner appropriated goals, actions, perspectives, and meanings that are part of mathematical practices, and how the learner was active in transforming several of the goals that she appropriated. The paper describes how a learner appropriated two aspects of mathematical practices that are crucial for working with functions (Breidenbach, Dubinsky, Nichols and Hawks, 1992; Even, 1990; Moschkovich, Schoenfeld and Arcavi, 1993; Schwarz and Yerushalmy, 1992; Sfard, 1992): a perspective treating lines as objects and the action of connecting a line to its corresponding equation in the form y = mx + b. I use examples from the analysis of two tutoring sessions to illustrate how the tutor introduced three tasks (estimating y-intercepts, evaluating slopes, and exploring parameters) that reflect these two aspects of mathematical practices in this domain and describe how the student appropriated goals, actions, meanings, and perspectives for carrying out these tasks. I describe how appropriation functioned in terms of the focus of attention, the meaning for utterances, and the goals for these three tasks. I also examine how the learner did not merely repeat the goals the tutor introduced but actively transformed some of these goals.

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 uses an evolutionary perspective of conceptual change to consider in detail a conception in the domain of linear functions. The analysis focuses on the nature of students' use of the x-intercept in equations of the form y = mx + b by summarizing the results of written assessments and presenting two case studies of students exploring and discussing linear equations and their graphs. I argue that the uses of the x-intercept documented in this study are not a superficial error, a simple mismatch with convention, or a misconception. Instead, this student conception is analyzed as an instance of a ‘transitional conception:’ a conception which is the result of sense-making, reflects the complexity of the domain, is productive in some contexts, and has the potential for refinement. The participants in the study were nine pairs of ninth and tenth grade students from an exemplary first-year algebra course. These students participated in videotaped discussion sessions with a peer of their choice where they used graphing software to explore linear equations and their graphs. The discussion sessions involved problems designed on the basis of student conceptions suggested in previous research (Moschkovich, 1989; Schoenfeld, Smith & Arcavi, 1993) and in classroom observations (Moschkovich, 1990). Protocol analysis of the videotaped discussion sessions was used to explore the nature and transformation of students' conceptions in this domain. Several uses of the x-intercept were documented in the written assessments and in the videotaped discussions. I summarize the results of the written assessments and present an analysis of the discussions for two pairs of students to show that the use of the x-intercept can be framed as a ‘transitional conception’

In this article, I describe the refinement of a conception in the domain of linear functions, the use of the x-intercept in equations of the form y = mx + b, to illustrate the resources used by students to refine a conception. The examples presented show how students refined their use of the x-intercept by (a) narrowing the problem contexts in which they applied this conception, (b) making connections between conceptions, (c) using a mathematical procedure, and (d) refining their verbal descriptions of how the graph of a line changes as a result of a change in the equation. The analysis draws on diSessa's (1993) theory of science learning and the recommendations made by Smith, diSessa, and Roschelle (1993) for describing conceptual change. The results extend this prior work from science to mathematics and identify resources used by students that teachers can draw on to support the learning of mathematical concepts.

This study examines mathematics learning in the context of peer discussions by focusing on students' descriptions of lines graphed on a computer screen. The article describes how these discussions provided a rich context for negotiating shared descriptions, supported conceptual change, and resulted in convergent meanings through reciprocal elaborations and clarifications. The participants in the study used graphing software to explore the connections between linear equations and their graphs with a peer. The article presents the analysis of three case studies examining how students negotiated shared descriptions of lines. These conversations supported students' construction of shared descriptions, but not necessarily by presenting conflicting ideas or through one student guiding another. Rather, negotiation functioned through local conversational resources such as the use of reference objects, spatial metaphors, and coordinated gestures and talk. These case studies also point to an important role for instruction in orchestrating and supporting peer discussions by modeling how to resolve negotiations and maintaining students' focus on mathematically productive learning trajectories.