This chapter lays out the analytic framework for the NSF research work, based on 1) broadening the definition of mathematical discourse to include more than the mathematics register and 2) moving from a focus on obstacles to an analysis of the resources bilingual students use to construct mathematical meaning. Rather than framing the relationship between learning mathematics and language in terms of discontinuities-from first language to second language, from social talk to academic talk (Cummins, 1981), or from the everyday to the mathematics register (Halliday, 1978)-the research will proceed from a situated perspective (Greeno, 1994). From a situated perspective learning mathematics is viewed as constructing conceptions and meanings, developing classroom norms (Cobb, Wood and Yackel, 1993), and participating in discourse practices (Gee, 1992; Lemke, 1990; Rosebery, Warren, and Conant, 1992) while using material and social resources. In contrast to cognitivist or constructivist perspectives, a situated perspective includes a consideration of both social and material aspects of a situation.

This chapter contributes a new way to frame the study of how bilingual students learn mathematics. A focus on the discontinuities that bilingual students face as they move from their first language or the everyday register to mathematical conversations in English is easily turned into a focus on the obstacles these students face. A focus on obstacles risks presenting a deficiency model (Garcia and Gonzalez, 1995; Gonzalez, 1995) of these students. Instruction needs to consider not only the obstacles that bilingual students face, but also the resources these students use to communicate mathematically. This chapter describes how the everyday register and students’ first language can, in fact, be used as resources for communicating mathematically.

This chapter is part of a volume designed as a resource for graduate students in mathematics and science education. We review the main principles for using a naturalistic paradigm to study mathematics and science cognition and learning, describe two studies that used a spiraling design for integrating this paradigm with other research methods, and outline standards of quality for naturalistic research studies. Our claims are that this paradigm can be integrated into studies of cognition, that naturalistic and cognitive (or experimental) methods can be combined in complementary ways, and that this integration and combination can move mathematics and science education research forward. We present two research projects as examples of how this integration is possible and what it can contribute. This chapter was written collaboratively and the names of the authors appear in reverse alphabetical order. Each author contributed equally to all sections of the chapter, except for the section on each of the two particular studies, which were written separately by each author. My role in producing this chapter was equally sharing the responsibility for the sections we wrote together (pages 457-466 and 474-477) and full responsibility for the section I wrote alone (pages 467-470).

Research and curriculum guidelines in mathematics education have outlined the characteristics of reform-oriented mathematics classrooms (NCTM, 1989). These characteristics include an increased emphasis on student communication and collaborative work. This chapter describes how these two new emphases, a focus on mathematical discourse and new forms of student participation, might intersect with the needs of Latino students and impact their experiences in the mathematics classroom. I also consider the limitations of research on how Latino students learn mathematics and productive areas for further study.

This volume is part of an NCTM series that summarizes issues and research in mathematics learning for different groups and is the first NCTM publication that specifically discusses Latino students.

This article based on the analysis of the first classroom study for the NSF project. The aim of the article is to explore how teachers can support English language learners in learning mathematics and not only English. The focus of the analysis is on teacher and student participation in mathematical discussions. The article uses a discourse perspective (Gee, 1990) on learning mathematics to define mathematical discourse as ways of talking about and points of view of mathematical situations. I examine a lesson from a third grade mathematics discussion of the geometric shapes from a tangram puzzle to illustrate how one teacher supported mathematical discussion and describe how students participated by talking about mathematical situations in different ways. I address several questions central to mathematics instruction for students who are learning English (as well as for students who are native English speakers): What is the teacher doing to facilitate student participation in a mathematical discussion? How is the teacher supporting students in speaking mathematically? What are the variety of ways that students talk about mathematical objects? What are the different points of view students bring to mathematical situations? Excerpts from the transcript of this lesson are used to exemplify supportive teaching strategies and describe the variety of ways that students communicate mathematically. In particular, the teacher did not focus primarily on vocabulary development but instead on mathematical content and arguments, as he interpreted, clarified, and rephrased what students were saying.

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 that is the result of sense-making, reflects the complexity of the domain, is productive in some contexts, and has the potential for refinement.