The transformation to reform mathematics teaching is a daunting task. It is often unclear to teachers what such a classroom would really look like, let alone how to get there. This article addresses this question: How does a teacher, along with her students, go about establishing the sort of classroom community that can enact reform mathematics practices? An intensive year-long case study of one teacher was undertaken in an urban elementary classroom with Latino children. Data analysis generated developmental trajectories for teacher and student learning that describe the building of a math-talk learning community—a community in which individuals assist one another's learning of mathematics by engaging in meaningful mathematical discourse. The developmental trajectories in the *Math-Talk* Learning Community framework are (a) questioning, (b) explaining mathematical thinking, (c) sources of mathematical ideas, and (d) responsibility for learning.

### Kimberly Hufferd-Ackles, Karen C. Fuson and Miriam Gamora Sherin

### Michael A. Forrester and Christopher D. Pike

In contrast to contemporary estimation researchers who have focused primarily on children's computational estimation abilities, we examined the ideas surrounding the teaching and learning of measurement estimation in the classroom. Employing ethnomethodologically informed conversation analysis, we focused on 2 teachers' instructions during estimation lessons and on pupils' (aged 9-11 years) talk during small-group follow-up activities. The results indicated that estimation is understood as discursively interdependent with measurement and is associated both with teacher-formulated accountability and with vagueness, ambiguity, and guessing. Furthermore, the meaning of what it is to estimate is embedded in practical action. In concluding comments we consider the advantages of using conversational analysis as a method for highlighting the relationships between language and mathematics in the classroom.

### Jessica Pierson Bishop

The moment-to-moment dynamics of student discourse plays a large role in students' enacted mathematics identities. Discourse analysis was used to describe meaningful discursive patterns in the interactions of 2 students in a 7th-grade, technology-based, curricular unit (SimCalc MathWorlds®) and to show how mathematics identities are enacted at the microlevel. Frameworks were theoretically and empirically connected to identity to characterize the participants' relative positioning and the structural patterns in their discourse (e.g., who talks, who initiates sequences, whose ideas are taken up and publicly recognized). Data indicated that students' peer-to-peer discourse patterns explained the enactment of differing mathematics identities within the same local context. Thus, the ways people talk and interact are powerful influences on who they are, and can become, with respect to mathematics.

### Brent Davis and Elaine Simmt

Complexity science may be described as the science of learning systems, where *learning* is understood in terms of the adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex learning systems, we explore some of the possible contributions of complexity science to discussions of the teaching of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of mathematical communities and the application of principles of complexity to the teaching of mathematics. Through the course of this writing, we attempt to highlight compatible and complementary discussions that are already represented in the mathematics education literature.

### Denise S. Mewborn

Four preservice elementary teachers were studied during a field-based mathematics methods course. The purpose of the study was to investigate the elements of mathematics teaching and learning the preservice teachers found problematic and how they resolved those problems. Data were collected in the form of individual interviews, group discussions, and individual journals. The preservice teachers exhibited concerns about the classroom context, pedagogy of mathematics, children's mathematical thinking, and, to a lesser extent, the mathematics content. The data indicate a relationship between the preservice teachers' locus of authority and the reflective quality of their thinking.

### Mary Ann Huntley

Using curriculum-specific tools for measuring fidelity of implementation is an essential yet often overlooked aspect of examining relationships among textbooks, teaching, and student learning. This “Brief Report” describes the variety of ways that curriculum implementation is measured and argues that there is an urgent need to develop curriculum-sensitive tools for analyzing classroom practice. The report outlines the use of the Concerns-Based Adoption Model (CBAM) theory to develop analytical tools for measuring implementation of two middle-grades reform mathematics curricula: *Connected Mathematics and MathThematics*. The report also presents next steps in this program of research.

### Stacy A. Brown, Kathleen Pitvorec, Catherine Ditto and Catherine Randall Kelso

Recent research on mathematics reforms in the United States indicates that the reforms are not yet widely implemented. Generally, this claim results from looking at the extent to which teachers use curricular materials or engage in particular classroom practices. This article moves beyond disparate questions of use and practice to examine interactions between teachers and curricula as evidenced by their enactments of whole-number lessons from a *Standards*-based curriculum. Specifically, we analyze videorecorded 1st- and 2nd-grade classroom lessons in terms of students' opportunities to reason and communicate about mathematics. This analysis indicates that the level of fidelity to the written curriculum differs from the level of fidelity to the authors' intended curriculum during lesson enactments. Drawing on this analysis, this article explores how curricula support and hinder teachers as they engage students in opportunities to learn mathematics and how teachers' instructional moves and choices impact the enactment of curricula.

### Indigo Esmonde and Jennifer M. Langer-Osuna

In this article, mathematics classrooms are conceptualized as heterogeneous spaces in which multiple figured worlds come into contact. The study explores how a group of high school students drew upon several figured worlds as they navigated mathematical discussions. Results highlight 3 major points. First, the students drew on 2 primary figured worlds: a mathematics learning figured world and a figured world of friendship and romance. Both of these figured worlds were racialized and gendered, and were actively constructed and contested by the students. Second, these figured worlds offered resources for 1 African American student, Dawn, to position herself powerfully within classroom hierarchies. Third, these acts of positioning allowed Dawn to engage in mathematical practices such as conjecturing, clarifying ideas, and providing evidence.

### Patricio G. Herbst

This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of *didactical contract* and *double bind* to uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching—what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition—are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.

### Terry Woods, Gaye Williams and Betsy McNeal

The relationship between normative patterns of social interaction and children's mathematical thinking was investigated in 5 classes (4 reform and 1 conventional) of 7- to 8-year-olds. In earlier studies, lessons from these classes had been analyzed for the nature of interaction broadly defined; the results indicated the existence of 4 types of classroom cultures (conventional textbook, conventional problem solving, strategy reporting, and inquiry/argument). In the current study, 42 lessons from this data resource were analyzed for children's mathematical thinking as verbalized in class discussions and for interaction patterns. These analyses were then combined to explore the relationship between interaction types and expressed mathematical thinking. The results suggest that increased complexity in children's expressed mathematical thinking was closely related to the types of interaction patterns that differentiated class discussions among the 4 classroom cultures.