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Background: Current efforts to promote reasoning, problem solving, and discussion are often framed as advancing equity, but scholarship suggests individual students' opportunities to learn can vary considerably in classrooms that attempt to take up these approaches to teaching mathematics. Noticing students' mathematical strengths and positioning their contributions as competent is among aspects of instruction associated with more equitable learning outcomes for students from marginalized groups, but research has yet to comprehensively examine the range and nuance of this aspect of teachers' practice in classrooms that feature broad distributions of participation. Purpose: The purpose of this study was to examine teachers' instructional practice with respect to assigning competence in two mathematics classrooms that demonstrated high levels of student participation. We investigated the kinds of situations in which teachers positioned students as competent, and the ways assigning competence opened opportunities to participate. Setting: Data were collected at a public elementary school in a culturally, linguistically, and economically diverse neighborhood in southern California. Participants: Participants included two teachers and 45 students from two third-grade classrooms. Teachers had participated in ongoing professional development focused on leveraging children's mathematical thinking in instruction. Research Design: We drew from qualitative methods for analyzing video to investigate classroom interactions from 12 mathematics lessons. Data sources included video recordings, transcripts, and student work. We used Studiocode software to parse each lesson into phases and episodes. Drawing from previous studies, we identified a subset of episodes in which teachers explicitly positioned a student's contribution as competent. An iterative process of coding and discussion was used to analyze patterns with respect to student participation, teacher support, and the unfolding of rights and obligations related to participating in mathematical activity. Findings: Analyses revealed different kinds of situations in which students participated in mathematically substantive ways (in terms of providing detailed explanations of their ideas or engaging with the details of a peer's idea) and teachers positioned their contributions as competent. These situations included highlighting, clarifying, and amplifying contributions; supporting the specificity of student contributions; recognizing emergent ideas; and validating unprompted attention to mathematical details. Assignments of competence emerged in ways that were integrated into teachers' ongoing efforts to surface and make explicit the details of their mathematical ideas, while also broadening the kinds of contributions students could make to joint mathematical work. Conclusions: Helping students to know what it could look and sound like to participate in the moment while recognizing a wide range of contributions as competent created openings for students who in many classrooms might be excluded or relegated to the periphery of conversations. Making competence explicit was a contingent, relational practice that required teachers to find specific ways of leveraging student strengths to support their participation. Recommendations for advancing mathematics teaching must attend to the nuances with which particular practices unfold to open or constrain individual students' opportunities to learn.

The students in Ms. Moscoso's second-grade class gather on the rug after recess, ready for one of their favorite math warm-ups: Choral Counting. Counting is an important part of doing mathematics throughout the school; students count collections (Schwerdtfeger and Chan 2007) and solve problems using a variety of strategies, many of which are grounded in different ways of counting (Gelman and Gallistel 1986). In this article, the authors share a specific instructional activity that extends typical ways of counting out loud (by ones to thirty; by twos, fives, and tens; etc.) and provides opportunities for students to consider "what they notice" about the number sequence, leveraging student ideas to build conceptual understanding of number.

In recent years, work in practice-based teacher education has focused on identifying and elaborating how teacher educators (TEs) use pedagogies of enactment to learn in and from practice. However, research on these pedagogies is still in its early development. Building on prior analyses, this article elaborates a particular pedagogy of enactment, "rehearsal", developed through a collaboration of elementary mathematics TEs across three institutions. Rehearsals are embedded within learning cycles that provide repeated opportunities for novice teachers (NTs) to investigate, reflect on, and enact teaching through coached feedback. This article shares a set of insights gained from 5 years of developing, studying, and learning how to support NTs' enactment in rehearsal. The insights we share in this article contribute to building a knowledge base for pedagogies of teacher education.

We analyze a particular pedagogy for learning to interact productively with students and subject matter, which we call “rehearsal.” Our goal is to specify a way in which teacher educators (TEs) and novice teachers (NTs) can interact around teaching that is both embedded in practice and amenable to analysis. We address two main research questions: (a) What do TEs and NTs do together during the kind of rehearsals we have developed to prepare novices for the complex, interactive work of teaching? and (b) Where, in what they do, are there opportunities for NTs to learn to enact the principles, practices, and knowledge entailed in ambitious teaching? We detail what happens in rehearsals using quantitative and qualitative methods. We begin with the results of our quantitative analyses to characterize how typical rehearsals were structured and what was worked on. We then show how NTs and TEs worked together to enable novices to study principled practice through qualitative analyses of a particularly salient aspect of ambitious teaching, namely, eliciting and responding to students' performance. (Contains 3 tables, 2 figures, and 5 notes.)

In recent years, work in practice-based teacher education has focused on identifying and elaborating how teacher educators (TEs) use pedagogies of enactment to learn in and from practice. However, research on these pedagogies is still in its early development. Building on prior analyses, this article elaborates a particular pedagogy of enactment, rehearsal, developed through a collaboration of elementary mathematics TEs across three institutions. Rehearsals are embedded within learning cycles that provide repeated opportunities for novice teachers (NTs) to investigate, reflect on, and enact teaching through coached feedback. This article shares a set of insights gained from 5 years of developing, studying, and learning how to support NTs’ enactment in rehearsal. The insights we share in this article contribute to building a knowledge base for pedagogies of teacher education. [ABSTRACT FROM AUTHOR]