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Instructional improvement initiatives in many districts include instructional coaching as a primary form of job-embedded support for teachers. However, the coaching literature provides little guidance about what activities coaches should engage in with teachers to improve instruction. When researchers do propose activities, they rarely justify why those activities might support teacher learning. Drawing on the preservice and inservice teacher education literatures, we present a conceptual analysis of learning activities that have the potential to support mathematics and science teachers to improve practice. We argue that our analysis can inform research on mathematics and science coaching, coaching policies, and the design of professional learning for coaches.

In the United States, teachers are often expected to use student assessment data to inform their instructional decisions. This study explores how teachers' collective reflections on students' mathematical thinking are influenced by the specific contexts of their classroom, school, and current events like the COVID-19 pandemic. The study uses audio recordings of a group of elementary teachers' reflections on student thinking through Cognitive Interview Reports. The results highlight how teachers' reflections on student data are heavily influenced by the shifting contexts of what is important in the world of teaching due to the COVID-19 pandemic. This ultimately shaped what the teachers were able to pay attention to in student data and what they identified as possible in their future practice. [ABSTRACT FROM AUTHOR]

Curiosity about student thinking lies at the heart of great mathematics teaching. When teachers are curious about and grounded in their students' thinking, their proficiency at eliciting, interpreting, and responding to student ideas increases (NCTM 2014). Coaching conversations that focus on students' thinking can support teachers in developing their instructional practices. When coaches frame their interactions with teachers by first examining students' thinking together, they can then problem solve together about next steps in instruction and engage in rich conversations about teaching and learning mathematics. In this article, the authors consider two related coaching conversations that arose from examining students' thinking during one-on-one coaching and during teacher collaborative time.

We present a framework for considering principals' knowledge and actions to support high-quality instruction in a specific content area (mathematics). Using design research, we engaged principals in professional development and assessed principals' ability to identify aspects of high-quality mathematical tasks and instruction through pre-post task sort analyses and classroom video analyses. Significant differences occurred in principals' identification of high-quality mathematics tasks and instruction, students' thinking, and teachers' actions. Subsequent data identified changes in principals' feedback to mathematics teachers; however, this change was not sustained in following years. We hypothesize necessary conditions for supporting principals as instructional leaders in specific content areas.

Many districts are using content-focused coaching as a strategy to provide job-embedded support to teachers. However, the current coaching literature provides little guidance on what coaches need to know and be able to do to engage teachers in activities that will support their development of ambitious instructional practices. Furthermore, little is known about how and why effective coaches choose to design particular types of activities with certain teachers. In this article, we report an exploratory case study that examined a mathematics coach who consistently engaged teachers in coaching activities that had the potential to support their development when she worked with them one-on-one in their classrooms. In presenting an analysis of the coach's design of coaching activities, we describe five aspects of her planning practice and delineate the knowledge implicated in those practices. The findings clarify goals for coaches' professional learning, and therefore have implications for school and district coaching policies.

Eliciting, responding to, and advancing students' mathematical thinking all lie at the heart of great teaching. In this article, the authors describe a formative assessment approach that teachers can use to learn more about their students' mathematical thinking and inform their instructional decisions. This assessment approach draws on a widely known set of frameworks for children's thinking, Cognitively Guided Instruction (CGI) (Carpenter et al. 2014). It enables teachers to learn from students by giving them time to voice their understandings and confusions. By listening carefully, teachers convey to students that their experience matters. The primary purpose of assessment is to inform and improve the teaching and learning of mathematics. The assessment approach described in this article gives teachers opportunities to talk with students about their mathematical thinking and gain a better understanding of their students as mathematicians.

Teachers are regularly asked to use data to inform their instruction. In the past, teachers examined student work in isolation. Now, however, teachers increasingly have dedicated meeting times. So how can teachers collaboratively examine student work and use their findings to improve instruction? In this article, a team of teachers at Hilltop Elementary School in the Pacific Northwest demonstrates the power of collaborative analysis of student work as teachers and school leaders use student work to guide their instructional decisions and support their professional learning about teaching mathematics. How can six students share eight sandwiches equally? The team of teachers use students' responses to this question to guide their instructional decisions and support their professional learning about teaching mathematics.

Supporting students to develop an understanding of the meaning of fractions is an important goal of elementary school mathematics. This involves developing partitioning strategies, creating representations, naming fractional quantities, and using symbolic notation. This article describes how teachers can use a formative assessment problem to elicit and understand students' ideas about the meaning of fractions. The fraction problem described in this article was designed using research on children's thinking in elementary mathematics, and it enables teachers to uncover students' understanding of the following essential aspects of understanding the meaning of fractions: (1) partitioning strategies; (2) representations; and (3) language and notation.

Mathematics lessons can take a variety of formats. In this article, the authors discuss lessons organized around complex mathematical tasks. These lessons usually unfold in three phases. First, the task is introduced to students. Second, students work on solving the task. Third, the teacher "orchestrates" a concluding whole-class discussion in which students are supported to make mathematical connections between solutions and to develop conceptual understanding of significant mathematical ideas. Complex tasks invite students to generate multiple solutions and justify their reasoning. Such tasks live up to their potential only if students engage productively in the task. Therefore, the first phase, the task introduction, is crucial. The authors describe how teachers can set up, or "launch," complex tasks by leading introductory whole-class discussions in which they help students engage in the task and learn significant math. (Contains 2 figures.)

Many districts are using content-focused coaching as a strategy to provide job-embedded support to teachers. However, the current coaching literature provides little guidance on what coaches need to know and be able to do to engage teachers in activities that will support their development of ambitious instructional practices. Furthermore, little is known about how and why effective coaches choose to design particular types of activities with certain teachers. In this article, we report an exploratory case study that examined a mathematics coach who consistently engaged teachers in coaching activities that had the potential to support their development when she worked with them one-on-one in their classrooms. In presenting an analysis of the coach's design of coaching activities, we describe five aspects of her planning practice and delineate the knowledge implicated in those practices. The findings clarify goals for coaches' professional learning, and therefore have implications for school and district coaching policies. [ABSTRACT FROM AUTHOR]