Explore the vast range of titles available.

In this commentary, I challenge the field to take seriously the idea of incremental professional development (PD) for teachers of mathematics. I begin by briefly describing the history of PD in the United States since the publication of NCTM’s Curriculum and Evaluation Standards, what we have learned about effective PD through research, and then pose set of dilemmas we face in having a wide impact on the teaching of mathematics in the United States. Through reflecting on my story of incremental and transformative change as a high school mathematics teacher and 28 years as a mathematics teacher educator and scholar, I argue that the concept of incremental PD is worthy of exploration.

\"Principles to Actions: Ensuring Mathematical Success for All\" (NCTM 2014, p. 10) contains eight research-informed teaching practices that have been shown to support students' mathematical thinking and learning. Two teaching practices highlighted herein are \"to elicit and use evidence of students' thinking\" and \"support students' productive struggle in learning mathematics.\" Through enacting these two teaching practices, teachers can support students in ways that do not take over the thinking of the students. Such communication involves teachers determining \"how\" students are thinking mathematically and, in the moment, being able to respond in a way that supports students building on their thinking. In this article, the authors share a set of communication moves that teachers can use to support efforts to \"elicit and use evidence of students' thinking\" and \"support students' productive struggle in learning mathematics\" and then illustrate how a teacher can implement such moves while working with students.

Parents in the United States expect their students to have homework; and students, especially in middle school and high school, expect daily homework assignments from their teachers. However, it is difficult to create effective homework assignments. Despite the challenges involved, the authors believe that homework \"can\" be an important component of mathematics learning. In this article, the authors first describe two important ideas about homework. They hope that these ideas will help teachers begin to rethink the purposes of homework and support them in making effective choices about assigning homework. They then describe specific strategies for choosing, revising, or creating assignments that help students learn from homework. Over time, keeping these big ideas in mind and following some of these suggestions can help make homework more productive for teachers and students.

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs, pictorial proofs, and so on. This article examines the effectiveness of a course on reasoning-and-proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

The recent push toward active learning – engaging students in the learning process – is meant to benefit students. Yet there is still much to learn about students’ perceptions of this phenomenon. We share results from an interview study of students’ perceptions of features of two active learning models institutionalized at a large doctoral-granting university – a model for teaching foundational mathematics courses and an early field experience model for teaching preservice secondary students to teach mathematics. These models were implemented simultaneously in a single precalculus course. Interviews were conducted with both student populations (i.e., precalculus students and preservice teachers) to understand which in-class features of the models students noticed and identified as beneficial to their learning. Precalculus students identified specific opportunities related to active learning in the undergraduate mathematics teaching model – working in groups on mathematics tasks that engaged students in sensemaking and interacting with their instructor around mathematics. Preservice teachers identified specific opportunities related to three features of the university field experience model – observing a mathematics instructor enacting ambitious instructional practices, planning and teaching a “real” lesson, and observing student thinking and practicing teaching moves during groupwork. We conclude with pedagogical recommendations about particular features of the models that may help mitigate student resistance to active learning.

In the research reported in this article, we sought to understand the instructional practices of 26 secondary teachers from one district who use a problems-based mathematics textbook series (Core-Plus). Further, we wanted to examine beliefs that may be associated with their instructional practices. After analyzing data from classroom observations, our findings indicated that the teachers’ instructional practices fell along a wide continuum of lesson implementation. Analysis of interview data suggested that teachers’ beliefs with regard to students’ ability to do mathematics were associated with their level of lesson implementation. Teachers also differed, by level of instructional practices, in their beliefs about appropriateness of the textbook series for all students. Results strongly support the need for professional development for teachers implementing a problems-based, reform mathematics curriculum. Further, findings indicate that the professional development be designed to meet the diverse nature of teacher needs.

In the last decade, mathematics teacher educators have begun to design learning opportunities for preservice mathematics teachers using a pedagogies-of-practice perspective. In particular, learning cycles provide a structure for engaging PSTs in learning to teach through the use of representations, approximations, and decompositions of practice (Grossman et al., 2009). In this article, we provide details of one learning cycle designed to support secondary mathematics preservice teachers' learning to elicit and use evidence of student thinking and pose purposeful questions (National Council of Teachers of Mathematics, 2014). Through qualitative analyses conducted on learning reflections, we provide evidence of the impact on engagement of this cycle through the lens of the Framework for Learning to Teach (Hammerness et al., 2005).

Recent research efforts (Schmidt et al. in The preparation gap: teacher education for middle school mathematics in six countries, MSU Center for Research in Mathematics and Science Education, 2007 ) demonstrate that teacher development programs in high-performing countries offer experiences that are designed to develop both mathematical knowledge and pedagogical knowledge. However, identifying the nature of the mathematical knowledge and the pedagogical content knowledge (PCK) required for effective teaching remains elusive (Ball et al. in J Teacher Educ 59:389–407, 2008 ). Building on the initial conceptual framework of Magnusson et al. (Examining pedagogical content knowledge, Kluwer, Dordrecht, pp 95–132, 1999 ), we examined the PCK development for two beginning middle and secondary mathematics teachers in an alternative certification program. The PCK development of these two individuals varied due to their focus on developing particular aspects of their PCK, with one individual focusing on assessment and student understanding, and the other individual focusing on curricular knowledge. Our findings indicate that these individuals privileged particular aspects of their knowledge, leading to differences in their PCK development. This study provides insight into the specific aspects of PCK that developed through the course of actual instructional practice, providing a lens for future research in this area.

This study draws on the theoretical underpinnings of the research literature in identity and investigates the transition from experienced teacher to novice mathematics coach. The 4 components of a math coach's identity (coach as supporter of teachers, coach as supporter of students, coach as learner, and coach as supporter of the school-at-large) that this study highlights were enacted by the beginning coaches on their school stage and negotiated with their audience (i.e., teachers, principals) as they attempted to fulfill these roles within the school environment. By examining the roles, expectations, and interactions of first-year mathematics coaches, we deepen our understanding of the demands placed on novice mathematics coaches as they assume new roles and identities.

In this study we investigate a strategy for engaging high school mathematics teachers in an \"initial\" examination of their teaching in a way that is nonthreatening and at the same time effectively supports the development of teachers' pedagogical content knowledge [Shulman (1986). \"Educational Researcher,\" 15(2), 4-14]. Based on the work undertaken by the QUASAR project with middle school mathematics teachers, we engaged a group of seven high school mathematics teachers in learning about the Levels of Cognitive Demand, a set of criteria that can be used to examine mathematical tasks critically. Using qualitative methods of data collection and analysis, we sought to understand how focusing the teachers on critically examining mathematical tasks influenced their thinking about the nature of mathematical tasks as well as their choice of tasks to use in their classrooms. Our research indicates that the teachers showed growth in the ways that they consider tasks, and that some of the teachers changed their patterns of task choice. Further, this study provides a new research instrument for measuring teachers' growth in pedagogical content knowledge.