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Jennifer Pfotenhauer, Rick Kleine, Yasmin Sitabkhan, and Darrell Earnest

Students had been learning about integers and fractions on the number line. For a lesson on mixed numbers, they solved an assessment problem at the beginning of the lesson. After the lesson, the authors interviewed two students individually and asked each girl to solve the same problem again.

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Hoyun Cho and Carolyn Osborne

Postscript items are designed as rich grab-and-go resources that any teacher can quickly incorporate into his or her classroom repertoire with little effort and maximum impact. A personal timeline provides a rich and relevant context this month for students' to investigate numbers and number relationships.

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Jinfa Cai, Anne Morris, Charles Hohensee, Stephen Hwang, Victoria Robison, and James Hiebert

In our March editorial (Cai et al., 2018), we considered the problem of isolation in the work of teachers and researchers. In particular, we proposed ways to take advantage of emerging technological resources, such as online archives of student data linked to instructional activities and indexed by learning goals, to produce a professional knowledge base (Cai et al., 2017b, 2018). This proposal would refashion our conceptions of the nature and collection of data so that teachers, researchers, and teacher-researcher partnerships could benefit from the accumulated learning of ordinarily isolated groups. Although we have discussed the general parameters for such a system in previous editorials, in this editorial, we present a potential mechanism for accumulating learning into a professional knowledge base, a mechanism that involves collaboration between multiple teacher-researcher partnerships. To illustrate our ideas, we return once again to the collaboration between fourth-grade teacher Mr. Lovemath and mathematics education researcher Ms. Research, who are mentioned in our previous editorials(Cai et al., 2017a, 2017b).

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Emily R. Fagan, Cheryl Rose Tobey, and Amy R. Brodesky

Start with a strategic process to gather and interpret evidence of students' mathematical understandings and misconceptions; then aim your teaching to address identified needs.

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Margaret Smith, Victoria Bill, and Mary Lynn Raith

This article provides an overview of the eight effective mathematics teaching practices first described in NCTM's Principles to Actions: Ensuring Mathematical Success for All.

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Pamela Edwards Johnson, Melissa Campet, Kelsey Gaber, and Emma Zuidema

Three preservice teachers used virtual manipulatives during clinical interviews with students of elementary school age. The technology exposed students' problem-solving strategies and mathematical understanding, promoting just-in-time teaching about the target content. The process of completing and reflecting on the interviews contributed to growth of the preservice teachers' technological pedagogical content knowledge.

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Andrew Izsák, Sybilla Beckmann, and Torrey Kulow

This article explores teaching practices described in NCTM's Principles to Actions: Ensuring Mathematical Success for All. Common factors, common multiples, strip diagrams, and double number lines are discussed in this, the third installment in the series.

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Andrew Izsák, Erik Jacobson, and Laine Bradshaw

We report a novel survey that narrows the gap between information about teachers' knowledge of fraction arithmetic provided, on the one hand, by measures practical to administer at scale and, on the other, by close analysis of moment-to-moment cognition. In particular, the survey measured components that would support reasoning directly with measured quantities, not by executing computational algorithms, to solve problems. These components—each of which was grounded in past research—were attention to referent units, partitioning and iterating, appropriateness, and reversibility. A second part of the survey asked about teachers' professional preparation and history. We administered the survey to a national sample of in-service middle-grades mathematics teachers in the United States and received responses from 990 of those teachers. We analyzed responses to items in the first part of the survey using the log-linear diagnostic classification model to estimate each teacher's profile of strengths and weaknesses with respect to the four components of reasoning. We report on the diversity of profiles that we found and on relationships between those profiles and various aspects of teachers' professional preparation and history. Our results provide insight into teachers' knowledge resources for enacting standards-based instruction in fraction arithmetic and an example of new possibilities for mathematics education research afforded by recent advances in psychometric modeling.