### Karen S. Karp, Sarah B. Bush, Barbara J. Dougherty, and Introduction by: Jeffrey Shih

From the Archives highlights articles from NCTM’s legacy journals, as chosen by leaders in mathematics education.

### Lara K. Dick, Mollie H. Appelgate, Dittika Gupta, and Melissa M. Soto

A group of mathematics teacher educators (MTEs) began a lesson study to develop a research-based lesson to engage elementary preservice teachers with professional teacher noticing within the context of multidigit multiplication. Afterward, MTEs continued teaching and revising the lesson, developing an integrated process that combined lesson study with the continuous improvement model. This article introduces the continuous improvement lesson study process, shares an example of how the process was used, and discusses how the process serves as a collaborative professional development model for MTEs across institutions.

### Emily Elrod, Emily Elrod, and Valerie Faulkner

This editorial explores relationships between reviewers and authors within the *Mathematics Teacher Educator* community and provides reasons why and ideas for how to write a strong review.

Why take on the role of reviewer? That is a question we have asked ourselves as we generate emails asking our peers to do just that. Why do we spend our time reviewing other people’s work? Perhaps we have a sense of obligation, a sense of fairness or duty: “Others have reviewed my work; I need to put in my time." This, perhaps, draws us in. But, from there, we hope many of you find there is something more, something special, in the process.

### Jared Webb and P. Holt Wilson

## ABSTRACT

In this article, we describe rehearsals designed for use in professional development (PD) with secondary mathematics teachers to support them in reimagining and refining their practice. We detail a theoretical framework for learning in PD that informs our rehearsal design. We then share evidence of secondary mathematics teachers’ improvements in classroom practice from a broader study examining their participation in a PD that featured the use of rehearsals and provide examples of the ways two teachers’ rehearsals of the practice of monitoring students’ engagement with mathematics corresponded to changes in their practice. We conclude with a set of considerations and revisions to our design and a discussion of the role of mathematics teacher educators in supporting teachers in expanding their practice toward more ambitious purposes for students’ mathematical learning.

### Megan H. Wickstrom

Preservice elementary teachers (PSTs) often enter their teacher preparation programs with procedural and underdeveloped understandings of area measurement and its applications. This is problematic given that area and the area model are used throughout K–Grade 12 to develop flexibility in students’ mathematical understanding and to provide them with a visual interpretation of numerical ideas. This study describes an intervention aimed at bolstering PSTs’ understanding of area and area units with respect to measurement and number and operations. Following the intervention, results indicate that PSTs had both an improved ability to solve area tiling tasks as well as increased flexibility in the strategies they implemented. The results indicate that PSTs, similar to elementary students, develop a conceptual understanding of area from the use of tangible tools and are able to leverage visualizations to make sense of multiplicative structure across different strategies.

### Hyunyi Jung, Ji-Won Son, and Ji-Yeong I

Use a COVID-19 lesson as an example of how to apply a framework aligned with research recommendations to support students as they apply mathematics to real life.

### Kate Degner

Using question 28 from the May Problems to Ponder in volume 114, the author and her seventh- and eighth-grade students launched into a discussion of creativity, linearity, piecewise, and recursive definitions of functions. This pattern to ponder provided rich mathematical opportunities for all students in my middle school classroom.

### Jonee Wilson and Erin Smith

Instructional practices are often implemented in attempts to support multilingual learners. We discuss reasons that some of these well-intended practices may be problematic, and we share productive alternatives that we have observed in classrooms that effectively support multilingual learners.

### Jere Confrey, Meetal Shah, and Alan Maloney

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.