Mathematics teacher educators play a key role in supporting secondary mathematics teachers’ development of effective, research-based formative assessment (FA) practices. We used qualitative research synthesis as a tool to identify actionable recommendations for mathematics teacher educators as they work with teachers on FA practices in secondary classrooms. These recommendations can strengthen the research-based practices of mathematics teacher educators as they support teachers’ collections and uses of FA data to move student thinking forward in secondary mathematics. We share and discuss recommendations for mathematics teacher educators to connect pedagogical content knowledge of students, teaching, and curriculum to FA practices. We also highlight the usefulness of the qualitative synthesis method, meta-aggregation, for generating research-based connections between theory and practice in mathematics education.

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### Rachael H. Kenney, Michael Lolkus, and Yukiko Maeda

### Sara Gartland, Shellee Wong, and Laurie Silverstein

Co-teachers in a ninth-grade algebra 1 class offered instruction that integrates mathematical learning with social and emotional learning during hybrid (online and face-to-face) class meetings, promoting healing and positive identity development among students.

### Evthokia Stephanie Saclarides, Brette Garner, Gladys Krause, Claudia Bertolone-Smith, and Jen Munson

Learning to teach mathematics is a complex endeavor, requiring sustained focus and time. Yet time is especially scarce in elementary teacher education programs, where preservice teachers (PSTs) learn all content areas. Through a collaborative self-study, five teacher educators identified three time-related tensions in elementary mathematics methods courses: (a) teaching mathematics content and pedagogy; (b) connecting theory and practice; and (c) promoting social contexts in teaching mathematics. To address these tensions, we offer three design principles and illustrative examples: (a) addressing multiple goals for each course component; (b) developing PSTs’ dispositions over time; and (c) building on PSTs’ strengths to develop understanding of mathematics. We present a reflection tool to assist mathematics teacher educators in designing their courses to maximize their instructional time.

### Kevin Voogt and Kristen Bieda

This article explores one novice mathematics teacher educator’s initial use of the Mathematical Quality in Planning Protocol, an innovative tool that was developed to assist in providing feedback on the mathematical quality of novice mathematics teachers’ lesson plans. The protocol was devised to help mathematics teacher educators bridge the gap between prospective teachers’ mathematical content knowledge and their mathematical content knowledge for teaching. Results of our analysis on an initial use of the protocol point to its potential as a tool to help mathematics teacher educators direct their feedback from being overly focused on the pedagogical aspects of the lesson (e.g., timing, planned activities) to the mathematical content prospective teachers are attempting to teach (e.g., anticipated student solutions, problem-solving strategies).

### Amber G. Candela and Melissa Boston

In this article we detail a research study using the Instructional Quality Assessment (IQA) Rubrics () as the frame for a professional development with mathematics teachers in grades 3-8. We wanted to create a professional development around a tool that was typically used in research as a way to observe teachers, as a tool to use with teachers on their reflection of instruction. In this study we share both the researchers’ and teachers’ perspectives of affordances and constraints of the professional development and observational rubrics.

### Amanda T. Sugimoto

Mathematics standards and practices highlight the vital role that language plays in mathematics education. However, there remains a common misconception that mathematics is somehow language-free or less linguistically demanding than other content areas. This qualitative study describes an intervention implemented in six elementary mathematics methods courses. The intervention was designed to attune prospective teachers’ noticing to the language modalities and supports in mathematics teaching and learning. The intervention began with an observation tool that prospective teachers completed in their field placement classrooms. This article classifies prospective teachers’ noticings and explicates how these noticing became a pedagogical catalyst for further learning and discussion in subsequent mathematics methods classes.

### Madelyn W. Colonnese

A teacher implements this type of personal prose in the classroom to help students make sense of fractions and communicate ideas.

### Jenna R. O’Dell, Cynthia W. Langrall, and Amanda L. Cullen

An unsolved problem gets elementary and middle school students thinking and doing mathematics like mathematicians.

### 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.

### 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.