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## Math Chat Opportunities Abound!

This article describes how fortuitous mathematical moments should be noticed, encouraged, embraced, and capitalized upon.

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## Continuous Improvement Lesson Study: A Model of MTE Professional Development

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.

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## Developing Preservice Teachers’ Understanding of Area Through a Units Intervention

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.

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## Learning Trajectories for Vertical Coherence

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

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## Smiles: A Strengths-Based On-Ramp Mathematics Task

Use this activity to support students in working together, recognizing one another’s contributions, and leveraging their mathematical strengths to solve challenging problems.

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## Discovering Area Models: The Giant Sicilian Pizza Task

This hands-on task, featuring differentiation and open-ended learning, sets up students to discover area models for themselves. Organized around NCTM’s eight teaching practices from Principles to Actions, this article describes the task’s setup and implementation.

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## Multiplication with and in Nature

This article features a third-grade multiplication exploration that integrates materials from nature and outside spaces. Teaching and learning mathematics with and in nature foster connections—mathematical, interpersonal, and with the natural world.

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## Developing Property-Based Geometric Reasoning

Recognizing the complex nature of students’ geometric reasoning, we present guidelines and suggestions for implementing a Guess My Shape minilesson that focuses students’ attention on properties and attributes of geometric shapes.

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## Eight Unproductive Practices in Developing Fact Fluency

Basic fact fluency has always been of interest to elementary school teachers and is particularly relevant because a wide variety of supplementary materials of varying quality exist for this topic. This article unpacks eight common unproductive practices with basic facts instruction and assessment.

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## Developing Skills for Exploring Children’s Thinking From Extensive One-on-One Work With Students

In this article, we share results from a field experience model in which junior-year methods classes were held in an elementary school and preservice teachers (PSTs) worked with a single student (a “Math Buddy") on mathematics for 30 minutes per day. We focus on the development of PSTs’ skills for exploring children’s thinking and the structures and tools that we used to support this development. Data sources include screencast recordings of interactions with Math Buddies and written reflections completed by PSTs. Although the responsiveness of interactions varied across individuals and interactions, in general, PSTs showed improvements in exploring children’s thinking. We share implications of these findings for similar field experience models and for practice-based approaches to teacher education generally.