### Stacy Musgrave, Cameron Byerley, Neil Hatfield, Surani Joshua, and Hyunkyoung Yoon

The Common Core State Standards for Mathematical Practice asks students to look for and make use of structure. Hence, mathematics teacher educators need to prepare teachers to support students’ structural reasoning. In this article, we present tasks and rubrics designed and validated to characterize teachers’ structural reasoning for the purposes of professional development. Initially, tasks were designed and improved using interviews and small pilot studies. Next, we gave written structure tasks to over 600 teachers in two countries and developed and validated rubrics to categorize responses. Our work contributes to the preparation and support of mathematics teachers as they develop their own structural reasoning and their ability to help students develop structural reasoning.

### F. Paul Wonsavage

In this article, I share a design-based research intervention meant to help mathematics district leaders build their capacity to engage with research quality. I present my design (i.e., principles, key features, and intervention structure) and elaborate on how the features of the design allowed for mathematics district leaders’ sensemaking of educational research quality, especially regarding the process for collecting data and research implications. I conclude with recommendations for mathematics teacher educators on how they might adapt my design to their contexts.

### Richard Kitchen, Libni B. Castellón, and Karla Matute

By examining some of Ms. Hill’s instructional moves, we demonstrate how a fifth-grade teacher simultaneously developed her multilingual learners’ mathematical reasoning and mathematics register.

### Keith Dreiling

The Trammel of Archimedes traces an ellipse as the machine’s lever is rotated. Specific measurements of the machine are used to compare the machine’s actions on GeoGebra with the graph of the ellipse and an ellipse formed by the string method.

### Gülseren Karagöz Akar, Merve Saraç, and Mervenur Belin

In this study, we investigated prospective secondary mathematics teachers’ development of a meaning for the Cartesian form of complex numbers by examining the roots of quadratic equations through quantitative reasoning. Data included transcripts of the two sessions of classroom teaching experiments prospective teachers participated in, written artifacts from these teaching sessions, and their answers to pre-and-post written assessment questions. Results point toward prospective teachers’ improved meanings regarding the definition of complex numbers and the algebraic and geometrical meanings of the Cartesian form of complex numbers. Implications for mathematics teacher education include providing specific tasks and strategies for strengthening the knowledge of prospective and in-service teachers.

### Jana Dean

Two classic hands-on tasks address conceptual understanding of functions. The tasks center student discourse and rough draft mathematics as students grapple with the relationship between input and output.

### Katherine Ariemma Marin and Natasha E. Gerstenschlager

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.