Teachers can implement a mathematics language routine within in-person/hybrid and remote instructional contexts.

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### Wayne Nirode

The author alters the definitions of ellipses and hyperbolas by using a line and a point not on the line as the foci, instead of two points. He develops the resulting prototypical diagrams from both synthetic and analytic perspectives, as well as making use of technology.

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

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.

### Kathryn Early, K. Elizabeth Hammonds, Brea Ratliff, Mariya Rosenhammer, and W. Gary Martin

A high-leverage strategy first discussed more than 50 years ago, wait time has many benefits for both teachers and students yet is not used to its full potential. See how it can enhance your students’ mathematical discourse.

### Ethan P. Smith, Jennifer Kelly, Susan Sappington, Kareemah Warren, and Amanda Jansen

Language is a conduit for communicating and understanding mathematical ideas. This article explores how we can use judicious telling to attend to students’ written and spoken literacy in mathematics.

### Sheldon P. Gordon and Michael B. Burns

We introduce variations on the Fibonacci sequence such as the sequences where each term is the sum of the previous three terms, the difference of the previous two, or the product of the previous two. We consider the issue of the ratio of the successive terms in ways that reinforce key behavioral concepts of polynomials.

### Eric Milou and Steve Leinwand

The standard high school math curriculum is *not* meeting the needs of the majority of high school students and that serious consideration of rigorous alternatives is a solution whose time has come.

### Charles F. Marion

The simplest of prekindergarten equations, 1 + 2 = 3, is the basis for an investigation involving much of high school mathematics, including triangular numbers, arithmetic sequences, and algebraic proofs.

### Victor Mateas

How trigonometry is used and portrayed differently in mathematics and physics textbooks highlights potential sources for student struggle, constraints on our trigonometry curriculum, and lessons learned when looking across STEM disciplines.

### 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.