A one-degree angle would certainly be difficult to draw accurately, especially without the aid of other tools like a protractor. However, mentally making a one-degree angle in visualized imagination is not only possible, but it is important for students' emergent understandings of angular measure. When students begin learning about length,…

Researchers have suggested that initial attempts to change mathematics instruction often leave educators "with no framework for the kinds of specific, constructive pedagogical moves" they might make (Chazan & Ball, 1999, p. 2). The intent of this article is to introduce a framework for teaching and learning mathematics that is robust…

In their work, the authors study what it looks like to build on a high-leverage student mathematical contribution--one that provides an in-the-moment opportunity to engage the class in joint sense making about the contribution to better understand the important mathematics within it. They have noticed that some "go-to" teacher practices…

In this article, the authors provide some insight into how teachers can potentially facilitate among students' representations and activities on mathematical tasks: between or among -- (1) student representations and the context; (2) multiple forms of student representations; and (3) processes in student representations. The authors use the…

Decisions about how mathematics is communicated--and who gets to communicate that work--influence who has "mathematical authority" in a classroom. But what exactly does that mean, and in what ways can that authority be shared between students and their teacher? In this article, the authors expand on the idea of what it means to have…

Mathematics assessments should allow all students opportunities to demonstrate their knowledge and skills as problem solvers. Looking at textbook word problems, the authors share a process for revising them using Universal Design for Learning (UDL). Outlined in detail here are problems for kindergarten, grades 5 and 7, and high school (algebra).

Although many teachers recognize that they must adapt their instruction for the diverse and growing student population of multilingual learners across public schools in the United States, many report that they are unprepared to do this appropriately and productively (Banilower et al. 2013). This article aims to shed light on this issue.…

Learning trajectories (LTs) can inform teaching by contributing a variety of pedagogical and content-related insights and strategies. They support sequencing topic introduction and development. Because ideas evolve gradually and often rely on a careful introduction of new representations, operations, cases of numbers, structures, and definitions,…

Disciplinary literacy involves reading, writing, speaking, listening, and thinking in the context of specific disciplines. Word sorts are a way for students to acquire and examine vital features of mathematics vocabulary as they process and organize new content-specific ideas. Addressing disciplinary literacy (i.e., literacy skills specific to…

"Symbiosis," meaning "with" and "living," describes an ecological relationship between two organisms from different species that is sometimes, but not always, beneficial to both parties. One such relationship is mutualism, which is a mutually beneficial symbiotic relationship. In a mutualistic symbiotic relationship,…

Off You Go is a PK-12 mathematical routine that leverages children's home resources and assets to help students take a concept with which they have initial or informal understanding and explore and test its boundaries to support defining or estimating with precision. The authors provide a guide for how to adapt this routine to engage students at…

In this article, the authors define numberless math stories and highlight three strategies for supporting equitable problem solving: (1) The Turn-and-Tell Strategy; (2) Storyboard and Story Mat; and (3) The Act-It-Out Strategy. These three strategies open up problem solving and give every student access to a challenging mathematics curriculum…

This article aims to share five practical approaches for how teachers can identify, adapt, and create their own writing prompts they assign to students: (1) Promote Students' Solution Paths; (2) Go beyond Asking Students to Simply "Explain"; (3) Prompt Students to Share Their Reasoning; (4) Have Students Consider the Validity of a Given…

Typically, mathematics lessons are shaped around mathematical tasks. However, teachers frequently modify or enhance school- or district-adopted curriculum materials, often drawing on multiple sources to develop lessons. Research shows that teachers with a large percentage of students classified as English learners report modifying their materials…

When thinking about digital equity, often what comes to mind are issues of access to digital technologies. Although that is a divide that must be conquered, once students have access to digital technologies, the next step is to ensure that they wield these technologies in such ways that they help to address the existing inequities in mathematics…