Mistakes are a window into students' mathematical understanding. Taking the time to analyze mistakes can inform you of where students are in their thinking. They can also guide you with respect to how to proceed with instruction. Introduced here are two categories of mistakes: meaty mistakes and minor mistakes. Meaty mistakes demonstrate gaps in…

Teachers' questions shape the instructional environment and the direction of mathematics discussions in critical ways. This article is the product of a research study on the relationship between teachers' questions and student engagement in the Standards for Mathematical Practice (SMP) during whole-class discussions. Using classroom video gathered…

A single fact-focused number-string discussion is unlikely to result in all students adopting more flexible and efficient fact strategies; however, these authors have found that repeated engagement in such discussions over time does promote student reasoning about relationships, which in turn leads to increased fact fluency. By cultivating a…

The standards from the Operations and Algebraic Thinking (OA) domain in the Common Core State Standards for Mathematics (CCSSM) build on one another across the grades (CCSSI 2010). Understanding each grade level's OA standards is necessary within the broader context of the K-grade 5 OA standards. In this article, the authors share their…

This article focuses on one common type of many different classroom discussions, open-strategy sharing (OSS), because it offers a starting point for teachers new to mathematical discussions. The goal of an OSS discussion is to explore multiple possible ways to solve a single mathematical task. Students share strategies, listen to others'…

The right types of technology can create new opportunities for students to demonstrate mathematical understandings and can strengthen existing formative assessment practices. The authors write that they frequently use the Math Learning Center apps--covering the range of manipulatives and models--to empower students to explain their thinking. The…

Early career K-grade 2 teachers in a yearlong project explored discourse and mathematical argumentation through monthly professional development (PD) sessions. The goal of the project was to develop an understanding of the types of K-grade 2 tasks that promote argumentation, the types of discourse and argumentation that K-grade 2 students engage…

As part of the Foundations for Success in STEM project at Florida State University, more than thirty teams of teachers have used the Formative Assessment Collaborative Team (FACT) meeting model to work together toward better understanding of their students' mathematical thinking and its implications for instruction. This article provides a brief…

In many prekindergarten classes, children begin the day by singing a song as part of their opening routine. In reflecting on opening routines in public prekindergarten classrooms across Michigan, the authors have been struck by how often small changes in teachers' routines or materials could enrich the mathematical engagement of children. The goal…

This article highlights four routines that use visual images and open questions to invite students to discuss, justify, make arguments, and question their own thinking as well as their classmates' thinking. The authors share specific classroom examples from each routine that highlight the meaningful discourse they can inspire. Teacher prompts and…

Thinking about students in terms of participation instead of "high" or "low" achievers is more than replacing one term with another; it encourages us to shift our focus to student activity as well as barriers to and supports for productive activity. It encourages us to create classroom spaces that draw on student strengths and…

Classroom discourse is critical to helping students learn mathematics. Interacting with peers and the teacher within a classroom community assists students in building their understanding of mathematical concepts. A discourse community, sometimes referred to as a math-talk learning community, is "a classroom community in which the teacher and…

There are fundamental questions that teachers should ask themselves and their students as they prepare lessons on calculating area. Area measurement is commonly used in everyday life (perhaps to carpet a room or organize a space) and plays a foundational role in mathematics from multiplication all way up to calculus. Despite the usefulness of area…

Word problems are an essential component of providing a context for computation but are also recognized as an enduring struggle for students. Students are challenged to solve these story-based problems, often resorting to trial and error. This article presents meaningful alternative approaches to helping students make sense of word problems.

The sandwich problem described in this article would likely be considered more appropriate for third grade or above. However, as members of the research team for a lesson study involving teachers from kindergarten through fourth grade, the authors saw kindergarten students rise to the challenge. The authors invite readers to review two scenarios,…