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…

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…

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…

Children's literature can provide useful contexts for teaching mathematics. In this article, the authors discuss the use of the children's book "Measuring Penny" (Leedy 1997) to support Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) related to length measurement. The authors first describe three types of misconceptions…

Composing and decomposing numbers with base-ten blocks depends on children being able to see ten both as ten units and as one group of ten units (a long), cognizant that its value is the same in either case. Being able to see, or deciding when to see, an object or collection of objects as a unit is a key skill that children must develop to solve…

The first author's initial attempts at a whole-class mathematics discussion were woefully dysfunctional. Ten second graders showing their answers to a missing addend question about the number of children left riding a school bus did not result in the mathematical epiphany that the textbook promised. She did not give up on using mathematical…

One of the most important jobs mathematics teachers have is supporting students and helping them to develop confidence in their own capabilities as mathematical thinkers. Children who do not believe in themselves are unlikely to share their ideas, persevere through challenges, or take risks that lead to new insights. Yet many students have no…

When measurement is taught without a context, students may struggle to make sense of the numbers and units involved. To master measurement, elementary school students need opportunities to discuss how they measured and applied their developing skills and reasoning in context. Measurement is an essential life skill, required to do everything from…

Overgeneralizing commonly accepted strategies, using imprecise vocabulary, and relying on tips and tricks that do not promote conceptual mathematical understanding can lead to misunderstanding later in students' math careers. In this article, the authors present thirteen pervasive mathematics rules that "expire." With the…

The authors recently conducted a randomized controlled trial that showed a significant impact of teachers' lesson study, supported by mathematical resources, on both teachers' and students' understanding of fractions. The research and mathematical resources are described in the second part of this article. First the authors examine some of the…

During the author's everyday math discussions with her class, young children talk about mathematical ideas, theories, and concepts within a predictable structure. These discussions include many concepts from--and beyond--the first-grade math curriculum, and their depth and complexity build throughout the school year. Concepts and skills include…

Building physical models is a natural activity for children and provides a context for lessons on two-dimensional (2D) and three-dimensional (3D) shapes. However, children do not think of their models as prisms, cylinders, and pyramids. Their creations are spaceships, tunnels, castles, and nets on the floor. Creating these types of shapes with…

With rigor outlined in state and Common Core standards and the addition of constructed-response test items to most state tests, math constructed-response questions have become increasingly popular in today's classroom. Although constructed-response problems can present a challenge for students, they do offer a glimpse of students' learning through…

In many classrooms, students solve problems posed by others--teachers, textbooks, and test materials. These problems typically describe a contrived situation followed by a question about an unknown that students are expected to resolve. Unsurprisingly, many students avoid reading these problems for meaning and instead engage in a suspension of…