The foundation of geometric knowledge and understanding starts in primary school, and in contemporary mathematics education there is an emphasis on identifying learning progressions and trajectories as a way of moving students forward. The van Hiele (1986) theory is well-documented and provides insights into the progression and differences of…

"Principles to Actions: Ensuring Mathematical Success for All" (NCTM 2014) emphasizes eight teaching practices for effective mathematics teaching, one of which is to "use and connect multiple representations" (NCTM 2014, p. 24). An action that describes how teachers might promote this practice is to "allocate substantial…

Long before the release of the Common Core State Standards (CCSSI 2010), the Mathematical Tug-of-War was engaging students in the type of reasoning and problem solving described by the Standards for Mathematical Practice (SMP). In this updated version of a Marilyn Burns task, students use algebraic reasoning to determine the outcome of a contest…

In this article the authors describe a project during which they unpacked fraction standards, created rigorous tasks and lesson plans, and developed formative and summative assessments to analyze students' thinking about fraction multiplication. The purpose of this article is to (1) illustrate a process that can be replicated by educators…

Open-ended mathematical tasks provide great opportunities for students to engage in authentic mathematical practices, such as conjecturing, generalizing, and justifying. Supporting students in open-ended tasks can be challenging. Appropriate scaffolding of a task has been linked to more opportunities for student learning and better student…

This article discusses how teaching via problem solving helps enact the Mathematics Teaching Practices and supports students' learning and development of the Standards for Mathematical Practice. This approach involves selecting and implementing mathematical tasks that serve as vehicles for meeting the learning goals for the lesson. For the lesson…

Shrinking the achievement gap in mathematics is a tall order. One way to approach this challenge is to think about how the achievement gap manifests itself in the classroom and take concrete action. For example, opportunities to participate in activities that involve mathematical reasoning and argumentation in a safe and supportive manner are…

Do your students think a triangle can be constructed from any three given line segments? Do they believe that a transformation affects only the pre-image--not the whole plane? Do they understand that examples--no matter how many they find--cannot prove a conjecture but one counterexample is sufficient to disprove it? "What tasks can you…

An important question for mathematics teachers is this: "How can we help students learn mathematics to solve everyday problems, rather than teaching them only to memorize rules and practice mathematical procedures?" Teaching students using modeling activities can help them learn mathematics in real-world problem-solving situations that…

As current mathematics standards, such as the Common Core, are being implemented throughout the United States, it has become evident that teachers need support to enact the tenets of those standards. To help in this endeavor, this article was published as a guideline to emphasize to mathematics education stakeholders that "effective teaching…

How can geometry teachers design great tasks that allow students to make connections among interrelated concepts and expand their geometric reasoning skills? Many curricular materials provide problems for students to apply a single geometric concept. However, these problems do not always promote reasoning opportunities for students, because…

Researchers have extensively documented, and math teachers know from experience, that algebra is a "gatekeeper" to more advanced mathematical topics. Students must have a strong understanding of fundamental algebraic concepts to be successful in later mathematics courses. Unfortunately, algebraic misconceptions that students may form or…

Students around the world have difficulties in learning about fractions. In many countries, the average student never gains a conceptual knowledge of fractions. This research guide provides suggestions for teachers and administrators looking to improve fraction instruction in their classrooms or schools. The recommendations are based on a…

Noticing relationships between similar things and developing rules for treating things the same or differently, based on characteristics like color, size, shape, and texture provides the basis of beginning mathematics learning. Based on the view that teachers and parents of young children should provide children with age-appropriate words,…