The twenty-first century has seen an increased call to train students to craft mathematical arguments. The third of the Common Core's (CCSS) Standards for Mathematical Practice (SMP 3) (CCSSI 2010) calls for all mathematically proficient students to "construct viable arguments" to support the truth of their ideas and to "critique…

Recognizing the challenges that students face when they may lack background and experiential knowledge with regard to coins is important so that teachers can address this during instruction. Exposure to coins is one of the basic necessities to learning about the coins, but that is just the initial part of the learning sequence. Unless students are…

Imagine a mathematics classroom in which students engage in sharing ideas and reasoning through solutions to interesting mathematical problems. They are excited about mathematics and working on challenging problems that encourage collaboration and critical thinking. These are things that teachers want, but sometimes they do not know how to achieve…

Tape diagrams, also known as strip diagrams, bar models, fraction strips, or length models, are linear representations that can be used to effectively model quantitative relationships during problem solving. Tape diagrams are a powerful model recommended by the Common Core, but they are relatively new to teachers and students in the United States.…

Effective use of mathematical representation is key to supporting student learning. In "Principles to Actions: Ensuring Mathematical Success for All" (NCTM 2014), "use and connect mathematical representations" is one of the effective Mathematics Teaching Practices. By using different representations, students examine concepts…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) outlines ambitious goals for fraction learning, starting in third grade, that include the use of the number line model. Understanding and constructing fractions on a number line are particularly complex tasks. The current work of the authors centers on ways to successfully…

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…

Teachers are on the front line in any educational controversy. Increasingly, some bloggers, newspaper articles, and other media have criticized the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) as being inappropriate for children in kindergarten and first grade. However, both research and expert practice reveal that children are…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

Teachers face many challenges helping students meet the expectations set forth in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010). CCSSM content standards not only introduce new concepts at each grade level but also require teachers to learn new terminology for old concepts and consider how students' understanding of concepts…

Many elementary schools across the nation are scaling back or cutting arts curriculum altogether. This article presents classroom teachers with an activity for integrating art with mathematics. Through a three-phase exploration of reasoning with shapes and their attributes, students will get to know a classmate better, work with defining and…

During the past few years, several of the authors have incorporated student problem posing as a regular instructional feature in their classrooms. When they offer their students the opportunity to construct their own problems, particularly during the course of an entire school year, they create many novel tasks. Student-created tasks not only…

Measurement is an important component of K-grade 12 mathematics curricula. The concepts of area and perimeter of polygons are first introduced in third grade and serve as the basis for teaching in the upper grades. Without a strong understanding of measurement, students will struggle to meaningfully grasp three-dimensional measurement concepts…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…

Twenty-one fourth graders are actively listening and constructively critiquing mathematical statements. Each student is eagerly participating in mathematical discussions that involve multiple strategies to discover solutions. What a dramatic change since the first months of school when the same students sat as quiet as stone statues, hoping their…