Ratio, rate, and proportion are central ideas in the Common Core State Standards (CCSS) for middle-grades mathematics (CCSSI 2010). These ideas closely connect to themes in earlier grades (pattern building, multiplicative reasoning, rational number concepts) and are the foundation for understanding linear functions as well as many high school…

Parents in the United States expect their students to have homework; and students, especially in middle school and high school, expect daily homework assignments from their teachers. However, it is difficult to create effective homework assignments. Despite the challenges involved, the authors believe that homework "can" be an important…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) identifies the strategic use of appropriate tools as one of the mathematical practices and emphasizes the use of pictures and diagrams as reasoning tools. Starting with the early elementary grades, CCSSM discusses students' solving of problems "by drawing." In later…

This article describes how teachers address issues and tensions that students meet in learning division of fractions. First, students must make sense of division of fractions on their own by working individually and in small groups, using concrete or pictorial representations, inventing their own processes, and presenting and justifying their…

Much attention has been devoted in the literature to students' struggles with fraction division. With regard to the traditional invert and multiply algorithm, researchers and educators have examined such areas as typical errors, how to help students understand why this method works, and what alternatives are available that might be more intuitive.…

An interesting change has been occurring in the author's middle school mathematics classes. Some students have become creative, free-thinking inventors. If they have had little experience with an operation, they invent strategies of their own. Sometimes these strategies have been well informed and supported by strong mental math skills and a good…

On entering Mr. Wilson's seventh-grade classroom, the authors immediately noticed the students talking about mathematics and their teacher listening to them. To create an interactive learning environment that would stimulate and perpetuate discourse, Wilson discussed two main issues with his students: (1) What the students and teacher would do in…

This article proposes four strategies for posing mathematics problems that raise the cognitive demands of the tasks given to students. Each strategy is illustrated with three common middle school mathematics examples: finding the greatest common factor, finding area or perimeter, and finding the equation of a line. Posing these types of problems…

Reports on middle school students' use of dynamic physical models to interpret functions. Includes student creations of symbolic variables and function equations. (KHR)

Describes an approach to algebraic thinking based on an extension of the problem-solving tasks typically investigated by elementary and middle school students. Demonstrates how elements of algebraic thinking may be developed in a systematic and coherent way through problem-solving experiences. States that students' pattern-building capabilities…

Presents some strategies for successfully implementing tasks that engage students at high cognitive levels. (YDS)

Discusses examples of the detailed explanations that students offered in response to a written, open-ended geometry task, especially the ways in which students communicated their knowledge. Illustrates how different students may select different methods of communication such as using text, diagrams, or mathematical symbols to display their…

Discusses several solutions--correct and incorrect--for a "Food for Thought" problem from the May 2000 issue. Emphasizes the importance of proportional reasoning. (KHR)

Presents some ideas about how teachers can foster mathematical communication by encouraging students to find multiple solutions to problems. Discusses three problems that help develop communication ability in mathematics. Views communication of multiple solutions as an instructional strategy to help students make sense of mathematics. (KHR)

Presents examples of students' work to illustrate the typical approaches used at sixth-, seventh-, and eighth-grade levels and four common threads that are consistent with student development as observed in a Connected Mathematics Project class. (ASK)