It is beneficial for students to discover intuitive strategies, as opposed to the teacher presenting strategies to them. Certain proportional reasoning tasks are more likely to elicit intuitive strategies than other tasks. The strategies that students are apt to use when approaching a task, as well as the likelihood of a student's success or…

More than 2,200 years ago, Eratosthenes, who was a Greek astronomer, geographer, and mathematician, used a simple proportion involving the distance between two ancient cities and measures of shadows cast in those cities during a summer solstice to estimate the circumference of Earth (Nicastro 2008, 25-28). Today, middle school students can use…

A renewed emphasis was placed on ratio and proportional reasoning in the middle grades in the Common Core State Standards for Mathematics (CCSSM). The expectation for students includes the ability to not only compute and then compare and interpret the results of computations in context but also interpret ratios and proportions as they are…

The combined seventh-grade and eighth-grade class began each day with a mathematical reasoning question as a warm-up activity. One day's question was: Is the product of two odd numbers always an odd number? The students took sides on the issue, and the exercise ended in frustration. Reflecting on the frustration caused by this warm-up activity,…

"Mathematical habits of mind" include reasoning by continuity, looking at extreme cases, performing thought experiments, and using abstraction that mathematicians use in their work. Current recommendations emphasize the critical nature of developing these habits of mind: "Once this kind of thinking is established, students can apply it in the…

An nth-term problem involves a sequence. Students must determine which expression will allow them to calculate the nth position of the sequence. To solve such problems, students are to find "a rule that determines the number of elements in a step from the step number." These types of problems help students develop concepts of functions, variables,…

This article provides the rationale for using Fermi questions with eighth graders studying algebra. It describes the students' reaction to such questions as well as their thinking processes and solutions to questions posed to them by their teacher. (Contains 1 figure.)

This article describes an activity that incorporates 9 of the 10 National Council of Teachers of Mathematics (NCTM) Standards. Students explore a stacking garden containing multiple layers. The garden allows students to focus on finding the amount of border material needed and the amount of space available for planting. The problem balances the…

Three pieces of literature are used to launch mathematical investigations. The tasks provide opportunities for students to apply proportional reasoning skills, explore linear relationships, calculate a unit rate, and find the relationship between the volume of a cylinder and a cone. A poem, a short story, and a young adult novel are discussed.…

Spatial visualization is an important skill that deserves instructional attention. Strong evidence supports the claim that "measures of mathematical ability tend to be strongly correlated with spatial ability" (Anderson 2000). Thus, there is every reason to believe that time spent helping students develop their spatial visualization skills will…

This article discusses relationships between the development of an understanding of algorithms and algebraic thinking. It also provides some sample activities for middle school teachers of mathematics to help promote students' algebraic thinking. (Contains 11 figures.)

This article describes an activity in measurement and geometry in which sixth-grade students explore the volume and surface area of a donut. Through calculation, creative thinking, and discovery, the students enhanced their geometric vocabulary and reasoning. (Contains 9 figures.)