Sharing "Cat Games" and Cookies: "Special Education Students Investigate Division"

Learning mathematics has traditionally been thought of as a sequential progression. Children learn to count to 10, then to 20, and then to 100. They learn to add without regrouping and then with regrouping. The authors teach addition before multiplication and the two-times table before the six-times table. They usually teach division as a separate unit after multiplication. Organizing mathematics textbooks as a sequential progression of skills promotes this perspective on teaching mathematics. When the authors joined forces to teach a unit on division to a class of 10 fourth graders, they purposefully adopted a very different perspective in planning the learning activities. Rather than preparing individualized instructional activities that targeted each student's misconceptions or computational difficulties, they selected learning activities that would give students many opportunities to investigate, in a variety of contexts and problem situations, a few major ideas related to multiplication and division. They focused on division as sharing, division as the inverse of multiplication, and using rectangular arrays to represent factors and products. They selected activities that had several correct solutions or solution paths so that students could successfully develop their individual knowledge and skills. As students shared their individual perspectives and solutions with other members of the class, they had opportunities to learn from each other. Because many of the activities were open ended, eliciting a variety of possible correct responses, the authors were able to assess students' knowledge and skills during each day's lesson. In particular, they were better able to understand how students were making sense of the problem situations and how they used models, pictures, or written symbols to represent problems and solutions. (Contains 7 figures.)