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GETTING STUDENTS TO THINK ABOUT MATHEmatics in ways that go beyond using procedures to solve routine problems is an important goal of mathematics reform. Manipulatives can be important tools in helping students to think and reason in more meaningful ways. By giving students concrete ways to compare and operate on quantities, such manipulatives as pattern blocks, tiles, and cubes can contribute to the development of well-grounded, interconnected understandings of mathematical ideas.
With the arrival of shiny buckets of manipulatives that now accompany many of the new middle school curricula, increasing numbers of teachers are becoming eager to create hands-on activities. Simply using manipulatives, however, does not guarantee a good mathematics lesson (Fennema 1972; NCTM 2000; Sowell 1989). In this article, we identify factors that are present when teachers create strong, mathematically sound lessons using manipulatives, then paint a portrait of a successful lesson and how those factors influenced it.
Factors Associated with Successful Manipulative Use
WE OBSERVED SEVERAL TEACHERS FROM A RURAL MIDdle school while each taught a lesson that used manipulatives to develop mathematical ideas. In the lessons, which supported the deepest levels of thinking and reasoning, students were given plenty of time to work with the manipulatives. The teacher assisted them as they constructed their own understandings on the basis of a well-designed set of manipulative activities. These lessons helped us to identify the factors that are present when manipulatives-- based lessons succeed.
The less successful lessons, however, helped us to understand ways in which the use of manipulatives can go astray. For example, even when manipulatives are planned as an integral part of the lesson, student thinking and reasoning can become routine and mechanical. This type of reasoning can occur when teachers show students how to work through problems "step by step," immediately "correct" any deviations from the prescribed procedure, and lead rather than guide students toward "discovery" of the mathematical ideas that the lesson was designed to teach. Rather than give students the time and latitude to think through and make sense of the manipulative activity on their own, teachers can shortcut student thinking by jumping in and supplying the "way to do it."
Another way in which manipulative activity can go astray is when teachers leave students too much...