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MIDDLE SCHOOL STUDENTS CONtinue to rely on concrete experiences to construct knowledge but are starting to develop the ability to think abstractly (NCTM 1989, p. 68). Thus, the middle school curriculum should provide a "bridge between the concrete elementary school curriculum and the more formal curriculum of the high school" (NCTM 1989, p. 102). This article describes a series of activities using attribute blocks designed to help middle school students construct knowledge about, and develop conceptual understanding of, probability. Depending on the ability levels of the students, these activities can be completed in a single two-- hour time block or can be spread over three or four sessions of about one hour each. These introductory probability explorations are appropriate for seventh graders but can be adapted for students at other levels. Attribute blocks are frequently used in the primary grades, but this article shows that they can be quite useful in the middle grades, as well.
Beginning the Exploration
EACH GROUP OF FOUR STUDENTS IS GIVEN A SET of attribute blocks. "Attribute blocks consist of various shapes, colors, sizes, and sometimes thicknesses, with exactly one piece of every possible combination of attributes" (Kutz 1991). The attribute blocks to be considered in these lessons vary in size, thickness, color, and shape. Each piece has the following characteristics: (1) large or small; (2) thick or thin; (3) red, blue, or yellow; and (4) a square, rectangular (i.e., nonsquare), triangular, hexagonal, or circular face (see fig. 1).
Although the students have used these manipulatives before, they are given a few minutes to become reacquainted with the various pieces. I ask them how many pieces their sets contain. The various counting strategies that they use to answer give me insight into the maturity level of their thinking. Many groups simply count the pieces item by item. Other groups see that three colors are involved and that the sets seem to have the same number of pieces of each color; these groups find the total by counting the pieces of one color and multiplying by 3. Occasionally, a group will deten-nine that the set has sixty pieces by multiplying 2 x 2 x 3 x 5, the number of possibilities for each characteristic.
The exploration process continues when the...