divorced from meaning. Virtually no time is spent in relating the various representations—decimals, fractions, percents—to each other.31

While these are all significant problems and oversights, however, there are more basic problems with traditional instruction. The central problem with most textbook instruction, many researchers agree,32 is the failure of textbooks to provide a grounding for the major conceptual shift to multiplicative reasoning that is essential to mastering rational number. To support this claim, let us look at how rational number is typically introduced in traditional practice.

Pie Charts and a Part–Whole Interpretation of Rational Numbers

Most of us learned fractions with the model of a pie chart, and for many people, fractions remain inextricably linked to a picture of a partly shaded shape. Instruction traditionally begins with the presentation of pictures of circles (pies) and rectangles (cakes) that are partitioned and partially shaded. First, students are asked to count the number of parts in the whole shape and then the number of parts shaded. They then use these counts as the basis for naming and symbolically representing fractions. They learn that the top number, the numerator, always indicates how many pieces are shaded and that the bottom number, the denominator, always tells how many pieces there are in all. Next, using these same sorts of pictures (see Figure 7-3), instruction continues with simple addition and subtraction operations: “Two shaded 1/4 pieces (the bottom half of the circle) + 1 shaded 1/4 piece (the top left piece of the circle) = 3 shaded 1/4 pieces or 3/4.”

From a psychological perspective, this introduction is sound because it is based on students’ present knowledge and aligned with their experiences both in and out of school. We know that students’ formal mathematics programs have been based on counting, and that from everyday experience, students know about cutting equal pieces of pies and cakes. Thus, the act of assessing partitioned regions is well within their experience.

From a mathematical point of view, the rationale for this introduction is


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