ing activities with integers has not been identified, but there are some promising elements that should be explored further. Students generally perform better on problems posed in the context of a story (debts and assets, scores and forfeits) or through movements on a number line than on the same problems presented solely as formal equations.60 This result suggests, as for other number domains, that stories and other conceptual structures such as a number line can be used effectively as the context in which students begin their work and develop meaning for the operations. Furthermore, there are some approaches that seem to minimize commonly reported errors.61 In general, approaches that use an appropriate model of integers and operations on integers, and that spend time developing these and linking them to the symbols, offer the most promise.

Beyond Whole Numbers

Although the research provides a less complete picture of students’ developing proficiency with rational numbers and integers than with whole numbers, several important points can be made. First, developing proficiency is a gradual and prolonged process. Many students acquire useful informal knowledge of fractions, decimals, ratios, percents, and integers through activities and experiences outside of school, but that knowledge needs to be made more explicit and extended through carefully designed instruction. Given current learning patterns, effective instruction must prepare for interferences arising from students’ superficial knowledge of whole numbers. The unevenness many students show in developing proficiency that we noted with whole numbers seems especially pronounced with rational numbers, where progress is made on different fronts at different rates. The challenge is to engage students throughout the middle grades in learning activities that support the integration of the strands of proficiency.

A second observation is that doing just that—integrating the strands of proficiency—is an even greater challenge for rational numbers than for whole numbers. Currently, many students learn different aspects of rational numbers as separate and isolated pieces of knowledge. For example, they fail to see the relationships between decimals, fractions, and percents, on the one hand, and whole numbers, on the other, or between integers and whole numbers. Also, connections among the strands of proficiency are often not made. Numerous studies show that with common fractions and decimals, especially, conceptual understanding and computational procedures are not appropriately linked. Further, students can use their informal knowledge of propor-

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