tionality or rational numbers strategically to solve problems but are unable to represent and solve the same problem formally. These discontinuities are of great concern because the research we have reviewed indicates that real progress along each strand and within any single topic is exceedingly difficult without building connections between them.

A third issue concerns the level of procedural fluency that should be required for arithmetic with decimals and common fractions. Decimal fractions are crucial in science, in metric measurement, and in more advanced mathematics, so it is important for students to be computationally fluent—to understand how and why computational procedures work, including being able to judge the order-of-magnitude accuracy of calculator-produced answers. Some educators have argued that common fractions are no longer essential in school mathematics because digital electronics have transformed almost all numerical transactions into decimal fractions. Technological developments certainly have increased the importance of decimals, but common fractions are still important in daily life and in their own right as mathematical objects, and they play a central role in the development of more advanced mathematical ideas. For example, computing with common fractions sets the stage for computing with rational expressions in algebra. It is important, therefore, for students to develop sound meanings for common fractions and to be fluent with ordering fractions, finding equivalent fractions, and using unit rates. Students should also develop procedural fluency for computations with “manageable” fractions. However, the rapid execution of paper-and-pencil computation algorithms for less frequently used fractions (e.g., ) is unnecessary today.

Finally, we cannot emphasize too strongly the simple fact that students need to be fully proficient with rational numbers and integers. This proficiency forms the basis for much of advanced mathematical thinking, as well as the understanding and interpretation of daily events. The level at which many U.S. students function with rational numbers and integers is unacceptable. The disconnections that many students exhibit among their conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning pose serious barriers to their progress in learning and using mathematics. Evidence from experimental programs in the United States and from the performance of students in other countries suggests that U.S. middle school students are capable of learning more about rational numbers and integers, with deeper levels of understanding.



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