often represent incomplete rather than incorrect knowledge.22 From the current research base, we can make several observations about the kinds of learning opportunities that instruction must provide students if they are to develop proficiency with rational numbers. These observations address both representing rational numbers and computing with them.

Representing Rational Numbers

As with whole numbers, the written notations and spoken words used for decimal and common fractions contribute to—or at least do not help correct— the many kinds of errors students make with them. Both decimals and common fractions use whole numbers in their notations. Nothing in the notation or the words used conveys their meaning as fractured parts. The English words used for fractions are the same words used to tell order in a line: fifth in line and three fifths (for ). In contrast, in Chinese, is read “out of 5 parts (take) 3.” Providing students with many experiences in partitioning quantities into equal parts using concrete models, pictures, and meaningful contexts can help them create meaning for fraction notations. Introducing the standard notation for common fractions and decimals must be done with care, ensuring that students are able to connect the meanings already developed for the numbers with the symbols that represent them.

Research does not prescribe a one best set of learning activities or one best instructional method for rational numbers. But some sequences of activities do seem to be more effective than others for helping students develop a conceptual understanding of symbolic representations and connect it with the other strands of proficiency.23 The sequences that have been shown to promote mathematical proficiency differ from each other in a number of ways, but they share some similarities. All of them spend time at the outset helping students develop meaning for the different forms of representation. Typically, students work with multiple physical models for rational numbers as well as with other supports such as pictures, realistic contexts, and verbal descriptions. Time is spent helping students connect these supports with the written symbols for rational numbers.

In one such instructional sequence, fourth graders received 20 lessons introducing them to rational numbers.24 Almost all the lessons focused on helping the students connect the various representations of rational number with concepts of rational number that they were developing. Unique to this program was the sequence in which the forms were introduced: percents, then decimal fractions, and then common fractions. Because many children



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