can be made about developing proficiency with them. First, students do have informal notions of sharing, partitioning sets, and measuring on which instruction can build. Second, in conventional instructional programs, the proficiency with rational numbers that many students develop is uneven across the five strands, and the strands are often disconnected from each other. Third, developing proficiency with rational numbers depends on well-designed classroom instruction that allows extended periods of time for students to construct and sustain close connections among the strands. We discuss each of these points below. Then we examine how students learn to represent and operate with rational numbers.

Using Informal Knowledge

Students’ informal notions of partitioning, sharing, and measuring provide a starting point for developing the concept of rational number.2 Young children appreciate the idea of “fair shares,” and they can use that understanding to partition quantities into equal parts. Their experience in sharing equal amounts can provide an entrance into the study of rational numbers. In some ways, sharing can play the role for rational numbers that counting does for whole numbers.

In some ways, sharing can play the role for rational numbers that counting does for whole numbers.

In view of the preschooler’s attention to counting and number that we noted in chapter 5, it is not surprising that initially many children are concerned more that each person gets an equal number of things than with the size of each thing.3 As they move through the early grades of school, they become more sensitive to the size of the parts as well.4 Soon after entering school, many students can partition quantities into equal shares corresponding to halves, fourths, and eighths. These fractions can be generated by successively partitioning by half, which is an especially fruitful procedure since one half can play a useful role in learning about other fractions.5 Accompanying their actions of partitioning in half, many students develop the language of “one half” to describe the actions. Not long after, many can partition quantities into thirds or fifths in order to share quantities fairly among three or five people.

An informal understanding of rational number, which is built mostly on the notion of sharing, is a good starting point for instruction. The notion of sharing quantities and comparing sizes of shares can provide an entry point that takes students into the world of rational numbers.6 Equal shares, for example, opens the concept of equivalent fractions (e.g., If there are 6 chil-



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