dren sharing 4 pizzas, how many pizzas would be needed for 12 children to receive the same amount?).

It is likely, however, that an informal understanding of rational numbers is less robust and widespread than the corresponding informal understanding of whole numbers. For whole numbers, many young children enter school with sufficient proficiency to invent their own procedures for adding, subtracting, multiplying, and dividing. For rational numbers, in contrast, teachers need to play a more active and direct role in providing relevant experiences to enhance students’ informal understanding and in helping them elaborate their informal understanding into a more formal network of concepts and procedures. The evidence suggests that carefully designed instructional programs can serve both of these functions quite well, laying the foundation for further progress.⁷

Proficiency with rational numbers, as with all mathematical topics, is signaled most clearly by the close intertwining of the five strands. Large-scale surveys of U.S. students’ knowledge of rational number indicate that many students are developing some proficiency within individual strands.⁸ Often, however, these strands are not connected. Furthermore, the knowledge students acquire within strands is also disconnected. A considerable body of research describes this separation of knowledge.⁹

As we said at the beginning of the chapter, rational numbers can be expressed in various forms (e.g., common fractions, decimal fractions, percents), and each form has many common uses in daily life (e.g., a part of a region, a part of a set, a quotient, a rate, a ratio).¹⁰ One way of describing this complexity is to observe that, from the student’s point of view, a rational number is not a single entity but has multiple personalities. The scheme that has guided research on rational number over the past two decades¹¹ identifies the following interpretations for any rational number, say : (a) a part-whole relation (3 out of 4 equal-sized shares); (b) a quotient (3 divided by 4); (c) a measure ( of the way from the beginning of the unit to the end); (d) a ratio (3 red cars for every 4 green cars); and (e) an operation that enlarges or reduces the size of something ( of 12). The task for students is to recognize these distinctions and, at the same time, to construct relations among them that generate a coherent concept of rational number.¹² Clearly, this process is lengthy and multifaceted.