The mathematics curriculum during the preschool, elementary school, and middle school years has many components. But at the heart of mathematics in those years are concepts of number and operations with numbers— the mathematical domain of number. In this report, much of our attention is given to issues associated with teaching and learning about number in pre-kindergarten through eighth-grade mathematics. Many controversies over the teaching of mathematics center on the understanding and use of numbers. The learning of concepts associated with number also has been more thoroughly investigated than the learning of other parts of the mathematics curriculum. And much of the rest of the mathematics curriculum, some of which we do address, is intertwined with number concepts.

Number is a rich, many-sided domain whose simplest forms are comprehended by very young children and whose far reaches are still being explored by mathematicians. Proficiency with numbers and numerical operations is an important foundation for further education in mathematics and in fields that use mathematics. Because much of this report attends to the learning and teaching of number, it is important to emphasize that our perspective is considerably broader than just computation. First, numbers and operations are abstractions—ideas based on experience but independent of any particular experience. Communication about numbers, therefore, requires some form of external representation, such as a graph or a system of notation. The usefulness of numerical ideas is enhanced when students encounter and use multiple representations for the same concept. Second, the numbers and operations of school mathematics are organized as number systems, such as the whole numbers, and the regularities of each system can help students learn with understanding. Third, numerical computations require algorithms—step-by-step procedures for performing the computations. An algorithm can be more or less useful to students depending on how it works and how well it is understood. And finally, the domain of number both supports and is supported by other branches of mathematics, including algebra, measure, space, data, and chance. Our decision to address the domain of number was a pragmatic one; in no way does it imply that the elementary and middle school curriculum should be limited to arithmetic.

The Committee on Mathematics Learning was established by the National Research Council at the end of 1998. It was formed at the request of the Division of Elementary, Secondary, and Informal Education in the