surement. These ideas, which are important preparation for school and for life, are also genuinely mathematical, with importance from a mathematician’s perspective. Moreover, they are interesting to children, who enjoy engaging with these ideas and exploring them.
The third section describes mathematical process goals, both general and specific. The general process goals are used throughout mathematics, in all areas and at every level, including in the mathematics for very young children. The specific process goals are common to many topics in mathematics. These process goals must be kept in mind when considering the teaching and learning of mathematics with young children.
The fourth section describes connections across the content described in the first two sections as well as to important mathematics that children study later in elementary school. These connections help to demonstrate the foundational nature of the mathematics described in the first two sections.
Number is a fundamental way of describing the world. Numbers are abstractions that apply to a broad range of real and imagined situations—five children, five on a die, five pieces of candy, five fingers, five years, five inches, five ideas. Because they are abstract, numbers are incredibly versatile ways of explaining the world. “Yet, in order to communicate about numbers, people need representations—something physical, spoken, or written” (National Research Council, 2001, p. 72). Understanding number and related concepts includes understanding concepts of quantity and relative quantity, facility with counting, and the ability to carry out simple operations. We group these major concepts into three core areas: number, relations, and operations. Box 2-1 summarizes the major ideas in each core area. Developing an understanding of number, operations, and how to represent them is one of the major mathematical tasks for children during the early childhood years.
The number core concerns the list of counting numbers 1, 2, 3, 4, 5, … and its use in describing how many things are in collections. There are two distinctly different ways of thinking about the counting numbers: on one hand, they form an ordered list, and, on the other hand, they describe cardinality, that is, how many things are in a set. The notion of 1-to-1 correspondence bridges these two views of the counting numbers and is also central to the notion of cardinality itself. Another subtle and important aspect of numbers is the way one writes (and says) them using the base 10