between the first and second floors of an apartment, dividing a cake so everyone gets the same amount, and figuring out how far it is to the bus stop.

By the time children reach kindergarten, many of them can use their counting skills to solve simple problems that call for adding, subtracting, multiplying, or dividing. It is only when they move beyond what they understand informally—to the base-10 system for teens and larger numbers, for example—that their fluency and strategic competencies falter. Young children also show a remarkable ability to formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities. They are positively disposed to do and to understand mathematics when they first encounter it. For the preschool child, the strands of mathematical proficiency are especially closely knit.

Although most U.S. children enter school with a basic understanding of number, their knowledge is limited to small whole numbers and heavily influenced by the context in which the numbers appear. Furthermore, not all children enter school with the informal understanding of number assumed by the elementary school curriculum.

Developing Proficiency with Whole Numbers

Whole numbers are the easiest numbers to understand and use. In the early grades, children begin by solving numerical problems using methods that are intuitive and concrete. They then proceed to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers.

For most of a century, learning single-digit arithmetic—the sums and products of single-digit numbers and their companion differences and quotients (e.g., 5+7=12, 12–5=7, 12–7=5 and 5×7=35, 35÷5=7, 35÷7=5) — has been characterized in the United States as “learning basic facts,” and the emphasis has been on memorizing those facts. Acquiring proficiency in single-digit arithmetic, however, involves much more than memorizing. Even in the early grades, students choose adaptively among different procedures, depending on the numbers involved and the context. We use the term basic number combinations to highlight the relational character of this knowledge.

For addition and subtraction, many children follow a well-documented progression of procedures. Counting becomes abbreviated and rapid, and students begin to use properties of arithmetic to simplify their computation. Basic multiplication and division combinations are more of a challenge. Learning these combinations seems to require much specific pattern-based



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