For addition and subtraction, there is a well-documented progression of procedures used worldwide^{37} by many children that stems from the sequential nature of the list of number words. This list is first used as a counting tool; then it becomes a representational tool in which the number words themselves are the objects that are counted.^{38} Counting becomes abbreviated and rapid, and students begin to develop procedures that take advantage of properties of arithmetic to simplify computation. During this progression, individual children use a range of different procedures on different problems and even on the same problem encountered at different times.^{39} Even adults have been found to use a range of different procedures for simple addition problems.^{40} Further, it takes an extended period of time before new and better strategies replace previously used strategies.^{41} Learning-disabled children and others having difficulty with mathematics do not use procedures that differ from this progression. They are just slower than others in moving through it.^{42}

Instruction can help students progress.^{43} Counting on is accessible to first graders; it makes possible the rapid and accurate addition of all single-digit numbers. Single-digit subtraction is usually more difficult than addition for U.S. children. If children understand the relationship between addition and subtraction, perhaps by thinking of the problem in terms of part-part-whole, then they recognize that counting up can be used to solve subtraction problems. This recognition makes subtraction more accessible.^{44}

The procedures of counting on for addition and counting up for subtraction can be learned with relative ease. Multiplication and division are somewhat more difficult. Even adults might not have quick ways of reconstructing the answers to problems like 6×8=? or if they have forgotten the answers. Learning these combinations seems to require much specific pattern-based knowledge that needs to be orchestrated into accessible and rapid-enough products and quotients. As with addition and subtraction, children derive some multiplication and division combinations from others; for example, they recall that 6×6=36 and use that combination to conclude that 6×7=42. Research into ways to support such pattern finding, along with the necessary follow-up thinking and practice, is needed if all U.S. children are to acquire higher levels of proficiency in single-digit arithmetic.

Acquiring proficiency with single-digit computations involves much more than rote memorization. This domain of number demonstrates how the different strands of proficiency contribute to each other. At this early point in