on with objects, they begin to use the counting words themselves as countable objects and keep track of how many words have been counted on by using fingers or auditory patterns. The counting list has become a representational tool. With time, children recompose numbers into other numbers (4 is recomposed into 3+1) and use thinking strategies in which they turn an addition combination they do not know into one they do know (3+4 becomes 3+3+1). In the United States, these strategies for derived number combinations often use a so-called double (2+2, 3+3, etc.). These doubles are learned very quickly.

As Box 6–1 shows, throughout this learning progression, specific sums move into the category of being rapidly recalled rather than solved in one of the other ways described above. Children vary in the sums they first recall readily, though doubles, adding one (the sum is the next counting word), and small totals are the most readily recalled. Several procedures for single-digit addition typically coexist for several years; they are used for different numbers and in different problem situations. Experience with figuring out the answer to addition problems provides the basis both for understanding what it means to say “5+3=8” and for eventually recalling that sum without the use of any conscious strategy.

Children in many countries often follow this progression of procedures, a natural progression of embedding and abbreviating. Some of these procedures can be taught, which accelerates their use,14 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition.15 In some countries, children learn a general procedure known as “make a 10” (see Box 6–2).16 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic. In some Asian countries this procedure is presumably facilitated by the number words.17 It has also been taught in some European countries in which the number names are more similar to those of English, suggesting that the procedure can be used with a variety of number-naming systems. The procedure is now beginning to appear in U.S. textbooks,18 although so little space may be devoted to it that some children may not have adequate time and opportunity to understand and learn it well.

There is notable variation in the procedures children use to solve simple addition problems.19 Confronted with that variation, teachers can take various steps to support children’s movement toward more advanced procedures. One technique is to talk about slightly more advanced procedures and why

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