to simple arithmetic problems. Furthermore, as children get older, they use the procedures more and more efficiently.4 Recent evidence indicates children can use such procedures quite quickly.5 Not all children follow the same path, but all children develop some intermediate and temporary procedures.

Most children continue to use those procedures occasionally and for some computations. Recall eventually becomes the predominant method for some children, but current research methods cannot adequately distinguish between answers produced by recall and those generated by fast (nonrecall) procedures. This chapter describes the complex processes by which children learn to compute with whole numbers. Because the research on whole numbers reveals how much can be understood about children’s mathematical development through sustained and interdisciplinary inquiry, we give more details in this chapter than in subsequent chapters.

Word Problems: A Meaningful Context

One of the most meaningful contexts in which young children begin to develop proficiency with whole numbers is provided by so-called word problems. This assertion probably comes as a surprise to many, especially mathematics teachers in middle and secondary school whose students have special difficulties with such problems. But extensive research shows that if children can count, they can begin to use their counting skills to solve simple word problems. Furthermore, they can advance those counting skills as they solve more problems.6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation.

Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing.7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use. The exact procedures children are likely to use have been well documented. Consider the following problems:

Sally had 6 toy cars. She gave 4 to Bill. How many did she have left?

Sally had 4 toy cars. How many more does she need to have 6?

Most young children solve the first problem by counting a set of 6, removing 4, and counting the remaining cars to find the answer. In contrast,



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