Preschool arithmetic: A wealth of strategies. Much research has described the diversity of strategies that children show in performing simple arithmetic, from preschool well into elementary school.29 Strategies for solving a problem such as “What is 3+5?” include counting all (“1, 2, 3,…4, 5, 6, 7, 8”), counting on from the larger addend (“5,…6, 7, 8”), deriving the sum (“3+5 is like 4+4, so it’s 8”), and recall. Some children will model the problem using available object or fingers; others will do it verbally. (These strategies are discussed in detail in Chapter 6.)

Kindergartners use all of these strategies, and second graders use all of them except for counting all.30 What changes with age is the distribution of strategies, not the use of completely new ones. When 5-year-olds were given four individual sessions over 11 weeks in which they solved more than 100 addition problems, most of them discovered the counting-on-from-larger strategy, which saves effort by requiring them to do less counting.31 The children typically first identified this strategy when they were working with small numbers, where it does not save much effort. They then were most likely to apply it to problems (e.g., “What is 2+9?”) in which it makes a big difference in the amount of work needed.

The diversity of strategies that children show in early arithmetic is a feature of their later mathematical development as well. In some circumstances the number of different strategies children show predicts their later learning.32 The fact that children are inventing their own diverse strategies for doing arithmetic creates its own educational issues, however, as teachers need to be able to help children understand why some strategies work and others do not and to help them move on to advanced strategies.

Solving word problems. Young children are able to make sense of the relationships between quantities and to come up with appropriate counting strategies when asked to solve simple word, or story, problems. Word problems are often thought to be more difficult than simple number sentences or equations. Young children, however, find them easier. If the problems pose simple relationships and are phrased clearly, preschool and kindergarten children can solve word problems involving addition, subtraction, multiplication, or division.33 Young children are extremely sensitive to context, however, so the way in which the problem is posed can make a big difference in their performance. For example, if a picture of five birds and four worms is shown to preschoolers, most of them can answer the following: “Suppose the birds all race over and each one tries to get a worm. Will every bird get a worm? How many birds won’t get a worm?” But fewer of them can answer the question, “How many more birds than worms are there?”34

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