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# Mathematics (page 2)

By Pearson Allyn Bacon Prentice Hall
Updated on Jul 20, 2010

### Mathematical Skills in Early Childhood

One of preschoolers' most obvious accomplishments is learning to count. Starting at about the age of 2, children begin to associate the counting words used in their language with the correct number of objects. They quickly become quite accurate in their counting and learn to count more and more things.

The structure of number words in a child's native language has an impact on some early mathematical competencies. For example, notice how we count in English. We count up through 10 ("one, two, ... ten"), but then we call the next numbers "eleven" and "twelve." Next come seven numbers with "teen" added to the end ("thirteen, fourteen, ... nineteen"). After the "teens:' we count up to 100 by naming the tens place followed by the ones place (as in "twenty-one, twenty-two"). Therefore, as they learn to count to 20 in English and many other Western languages, young children may not recognize the tens-ones system (called the base-10 system) that is the foundation of mathematics. In contrast, many Asian languages follow a much simpler rule (Miller, Smith, Zhu, & Zhang, 1995). After reaching 10 they go straight into naming the tens place followed by the ones place (essentially counting "ten-one, ten-two:' and so forth). In these languages children don't need to remember the inconsistent rules of giving special names to 11 and 12, then adding "teen" to the end of a few numbers before finally starting to count with the tens-ones system. As you can see in the figure below, it takes English-speaking children longer to learn to count than it does for their Chinese counterparts. By mastering the counting system at an earlier age, Chinese children get a head start on solving mathematical calculations and problems.

Language differences also affect young children's understanding of place value, and this understanding has a bearing on the strategies children use to solve arithmetic problems. For example, a common strategy for simple addition and subtraction problems is "decomposition." In a problem such as 6 + 7, a child might decompose the 7 into 4 + 3, then solve the problem by first adding 6 + 4 to get 10, then adding the remaining 3. Young Korean and Chinese children often use this strategy. But skill in decomposition depends on a solid understanding of the base-l 0 system, which Asian languages support better than English does (Fuson & Kwon, 1992; Geary, Bow-Thomas, Liu, & Siegler, 1996; Miller et al., 1995). Language differences also influence how quickly people can pronounce number names, which in turn affects how quickly children memorize basic math facts (Geary, Bow-Thomas, Fan, & Siegler, 1993; Geary et al., 1996). Similar language effects on learning fractions have also been found (Paik & Mix, 2003). So these seemingly simple differences in number words can contribute to more long-lasting differences in the development of math skills (Beaton et al., 1996; Geary, Liu, Chen, Saults, & Hoard, 1999; Stevenson, Chen, & Lee, 1993).

At about 4 years of age, children combine their developing counting skills with their knowledge of addition and subtraction. At this point they begin to use counting as a tool to solve simple arithmetic problems instead of relying on subitizing. This represents an important advance in their mathematical skills because a child can use counting with sets of any size and in the absence of concrete objects, whereas subitizing works only with small sets and visible objects. Children quickly begin to use several different counting strategies, approaches to solving problems that involve counting of the quantities. For example: A child figures out that 2 + 2 = 4 by first counting to 2 and then counting on (two more steps) to 3 and then 4. Preschoolers learn to solve simple problems whether the change in number is visible to them, screened from their view, or even described verbally in the absence of any concrete objects. Gradually, over the later preschool and elementary school years, children increase the complexity and sophistication of their counting strategies (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Geary, 2006).

### Mathematical Skills during the Elementary School Years

Over the course of the elementary school years, the counting strategies children use to solve addition and subtraction problems gradually become more efficient. The table below shows some of these strategies. Younger children tend to use strategies that require more counting. This is cumbersome and increases the opportunity for errors, but it reduces the burden on their working memory. As children gain experience, they tend to move to strategies that require more memory but can be executed more quickly. Children begin to store basic math facts and simply retrieve them from memory instead of relying on counting strategies. Once a child has memorized a fact, he or she has a tool for deriving the answers to other, nonmemorized problems. For example, given the problem 5 + 8, a child who knows that 5 + 5 = 10 may use this known fact, reason that 8 is three larger than 5, and add 3 to 10 to obtain the answer 13 (Ashcraft, 1992; for reviews, see Geary, 1994, and Ginsburg, Klein, & Starkey, 1998).

How do children memorize and retrieve math facts? One possibility is the strategy choice model (Siegler & Jenkins, 1989). According to this model, "Children tend to choose the fastest approach that they can execute accurately" (Siegler, 1998, p. 286). Using a fast strategy increases the likelihood that the problem and answer will both be present in working memory, and therefore increases their degree of association. Using the most accurate strategy helps ensure that the association will develop between the problem and its correct answer. The more often a correct answer (e.g., 7) is associated with a problem (e.g.,4 + 3), the greater the association between them and the more likely it is that the child will retrieve this answer on future occasions. If the degree of association is not strong, the child will use a backup strategy such as counting, guessing, or deriving an answer based on known facts or rules. According to the strategy choice model, discouraging children from using backup strategies (like counting on their fingers) may actually delay their memorizing basic math facts (Siegler, 1998). Can you explain why?

Though children gradually move from less to more efficient strategies, they do not rely on only a single strategy for solving arithmetic problems. Instead, they consistently use a variety of strategies. The strategy choice model helps explain why. The availability of multiple strategies increases the likelihood that a child can obtain a correct answer quickly. This helps the child succeed in solving problems and helps build strong associations between problems and their correct answers. Researchers have found evidence of multiple strategies for a variety of arithmetic operations and across a wide range of student ages, abilities, and nationalities (Fuson & Kwon, 1992; Geary, 1996; LeFevre & Bisanz, 1996; Mabbott & Bisanz, 2003).

For better or worse, elementary school children spend many hours working mathematical word problems, or verbal descriptions of mathematical situations. As you may recall from personal experience, word problems can cause even the best math students to groan with dread. Several factors contribute to the difficulty of a word problem—among them the number of words in the problem, the number of required arithmetic operations, and the number of mathematical terms. The context of the problem has an important effect as well. Sometimes problems are difficult because their content is simply not interesting to the child, their wording is confusing, or their context is unfamiliar. In one study, for example, Brazilian children experienced at selling products at street stands were quite good at solving word problems that had a "selling" context, even when the items named in the problems were not ones they had sold. Their performance was much worse when problems did not have the sales context, even though the required computations were identical to those in the other problems (Carraher, Carraher, & Schliemann, 1985). Unfamiliar details and situations may not provide effective cues to help children access and use relevant knowledge, or they may simply overload working memory. With familiar contexts, children have a greater chance of comprehending the situation being described, understanding what they are being asked to figure out, and being motivated to solve the problem (Mayer, Lewis, & Hegarty, 1992; Stern, 1993; Vlahovic-Stetic, Rovan, & Mendek, 2004).