From learning basic numbers to memorizing multiplication tables to solving algebraic equations, children progress through an amazing range of mathematical abilities as they grow. In this section we will explore how mathematical reasoning develops through childhood.
In the first months after birth, infants can already distinguish among small numbers of objects (e.g., among one, two, and three objects), whether the objects are similar or different, moving or still, or presented at the same time or in sequence. They can even match the number of objects they see with the number of sounds they hear. For example, when infants hear a sound track of two drumbeats, they prefer to look at a photo of two household objects rather than a photo of three objects. When they hear three drumbeats, however, their preference switches to the photo of three objects (Starkey, Spelke, & Gelman, 1983,1990). Impressive as these skills are, however, they apply only to very small sets. If researchers increase the number of objects in each set to five or more, then children don't show evidence that they recognize the quantities until they are about 3 or 4 years old (Canfield & Smith, 1996; Simon, Hespos, & Rochat, 1995; Starkey & Cooper, 1980; Strauss & Curtis, 1981; van Loosbroek & Smitsman, 1990; Wynn, 1992, 1995).
How are infants able to show such abilities? They clearly cannot count objects. They have no experience with a number system, and they don't have the language skills they would need to say the words that go with the numbers. Researchers propose that infants enumerate small sets by subitizing, a perceptual process that we all use to quickly and easily determine the basic quantity in a small set of objects. To see how subitizing works, try the following experiment. Have a friend toss three or four pennies onto a table while you have your eyes closed. Now open your eyes and, as quickly as you can, look to see how many pennies there are. Most people can "see" that there are three pennies, or four pennies, without needing to actually count each penny. There is something about the visual arrangement of the pennies that lets you know immediately how many there are. Of course we can't be sure that infants are subitizing object sets exactly the way adults do, but from the experimental evidence it does seem that they use a similar process. Somehow, without actually counting, they can determine that one set of objects has more items than another set, and they can match the number of things they see with the number of sounds they hear. Quite remarkable math skills for such a young age!
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 in Infancy
Amazing but true: Researchers have found evidence that even newborns have rudimentary mathematical skills. Humans seem to be "born with a fundamental sense of quantity" (Geary, 1994, p. 1). For example, researchers showed newborn infants less than one week old!) a card with two black dots (Antell & Keating, 1983). The newborns looked at the dots for a bit, then started looking away. Looking away signals boredom. When researchers then switched to a card that had three dots, the newborns regained interest in looking at the dots—they dishabituated. Newborns also showed dishabituation when the set size was changed from three dots to two. These patterns of habituation and dishabituation show that newborns can see the difference between two and three dots.In the first months after birth, infants can already distinguish among small numbers of objects (e.g., among one, two, and three objects), whether the objects are similar or different, moving or still, or presented at the same time or in sequence. They can even match the number of objects they see with the number of sounds they hear. For example, when infants hear a sound track of two drumbeats, they prefer to look at a photo of two household objects rather than a photo of three objects. When they hear three drumbeats, however, their preference switches to the photo of three objects (Starkey, Spelke, & Gelman, 1983,1990). Impressive as these skills are, however, they apply only to very small sets. If researchers increase the number of objects in each set to five or more, then children don't show evidence that they recognize the quantities until they are about 3 or 4 years old (Canfield & Smith, 1996; Simon, Hespos, & Rochat, 1995; Starkey & Cooper, 1980; Strauss & Curtis, 1981; van Loosbroek & Smitsman, 1990; Wynn, 1992, 1995).
How are infants able to show such abilities? They clearly cannot count objects. They have no experience with a number system, and they don't have the language skills they would need to say the words that go with the numbers. Researchers propose that infants enumerate small sets by subitizing, a perceptual process that we all use to quickly and easily determine the basic quantity in a small set of objects. To see how subitizing works, try the following experiment. Have a friend toss three or four pennies onto a table while you have your eyes closed. Now open your eyes and, as quickly as you can, look to see how many pennies there are. Most people can "see" that there are three pennies, or four pennies, without needing to actually count each penny. There is something about the visual arrangement of the pennies that lets you know immediately how many there are. Of course we can't be sure that infants are subitizing object sets exactly the way adults do, but from the experimental evidence it does seem that they use a similar process. Somehow, without actually counting, they can determine that one set of objects has more items than another set, and they can match the number of things they see with the number of sounds they hear. Quite remarkable math skills for such a young age!
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).
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