Mathematics (page 4)
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.
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).
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).
Examples of Common Counting Strategies for Addition
|Strategy||Sample Problem||Use of Strategy|
|Counting manipulatives||2 + 5||Child counts two manipulatives (e.g., blocks, coins, candies), then counts five more manipulatives, then counts all seven objects.|
|Counting fingers||2 + 5||Child raises two fingers on one hand, then five on the other hand. Child then counts all seven fingers one at a time, moving each one as it is counted.|
|Counting all (sum)||2 + 5||Child counts the first number aloud: "One, two." Child then continues counting on by the second number "three, four, five, six, seven"—and then states the answer as "seven."|
|Counting on first||2 + 5||Child states the first number, "two"; counts on by the second number—"three, four, five, six, seven"; then states the answer as "seven."|
|Counting on larger (minimum, or min)||2 + 5||Child states the biggest number, regardless of whether it is first or second. Here, child says "five," then counts on by the smaller number ("six, seven"), then states the answer as "seven."|
|Decomposition (deriving a fact)||2 + 5||Child decomposes the 6 into 3 + 3, adds 7 + 3 and gets 10, adds the other 3 to 10, and gives the answer: 13. (The child could choose to decompose the other number, 7, into 4 + 3, add 4 + 6 to get 10, then add the remaining 3 to 10.)|
|Fact retrieval||7 + 6||Child retrieves the answer from long-term memory and states the answer as "seven."|
|Regrouping||16 + 23||Child decomposes both numbers into tens and units (16 = 10 + 6; 23 = 20 + 3), sums the tens and units separately (10 + 20 = 30; 6 + 3 = 9), adds the subtotals (30 + 9 = 39), and gives the answer: 39.|
Source: Adapted from Geary (1994)
Mathematical Skills during Older Childhood and Adolescence
Children continue to increase their understanding of basic mathematical principles throughout their school years, although some principles take some time to develop (Geary, 2006). Sometimes misunderstandings about fundamental math principles lead to bugs, or systematic errors in children's problem-solving procedures. The table below presents examples of common bugs in multidigit subtraction (Brown & Burton, 1978; Brown & VanLehn, 1982). Buggy procedures are common in all arithmetic operations, and they can persist well into the late elementary school years.
As children learn about more complex math topics such as fractions, geometry, and algebra, they consistently attempt to build new understanding on what they already know. At times this leads to misunderstandings of new principles; but when students recognize their mistakes, they begin to debug their math knowledge and understand the accurate procedures. Instruction that builds on prior knowledge is typically more effective in helping students understand new mathematical concepts. In contrast, instruction that focuses on memorization of facts and rules gives students less opportunity to work out their buggy procedures. As a result, it can take longer for the child to gain correct understanding of a concept (Clement, 1982).
|Problems and Correct Answers||307 – 49 = 258||286 – 68 = 218||293 – 171 = 122|
|Bug: Borrow across zero
Child decrements first nonzero number correctly but does not alter the zero.
(combination of "borrow across zero" and "0 – N = N" bugs)
|Bug: Blank instead of borrow
When borrowing is needed, child skips that column and goes to the next.
|Bug: Smaller from larger
Child always subtracts the smaller number from the larger number.
|Bug: Smaller from larger instead of borrow from zero
Child does not borrow from zero but instead subtracts the smaller from the larger number.
|Bug: Borrow, no decrement
Child borrows but does not decrement the number being borrowed from.
|Bug: Borrow from zero
Child changes zero to a nine but does not decrement the next digit to the left.
|Bug: Zero minus N equals N
If top digit is a zero, child writes down the bottom number as the answer.
(combination of "borrow, no decrement" and "0 – N = N" bugs)
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