The words and symbols used to represent numbers may interfere with understanding of math concepts.
By Beth Azar, Monitor staff
Each time the government releases a new round of test scores, the United States laments the dismal performance of its children compared with children in other nations, particularly those from Asia.
And although differences in classroom instruction may be partly to blame, psychologists are finding that cultural differences in computational ability can begin before school and may have their roots in the words and symbols different cultures use to represent numbers.
For example, Asian children may get a head start in understanding that our number system is base 10 because their number words make that connection explicit whereas English does not. And fractions may pose a particular problem for all children in part because using the same numerals for fractions as for whole numbers may interfere with learning and in part because their brains are hard-wired to deal with whole numbers.
Classroom instruction may be able to address these inherent problems by explicitly teaching the concepts that children struggle with, says psychologist David Geary, PhD, of the University of Missouri, Columbia.
Words get in the way
For English-speaking children, number words may hamper learning before they even enter school: Studies by researchers including Kevin Miller, PhD, of the University of Illinois-Urbana- Champaign, consistently show that Asian children learn to count earlier and higher than their American counterparts and can do simple addition and subtraction sooner as well.
Researchers argue that differences in number words may be a major factor behind these differences. The culprit is the way English--as well as some other languages--treats numbers between 10 and 100. The teen numbers in these languages are irregular and difficult for children to learn and the rest of the count is separated into decades with words such as "twenty," "thirty" and "forty."
In most Asian languages the number words are far more consistent. In China, for example, the teen words are presented as 10 plus some ones: Eleven is simply "ten one," 12 is "ten two" and 13 is "ten three." This pattern continues into the decade numbers where 20 is "two ten," 30 is "three ten" and 45 is "four ten five." The language makes it obvious that the number, system is base 10.
This difference in language may partly explain why most Asian children learn by the mid-dle of first grade to subtract and add by thinking of numbers in as a 10 and some ones--an extremely helpful and efficient method of doing addition and subtraction, says Geary. In contrast, children in the United States, where much of the cross-cultural work has been conducted, rarely use such a method, even as they get older, Geary and other researcher find.
In fact, Chinese children who are good at counting at age 5 are already beginning to understand that teen words can be thought of as 10 plus some ones, find psychologists Karen Fuson, PhD and Connie Suk-Han Ho, PhD, of the Chinese University of Hong Kong. No children in the United States or England, regardless of their counting proficiency, understood this concept by age 5, they found in a series of studies published in the <I>Journal of Educational Psychology<P> (Vol. 90, No. 3, p. 536-544).
Children in the United States eventually learn that the number system is base 10 and that teens are tens plus ones, but only the most mathematically adept children ever learn to add by adding up to 10 and then adding the remaining ones (as with adding 7+8 by breaking 7 into 5 and 2, adding 2 to 8 to get 10 and then adding 5 for 15), says Fuson.
She has emphasized teaching about base 10 in a curriculum she's developed and is finding in preliminary evaluations that when taught this way children from poor inner-city schools districts quickly begin to outperform children from wealthier school districts. Countries that have similar language problems, but better math scores than the United States, may already use this kind of instruction.
Friction with fractions
Teachers may also need to work on the way they instruct children in fractions, which are notoriously difficult even for adults, say researchers.
Children may have trouble with fractions for several reasons, says Rochel Gelman, PhD, of the University of California, Los Angeles, whose research is geared toward understanding how children's notion of fractions develops. For one, because they learn to use numerals as whole numbers, it might confuse them to use the same symbols in a different way. It may also be that children have hard-wired mental structures that are designed to handle whole numbers and have trouble dealing with fractions.
A new study by University of Chicago psychologist Janellen Huttenlocher, PhD, and her colleagues supports the latter theory. They find that when numerals are removed from the equation and children are asked to calculate with fractions using nonverbal tasks, they do quite well.
In the study of 3- to 7-year-olds, instead of asking children to add numerical fractions, the researchers asked children to recreate fractional sums using wedge-shaped pieces of sponges that, when put together, formed a circle. As the children aged, they grew better at solving the nonverbal fraction problems. In fact, their skills improved in parallel, though at a slower rate, with their skills for manipulating whole numbers.
This finding indicates that children are able to manipulate fractions when they can form a mental model of the problem using real-world objects. They stumble only when they're asked to work with fractions represented as numerals, says Huttenlocher.
"With fractions the villain is putting 3's over 4's," says Huttenlocher, whose study is published in _Developmental Psychology_ (Vol. 35, No. 5, p. 164-174). "Children can mentally handle fractions, but the numbers get in their way--so with 4/5 and 4/8, children think 4/8 must be more because there's an 8 there."
Gelman agrees that interference is likely part of the problem. But she also believes--based on research in animals and humans--that there is an innate and hard-wired part of the brain that was designed through the course of evolution to handle whole numbers. This predisposition makes it easier for children to learn about whole numbers and hinders their learning of fractions, she argues.
"It's hard to think that we have trouble learning fractions just because we have no experience with numbers used this way," says Gelman. "That can't explain why the fraction problem extends well into college for some people."
Regardless of why children, and adults, have trouble with fractions, most researchers agree that teachers should introduce fractions in the context of real-world examples, including slices of a pie, pieces of an apple and portions of candy
* Geary, D. Reflections of evolution and culture in children's cognition: implications for mathematical development and instruction. American Psychologist, Vol. 50, No. 1, p. 24-37.