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Adults frequently make estimates--how much time it takes to get from point A to point B, how much a cart of groceries will cost--but children are notoriously inaccurate at such tasks, said Robert Siegler, PhD, at APA's 2005 Annual Convention. Sometimes their wild guesses stem from a lack of factual knowledge, but research presented by Siegler, a psychology professor at Carnegie Mellon University in Pittsburgh, suggests that children's basic conceptions of numbers may also contribute to their inaccurate estimations.

"We used tasks of pure numerical estimation, where you don't need real-world knowledge but you do need to know numbers," he said.

On one such task most kindergarten-aged children place numbers on an unmarked 0-100 number line in a logarithmic pattern, with lower numbers spaced further apart than the higher ones, according to research published by Siegler and Carnegie Mellon postdoc Julie Booth, PhD, in Child Development (Vol. 75, No. 2, pages 428-444). By second grade, most children space the numbers on the 0-100 line evenly, Siegler reported. However, when guessing the location of numbers on the 0-1,000 number line, the second-grade students make the same mistake as the kindergarten students--bunching together the larger numbers at the high end of the scale. Not until fourth grade can students accurately estimate the position of numbers on the 0-1,000 number lines, Siegler said.

While most children eventually develop a linear number line, the speed at which they do may affect their overall math achievement, said Siegler.

"We think this numerical representation skill is a key part of math learning in general," he noted.

In fact, Siegler has found math achievement correlates with children's ability to correctly space numbers on number lines. Moreover, it may be possible to bolster student's math achievement by helping them to acquire a linear conception of numbers, he said. In an as-yet-unpublished study, Siegler and other researchers accelerated children's development of mental number lines by playing a simple board game.

Developing number theories

Previous theories have suggested that children have either logarithmic or linear conceptions of number lines, but Siegler's studies show that they can have both at the same time.

"Children use multiple representations of numerical quantity," said Siegler.

Siegler and University of Michigan psychology professor John Opfer, PhD, used a simple procedure to test this idea. They presented 32 second-graders with number lines of identical length, labeled 0 at the left end and either 100 or 1,000 on the right. They then asked each child to estimate the position of a number--such as 71--by making a mark on the line.

Ninety-one percent of the second-graders generated a logarithmic pattern of estimates on the 0-1,000 line, placing numbers such as 71 much too high--often as much as halfway up the number line. On the 0-100 number line, only 44 percent of the same children generated logarithmic patterns of estimates, according to the study, which was published in Psychological Science (Vol. 14, No. 3, pages 237-243). Moreover, those children who were able to correctly estimate numbers' positions tended to score higher on the math portion of the Stanford Achievement Test, Ninth Edition.

In a second experiment, currently in press, Siegler and Booth tested children's ability to make other kinds of numerical estimates. For example, the children guessed at the number of dots on a screen and the length of a line. Again, younger children tended to make errors consistent with a logarithmic representation of numbers, and older children tended to show a linear grasp of numbers. Additionally, participants who tended to answer correctly on the two tasks later scored higher on standardized math tests than those who showed difficulty making estimates, Siegler said.

Math games

Because accurate numerical representation appears key to math achievement, Siegler and Carnegie Mellon postdoc Geetha Ramani, PhD, set out to teach young children to better estimate numbers' positions on number lines. In an as-yet-unpublished study, the researchers targeted 40 4- and 5-year-olds at a Head Start program.

When estimating the position of numbers on a 1-10 number line, children in Head Start programs tend to make many errors, unevenly spacing numbers and even putting them in the wrong order. Children from middle-income families tend to do better, he said.

"Before [low-income] children even enter school, they are behind on their numerical representations," Siegler said.

The researchers attempted to bridge this gap by helping the children to play a simple board game called The Great Race. Players progress across a board with 10 numbered spaces by flicking a spinner and moving their marker the allotted number of spaces. The players count out loud while moving their pieces, and the first person to reach 10 wins.

The researchers played the game with half of the children for four 20-minute sessions over a two-week period. The other children played a variation of The Great Race in which color, rather than number, distinguishes the spaces on the board.

After playing the number version of game, 61 percent of the students' answers fit into an accurate representation of number lines, while only 15 percent of the control group's answers did.

Siegler's future research will investigate whether games such as The Great Race can improve math achievement in addition to number-line estimation, and measure how long such improvements last, he said.