HARVARD GAZETTE ARCHIVES

Children are born with an ability to do arithmetic, whether they understand numbers or not, psychologist Elizabeth Spelke concludes. Such innate intuition could make arithmetic easier and more enjoyable when they have to learn it in school. (Staff file photo Rose Lincoln/Harvard News Office)

In these experiments, 5-year-olds, who had no real experience using number symbols, "added" two arrays of dots and compared them to a third array. When researchers replaced the third array of dots with beeps, the kids integrated the sight and sound quantities easily.

The children performed all these tasks successfully, without actual counting or having any knowledge of number symbols, notes Elizabeth Spelke, a professor of psychology who led the study.

Last year, Spelke and her colleagues reported on experiments with 5-month-old infants, which support the idea that thinking shapes language rather than the other way around. "Infants are born with a language-independent system for thinking about objects," Spelke concluded. "These intuitive concepts give meaning to the words they learn later." Her new findings suggest that the same can be said for numbers. Inborn intuition gives meaning to number symbols that kids learn later.

"This is a surprising finding, given that many school-age children have considerable difficulty learning symbolic arithmetic," Spelke comments. "Our results offer the promise that new strategies in elementary education may be devised: strategies that harness children's pre-existing arithmetic intuitions to foster the acquisition of knowledge about symbolic numbers and operations."

Spelke reports details of these arithmetic experiments in the latest issue of the Proceeding of the National Academy of Sciences, along with colleagues Hilary Barth, Kristen La Mont, and Jennifer Lipton.

Seeing and hearing 'many'

Other researchers, including Marc Hauser, a professor of psychology at Harvard, have shown that monkeys can count up to four. Monkeys also are aware that a picture with 30 dots displays a larger number than one with 20 dots. The same is true for human cultures that don't have a system of formal education. For example, Munduruku people, hunter-gatherers in Brazil, have words for "one" and "two," but none specifically for larger numbers. They can, however, compare and add numbers the way preschoolers did in the Harvard experiments.

"Rhesus monkeys as well as human adults and older children living in a remote Amazon village have been given comparison and addition tasks using arrays of dots, and they show the same abilities we find in 5-year-old Boston children," Spelke says.

Previously, she and others, including her colleague Susan Carey, demonstrated that infants come into the world with an inborn capability to count. For example, they know that 30 things are more than 20 things, even though they have no names for such numbers. To study this ability further, Spelke and her team decided to investigate the responses of preschoolers to both visual and audio inputs.

On the first test, the kids were shown some blue dots. After these were covered, they saw red dots. They were asked if there were more red or blue dots. The preschoolers had no trouble answering correctly even when the difference was only a matter of a few dots. On another test, the children had to visually add two arrays of blue dots and compare them with the number of red dots in a third array. This they did without problems.

Then sound was added. First, they compared numbers of dots to numbers of beeps. Then they added two arrays of dots and compared them to a sequence of beeps. To the researchers' amazement, the kids added and compared dots and beeps as easily as they did dots alone.

The last test was the trickiest. A researcher asked 33 5-year-olds questions like: "If your mom gave you 27 marshmallows, then she gave you 31 more, how many would you have?" The adult then asked: "would it be more like 58 or 33?"

If the youngsters actually had picked up some knowledge of number symbols, their correct answers should be higher than what you'd expect if they just guessed. In that case, such knowledge might account for their correct answers on the other dot and beep tests.

Their answers, however, showed that they came from intuition and not any knowledge of number symbols. "The children's poor performance... provides evidence that their ability to perform addition does not depend on knowledge of symbols," Spelke concludes.

Beeping-up math lessons

How can such findings be used to help the many children who experience trouble learning to add, subtract, multiply, and divide? Spelke and others are addressing that question now. She sees two possibilities.

First, youngsters who struggle with symbols for numbers might be encouraged and reassured if they discover that they can successfully play the kinds of games mastered by kids in the Harvard experiments. This play could show them that they already have the abilities they need to do the operations that their math teachers challenge them with.

Second, joining nonsymbolic play with symbolic arithmetic problems could help children master the symbol system. "Numerical symbols and operations," Spelke suspects, "may confuse children less if they are coupled with examples of sets of dots and being added, subtracted, multiplied, or divided, events they may already understand intuitively."

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