The New Math: Kids Can Add and Subtract without Arithmetic

Knowing how to count lets kindergartners do arithmetic before they learn its rules

ALL IN YOUR HEAD: Five- and six-year-olds who know how to count can solve simple addition and subtraction problems without knowing arithmetic. Image: © ISTOCKPHOTO/DON BAYLEY

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Young children can crudely add and subtract numbers before they have learned the rules of arithmetic, a new study finds. Researchers presented five- and six-year-old children with problems such as, "Sarah has 15 candies and she gets 19 more; John has 51 candies. Who has more?" To answer correctly, researchers say, the children must be harnessing an intuitive sense of how large different numbers are, which could help ease the pain of learning arithmetic.

Prior studies have shown that children as young as infants can judge simple mathematical relationships. When shown bunches of dots on a computer screen, for example, a preschool- or kindergarten-age child can tell that there are more dots combined in an image of 21 dots followed by one of 30 dots compared with a third 34-dot image.

A group including cognitive psychologist Elizabeth Spelke of Harvard University wondered if children could apply that ability, called nonsymbolic arithmetic, to Arabic numerals after learning to count but before they learned to add and subtract.

To find out, they gave several groups of children a laptop-based audiovisual test that asked whether one person had more or fewer candies or other objects than another person. The screen showed numbers to be added, such as 21 and 30, or subtracted, such as 64 and 13, followed by another number, such as 34, with which to compare the added or subtracted value.

The children answered correctly from 64 to 73 percent of the time, according to a report published online today by Nature. More affluent kids tested in the laboratory did better than their less well off peers tested in their classrooms, the group reports. The reason for the difference could be the testing environment, says Spelke, who adds that the important point is that kids from diverse backgrounds all showed the ability. "We never dreamed that you could simply give children the symbols and they will succeed," she says.

Consistent with the use of nonsymbolic arithmetic, their answers were more accurate the greater the difference was between the sum (or difference) of the first two numbers and the comparison number. Kids also did better when adding than when subtracting. (Subtraction requires a larger number than addition to arrive at the same comparison, and larger numbers have fuzzier sizes in nonsymbolic arithmetic, Spelke says.)

Spelke says the link between number symbols and nonsymbolic arithmetic might help kids learn math more easily. Teachers were skeptical of the experiment because arithmetic lessons easily frustrate children, but "the kids really loved these problems," she says. "It looks to us like a big part of the logic of addition and subtraction is already available to them."

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1. 1. owen 01:15 PM 8/22/08

In 1966 I used dots on top of the numbers to add and subtract and have found that many people instintively count dots . There are even some programs for sale that are a copy of my first version with dots on numbers one to five. I have now refined my "Dotmath for kids" to a fine art by fixing any errors that I found or that people have told me about. The worst problem was that the children got stuck on the dots and could not see the symbol without the dots.

In my first versions the dots and number combine to make a new symbol that look different than the universal standard number symbol, the children always wanted to put the dots on top of every number every time because they thought that is what a number looked like.

They saw the dot and number as one symbol but counted on the dots and ignored the universal symbol. They always wanted to count by ones and add by dots and not by groups. They learned the pattern of putting dots on all the ends of the number symbols 1,2,3 then got confused about how many dots to put on four. They wanted to put a dot on every end and intersection of the four. This would be six dots not four. They would get addicted to counting the dots and not want to translate back to the universal number symbols ( counting by ones slows them down). This is why I do not recommend the use of my old versions (1966-94) because it had dots on top of the number symbol and I found this to confuse the children.

It took me many years to recover from putting the dots on top of the number symbol and I knew that the dots and number were not one symbol. Children that do not understand that the dots and number are not one thing will not likely recover without years of expert help. There is no way for me to find everyone exposed to my first version so I have made the charts on the "DotMath for kids" web site free for personal and education use.

If the University or others are interested in this information they can find it with any seach box . Type "DotMath for kids" in any search engine and they will be able to find out more about using dots to understand math.

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