Campus & Community

# Children can perform approximate math without arithmetic instruction

## Study shows that children spontaneously show a sense of number when presented with symbolic math

Children are able to solve approximate addition or subtraction problems involving large numbers even before they have been taught arithmetic, according to a study conducted at Harvard University by researchers from the University of Nottingham and Harvard.

The study, published in the journal Nature, suggests that children do not need to master either the logic of place value or the addition table in order to perform approximate addition and subtraction. Children’s difficulty with learning school arithmetic may stem from the need to produce an exact number when solving problems. Elementary education in mathematics might be improved — and children’s interest in the subject enhanced — if children’s talent for approximate calculation could be built upon in the classroom, the authors suggest.

Researchers presented 5-year-old children with a series of illustrated problems, in the form of scenarios that involved the approximate addition and subtraction of symbolic numbers between five and 98. A subtraction question, for example, stated: “Sarah has 64 candies and gives 13 of them away, and John has 34 candies. Who has more?”

Even though the children had not yet been taught about symbolic arithmetic, and were yet to master the mechanics of symbolic addition and subtraction, they performed well above chance on the tests and without resorting to guessing. The children’s inability to provide an exact solution to the problems showed that their approximate performance was not dependent on precise knowledge of the numbers.

The authors — lead researcher Camilla Gilmore, now at the University of Nottingham, with Elizabeth Spelke, Marshall L. Berkman Professor of Psychology, and Shannon McCarthy, a research assistant in the department of psychology, both of the Faculty of Arts and Sciences at Harvard — found evidence for these abilities in children from a broad range of backgrounds, when studies were conducted in both a quiet “laboratory” setting and in the classroom.

The study also assessed whether children used their nonsymbolic number sense in order to perform the approximate addition and subtraction. Adults, children, and even infants are sensitive to number in arrays of dots and sequences of sounds. These number representations display characteristic limitations: arrays of dots can be numerically compared, added, or subtracted only approximately, subtraction is less precise than addition, and numerical comparison becomes more difficult when the ratio of the two numbers involved in the problem approaches one. The children involved in the study displayed these same characteristics with regard to the symbolic addition and subtraction problems.

“We’ve known for some time that adults, children, and even infants and nonhuman animals have a sense of number. We were surprised to see, however, that children spontaneously use their number sense when they’re presented with problems in symbolic arithmetic. These children haven’t begun to be taught place value or exact addition facts,” says Spelke. “Nevertheless, their natural sense of number gives them a way to think about arithmetic.”

The authors suggest their findings may be useful for the teaching of elementary mathematics.

Gilmore, who is a research fellow based in Nottingham’s Learning Sciences Research Institute (LSRI), says: “Exact symbolic arithmetic takes years to learn and poses difficulties for many children. For this reason, teachers were concerned that our problems would frustrate the children, and they were amazed at the children’s success and engagement. Our findings suggest new possible strategies for teaching primary mathematics and making it fun.”