Children's Numerical Skills

　　People appear to be born to compute. The numerical skills of children

　　develop so early and so inexorably that it is easy to imagine an internal

　　clock of mathematical maturity guiding their growth. Not long after

　　learning to walk and talk, they can set the table with impressive accuracy

　　-- one knife, one spoon, one fork, for each of the five chairs. Soon they

　　are capable of noting that they have placed five knives, spoons and forks

　　on the table and, a bit later, that this amounts to fifteen pieces of

　　silverware. Having thus mastered addition, they move on to subtraction.

　　It seems almost reasonable to expect that if a child were secluded on a

　　desert island at birth and retrieved seven years later, he or she could

　　enter a second-grade mathematics class without any serious problems of

　　intellectual adjustment.

　　Of course, the truth is not so simple. This century, the work of cognitive

　　psychologists has illuminated the subtle forms of daily learning on which

　　intellectual progress depends. Children were observed as they slowly

　　grasped -- or, as the case might be, bumped into -- concepts that adults

　　take for granted, as they refused, for instance, to concede that quantity

　　is unchanged as water pours from a short stout glass into a tall thin one.

　　Psychologists have since demonstrated that young children, asked to count

　　the pencils in a pile, readily report the number of blue or red pencils,

　　but must be coaxed into finding the total. Such studies have suggested that

　　the rudiments of mathematics are mastered gradually, and with effort. They

　　have also suggested that the very concept of abstract numbers - the idea of

　　a oneness, a twoness, a threeness that applies to any class of objects and

　　is a prerequisite for doing anything more mathematically demanding than

　　setting a table - is itself far from innate.