people appear to born to compute¹. The numerical skills of children develop so early and so inexorably that it is easy to imagine an internal clock of mathematical maturity² guiding their growth. Not long after learning to walk and talk, they can set the table with impress accuracy---one knife, one spoon, one fork, for each of the five chairs. Soon they are capable of nothing 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 subtraction³. It seems almost reasonable to expect that if a child were secluded⁴ on a desert island at birth and retrieved⁵ seven years later, he or she could enter a second 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 cognitive⁶ psychologists has illuminated⁷ 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 quantity is unchanged as water pours from a short 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 coaxed⁸ into finding the total. Such studies have suggested that the rudiments⁹ 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