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People appear to be born to compute. The numerical skills of childrendevelop so early and so inexorably that it is easy to imagine an internal clockof mathematical maturity guiding their growth. Not long after learning towalk and talk, they can set the table with impressive accuracy-- oneknife, one spoon, one fork, for each of the five chairs. Soon they arecapable of noting that they have placed five knives, spoons and forks on thetable and, a bit later, that this amounts to fifteen pieces ofsilverware. Having thus mastered addition, they move onto subtraction. It seems almost reasonable to expect that if achild were secluded on a desert island at birth and retrievedseven years later, he or she could enter a second-grade mathematics classwithout any serious problems of intellectual adjustment. Of course,the truth is not so simple. This century, the work of cognitive psychologistshas illuminated the subtle forms of daily learning on whichintellectual progress depends. Children were observed as theyslowly grasped -- or, as the case might be, bumped into -- concepts thatadults take for granted, as they refused, for instance, to concede that quantityis unchanged as water pours from a short stout glass into a tall thin one. Psychologistshave since demonstrated that young children, asked to count the pencils in apile, readily report the number of blue or red pencils, but must be coaxedinto finding the total. Such studies have suggested that the rudimentsof mathematics are mastered gradually, and with effort. They have alsosuggested that the very concept of abstract numbers - the idea of aoneness, a twoness, a threeness that applies to any class of objects andis a prerequisite for doing anything more mathematically demandingthan setting a table - is itself far from innate. | 
