People appear to born to compute1. The numerical skills of children develop so early and so inexorably（冷漠地，无情地） that it is easy to imagine an internal clock of mathematical maturity2（成熟，完备） 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（银器） . Having thus mastered addition, they move on to subtraction3（降低，差集） . It seems almost reasonable to expect that if a child were secluded4（隔绝） on a desert island at birth and retrieved5（取回，寻回） 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（认知的，认识的） psychologists has illuminated7（照明，启发） the subtle（微妙的，敏锐的） 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（哄骗，劝诱） into finding the total. Such studies have suggested that the rudiments9（基础知识，初步） 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（二重性） , a threeness that applies to any class of objects and is a prerequisite10（先决条件） for doing anything more mathematically demanding than setting a table -- is itself far from innate11.
