People appear to be born to compute. The digital skills of children 【M1】_________

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

clock of mathematical maturity guiding its growth. Not long after 【M2】_________

learning to walk and talk, they can make the table with impressive 【M3】_________

accuracy—one plate, 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 adds to 【M4】_________

fifteen pieces of silverware. Having thus mastered addition, they move

on to subtraction. It seems almost reasonable to expect that if a child

was secluded on a desert island at birth and retrieved seven years later, 【M5】_________

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. In this century, the work of

cognitive psychologists have illuminated the subtle forms of daily 【M6】_________

learning on which intellectual progress depends. Children observed as 【M7】_________

they slowly grasped—or, as the case might be, encountered—concepts

that adults take it for granted, and that they refused, for instance, to 【M8】_________

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, asking to count the pencils in a pile, readily report the 【M9】_________

number of blue or red pencils, but must be coaxed into finding a total. 【M10】________

Such studies have suggested that the basics of mathematics are mastered

gradually, and with effort.

【M1】