19.children’s numerical skills

　　people appear to 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 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 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 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 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 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

　　20 The Historical Significance of American Revolution

　　The ways of history are so intricate and the motivations of human actions so complex that it is always hazardous to attempt to represent events covering a number of years, a multiplicity of persons, and distant localities as the expression of one intellectual or social movement; yet the historical process which culminated in the ascent of Thomas Jefferson to the presidency can be regarded as the outstanding example not only of the birth of a new way of life but of nationalism as a new way of life. The American Revolution represents the link between the seventeenth century, in which modern England became conscious of itself, and the awakening of modern Europe at the end of the eighteenth century. It may seem strange that the march of history should have had to cross the Atlantic Ocean, but only in the North American colonies could a struggle for civic liberty lead also to the foundation of a new nation. Here, in the popular rising against a “tyrannical” government, the fruits were more than the securing of a freer constitution. They included the growth of a nation born in liberty by the will of the people, not from the roots of common descent, a geographic entity, or the ambitions of king or dynasty. With the American nation, for the first time, a nation was born, not in the dim past of history but before the eyes of the whole world.