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.
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.
如此这般地掌握了加法之后,他们又转向减法。
It seems almost reasonable to expect that if a child were secluded on a desert island at birth and retrieved seven years later,
有一种设想几乎顺理成章,那就是,即使一个孩子一出生就被隔绝到荒岛上,七年后返回世间,
he or she could enter a second enter a second-grade mathematics class without any serious problems of intellectual adjustment.
也能直接上小学二年级的数学课,而不会碰到任何智力调整方面的大麻烦。
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.
当然,事实并没有这么简单。本世纪认知心理学家的工作已经揭示了智力发展所依赖的日常学习的微妙形式。
Children were observed as they slowly grasped-----
他们观察到孩子们缓慢掌握那些成年人认为理所当然的概念的过程,
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.
此类研究表明:数学基础是经过逐渐努力后掌握的。
They have also suggested that the very concept of abstract numbers
他们还表示抽象的数字概念,
------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