English Original
People appear to be 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 impressive accuracy—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 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-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 granted, such as the principle that 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.
中文翻译
人类似乎天生就会计算。儿童的数字能力发展得如此之早、如此不可阻挡，以至于很容易想象有一个数学成熟度的内在时钟在指导他们的成长。学会走路和说话后不久，他们就能以令人印象深刻的准确度摆放餐具——为五把椅子中的每一把摆放一把刀、一个勺子、一把叉子。很快，他们就能意识到自己已经在桌上放了五把刀、五个勺子和五把叉子，再过一会儿，他们能算出这总共是十五件银器。就这样掌握了加法之后，他们便开始学习减法。人们似乎有理由期待，如果一个孩子出生时就被隔绝在荒岛上，七年后被找回，他或她也能直接进入二年级的数学课堂，在智力适应上不会遇到任何严重问题。
当然，事实并非如此简单。本世纪，认知心理学家的工作阐明了智力进步所依赖的那些微妙的日常学习形式。研究人员观察了儿童如何缓慢地掌握（或者，有时是偶然碰到）一些成年人认为理所当然的概念，例如水从一个矮杯子倒入一个细高杯子时，其总量保持不变的原则。
心理学家随后证明，当要求幼儿数一堆铅笔时，他们能轻松地说出蓝色或红色铅笔的数量，但必须经过引导才能得出总数。此类研究表明，数学的基础知识是逐步、通过努力掌握的。研究还表明，抽象数字这一概念本身——即适用于任何一类物体的“一性”、“二性”、“三性”的概念，它是进行比摆放餐桌更具数学挑战性任务的先决条件——远非与生俱来的。