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

兒童的數學能力

人似乎生來就會計算。孩子們使用數字的技能發展得如此之早和如此必然，很容易讓人想象有一個內在的精確而成熟的數字鐘在指導他們的成長。

孩子們在學會走路和說話后不久，就能以令人驚嘆的準確布置桌子--五把椅子前面分別擺上一把刀、一個湯匙、一把叉子。很快地，他們就能知道他們已在桌面上擺放了五把刀、五個湯匙、五把叉子。沒有多久，他們就又能知道這些東西加起來總共是15把銀餐具。

如此這般地掌握了加法之后，他們又轉向減法。有一種設想幾乎順理成章，那就是，即使一個孩子一出生就被隔絕到荒島上，七年后返回世間，也能直接上小學二年級的數學課，而不會碰到任何智力調整方面的大麻煩。當然，事實并沒有這么簡單。

本世紀認知心理學家的工作已經揭示了智力發展所依賴的日常學習的微妙形式。他們觀察到孩子們緩慢掌握那些成年人認為理所當然的概念的過程，或者是孩子們偶然遇到這些概念的過程。他們也觀察到孩子們拒絕承認某些常識的情況。比如：

孩子們拒絕承認當水從短而粗的瓶中倒入細而長的瓶子中時，水的數量沒有變化。心理學家們而后又展示一個例子，即：讓孩子們數一堆鉛筆時，他們能順利地報出藍鉛筆或紅鉛筆的數目，但卻需誘導才能報出總的數目。此類研究表明：數學基礎是經過逐漸努力后掌握的。

他們還表示抽象的數字概念，如可表示任何一類物品并且是在做比擺桌子有更高數學要求的任何事時都必備的一、二、三意識，遠遠不是天生就具備的。