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    Children's Numerical Skills

    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， as they refused， for instance， to 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， 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.

    全文翻譯：兒童的數(shù)學(xué)能力

    人似乎生來(lái)就會(huì)計(jì)算。  孩子們使用數(shù)字的技能發(fā)展得如此之早和如此必然，很容易讓 人想象有一個(gè)內(nèi)在的精確而成熟的數(shù)字鐘在指導(dǎo)他們的成長(zhǎng)。

    孩子們?cè)趯W(xué)會(huì)走路和說(shuō)話后 不久，就能以令人驚嘆的準(zhǔn)確布置桌子——五把椅子前面分別擺上一把刀、一個(gè)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(gè)湯匙、五把叉子。  沒(méi)有多 久，他們就又能知道這些東西加起來(lái)總共是 15 把銀餐具。

    如此這般地掌握了加法之后，他 們又轉(zhuǎn)向減法。  有一種設(shè)想幾乎順理成章，那就是，即使一個(gè)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(xué)二年級(jí)的數(shù)學(xué)課，而不會(huì)碰到任何智力調(diào)整方面的大 麻煩。當(dāng)然，事實(shí)并沒(méi)有這么簡(jiǎn)單。

    本世紀(jì)認(rèn)知心理學(xué)家的工作已經(jīng)揭示了智力發(fā)展所依 賴的日常學(xué)習(xí)的微妙形式。  他們觀察到孩子們緩慢掌握那些成年人認(rèn)為理所當(dāng)然的概念的 過(guò)程，或者是孩子們偶然遇到這些概念的過(guò)程。  他們也觀察到孩子們拒絕承認(rèn)某些常識(shí)的 情況。 比如： 孩子們拒絕承認(rèn)當(dāng)水從短而粗的瓶中倒入細(xì)而長(zhǎng)的瓶子中時(shí)，水的數(shù)量沒(méi)有 變化。  心理學(xué)家們而后又展示一個(gè)例子， 即：讓孩子們數(shù)一堆鉛筆時(shí)，他們能順利地報(bào)出 藍(lán)鉛筆或紅鉛筆的數(shù)目，但卻需誘導(dǎo)才能報(bào)出總的數(shù)目。  此類研究表明：數(shù)學(xué)基礎(chǔ)是經(jīng)過(guò) 逐漸努力后掌握的。

    他們還表示抽象的數(shù)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(shù)學(xué)要求的任何事時(shí)都必備的一、二、三意識(shí)，遠(yuǎn)遠(yuǎn)不是天生就具備的。