| 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 | |
| 人似乎生来就会计算。孩子们使用数字的技能发展得如此之早和如此必然，很容易让人想象有一个内在的精确而成熟的数字钟在指导他们的成长。 孩子们在学会走路和说话后不久，就能以令人惊叹的准确布置桌子--五把椅子前面分别摆上一把刀、一个汤匙、一把叉子。很快地，他们就能知道他们已在桌面上 摆放了五把刀、五个汤匙、五把叉子。没有多久，他们就又能知道这些东西加起来总共是15把银餐具。如此这般地掌握了加法之后，他们又转向减法。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，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。 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。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，或者是孩子们偶然遇到这些概念的过程。他们也观察到孩子们拒绝承认某些常识的情况。比如：孩子们拒绝承认当水从短而粗的瓶中倒入细而长的瓶子中时， 水的数量没有变化。心理学家们而后又展示一个例子，即：让孩子们数一堆铅笔时，他们能顺利地报出蓝铅笔或红铅笔的数目，但却需诱导才能报出总的数目。此类 研究表明：数学基础是经过逐渐努力后掌握的。他们还表示抽象的数字概念，如可表示任何一类物品并且是在做比摆桌子有更高数学要求的任何事时都必备的一、 二、三意识，远远不是天生就具备的。 |
| 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 accuracyone 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 secondgrade 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 graspedor, as the case might be, bumped intoconcepts 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 numbersthe 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 tableis itself far from innate | |
| ren si hu sheng lai jiu kuai ji suan. hai zi men shi yong shu zi de ji neng fa zhan de ru ci zhi zao he ru ci bi ran, hen rong yi rang ren xiang xiang you yi ge nei zai de jing que er cheng shu de shu zi zhong zai zhi dao ta men de cheng zhang. hai zi men zai xue hui zou lu he shuo hua hou bu jiu, jiu neng yi ling ren jing tan de zhun que bu zhi zhuo zi wu ba yi zi qian mian fen bie bai shang yi ba dao yi ge tang chi yi ba cha zi. hen kuai di, ta men jiu neng zhi dao ta men yi zai zhuo mian shang bai fang le wu ba dao wu ge tang chi wu ba cha zi. mei you duo jiu, ta men jiu you neng zhi dao zhei xie dong xi jia qi lai zong gong shi 15 ba yin can ju. ru ci zhe ban di zhang wo le jia fa zhi hou, ta men you zhuan xiang jian fa. you yi zhong she xiang ji hu shun li cheng zhang, na jiu shi, ji shi yi ge hai zi yi chu sheng jiu bei ge jue dao huang dao shang, qi nian hou fan hui shi jian, ye neng zhi jie shang xiao xue er nian ji de shu xue ke, er bu hui peng dao ren he zhi li tiao zheng fang mian di da ma fan. dang ran, shi shi bing mei you zhe me jian dan. ben shi ji ren zhi xin li xue jia de gong zuo yi jing jie shi le zhi li fa zhan suo yi lai de ri chang xue xi de wei miao xing shi. ta men guan cha dao hai zi men huan man zhang wo nei xie cheng nian ren ren wei li suo dang ran de gai nian de guo cheng, huo zhe shi hai zi men ou ran yu dao zhei xie gai nian de guo cheng. ta men ye guan cha dao hai zi men ju jue cheng ren mou xie chang shi de qing kuang. bi ru: hai zi men ju jue cheng ren dang shui cong duan er cu de ping zhong dao ru xi er zhang de ping zi zhong shi, shui de shu liang mei you bian hua. xin li xue jia men er hou you zhan shi yi ge li zi, ji: rang hai zi men shu yi dui qian bi shi, ta men neng shun li di bao chu lan qian bi huo hong qian bi de shu mu, dan que xu you dao cai neng bao chu zong de shu mu. ci lei yan jiu biao ming: shu xue ji chu shi jing guo zhu jian nu li hou zhang wo de. ta men hai biao shi chou xiang de shu zi gai nian, ru ke biao shi ren he yi lei wu pin bing qie shi zai zuo bi bai zhuo zi you geng gao shu xue yao qiu de ren he shi shi dou bi bei de yi er san yi shi, yuan yuan bu shi tian sheng jiu ju bei de. |
| 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 accuracyone 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 secondgrade 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 graspedor, as the case might be, bumped intoconcepts 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 numbersthe 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 tableis itself far from innate | |
| rén sì hū shēng lái jiù kuài jì suàn. hái zi men shǐ yòng shù zì de jì néng fā zhǎn dé rú cǐ zhī zǎo hé rú cǐ bì rán, hěn róng yì ràng rén xiǎng xiàng yǒu yí gè nèi zài de jīng què ér chéng shú de shù zì zhōng zài zhǐ dǎo tā men de chéng zhǎng. hái zi men zài xué huì zǒu lù hé shuō huà hòu bù jiǔ, jiù néng yǐ lìng rén jīng tàn de zhǔn què bù zhì zhuō zi wǔ bǎ yǐ zi qián miàn fēn bié bǎi shàng yī bǎ dāo yí gè tāng chí yī bǎ chā zǐ. hěn kuài dì, tā men jiù néng zhī dào tā men yǐ zài zhuō miàn shàng bǎi fàng le wǔ bǎ dāo wǔ gè tāng chí wǔ bǎ chā zǐ. méi yǒu duō jiǔ, tā men jiù yòu néng zhī dào zhèi xiē dōng xī jiā qǐ lái zǒng gòng shì 15 bǎ yín cān jù. rú cǐ zhè bān dì zhǎng wò le jiā fǎ zhī hòu, tā men yòu zhuǎn xiàng jiǎn fǎ. yǒu yī zhǒng shè xiǎng jī hū shùn lǐ chéng zhāng, nà jiù shì, jí shǐ yí gè hái zi yī chū shēng jiù bèi gé jué dào huāng dǎo shàng, qī nián hòu fǎn huí shì jiān, yě néng zhí jiē shàng xiǎo xué èr nián jí de shù xué kè, ér bú huì pèng dào rèn hé zhì lì tiáo zhěng fāng miàn dí dà má fán. dāng rán, shì shí bìng méi yǒu zhè me jiǎn dān. běn shì jì rèn zhī xīn lǐ xué jiā de gōng zuò yǐ jīng jiē shì le zhì lì fā zhǎn suǒ yī lài de rì cháng xué xí de wēi miào xíng shì. tā men guān chá dào hái zi men huǎn màn zhǎng wò nèi xiē chéng nián rén rèn wéi lǐ suǒ dāng rán de gài niàn de guò chéng, huò zhě shì hái zi men ǒu rán yù dào zhèi xiē gài niàn de guò chéng. tā men yě guān chá dào hái zi men jù jué chéng rèn mǒu xiē cháng shí de qíng kuàng. bǐ rú: hái zi men jù jué chéng rèn dāng shuǐ cóng duǎn ér cū de píng zhōng dào rù xì ér zhǎng de píng zi zhōng shí, shuǐ de shù liàng méi yǒu biàn huà. xīn lǐ xué jiā men ér hòu yòu zhǎn shì yí gè lì zi, jí: ràng hái zi men shù yī duī qiān bǐ shí, tā men néng shùn lì dì bào chū lán qiān bǐ huò hóng qiān bǐ de shù mù, dàn què xū yòu dǎo cái néng bào chū zǒng de shù mù. cǐ lèi yán jiū biǎo míng: shù xué jī chǔ shì jīng guò zhú jiàn nǔ lì hòu zhǎng wò de. tā men hái biǎo shì chōu xiàng de shù zì gài niàn, rú kě biǎo shì rèn hé yī lèi wù pǐn bìng qiě shì zài zuò bǐ bǎi zhuō zi yǒu gèng gāo shù xué yāo qiú de rèn hé shì shí dōu bì bèi de yī èr sān yì shí, yuǎn yuǎn bú shì tiān shēng jiù jù bèi de. |
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