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Teaching mathematics 教數(shù)學(xué) Time for a ceasefire 暫時(shí)停火 Technology and fresh ideas are replacing classroom drill—and helping pupils to learn 科技和新鮮的想法正代替課堂訓(xùn)練，幫助學(xué)生們?nèi)W(xué)習(xí) Feb 1st 2014 | SHANGHAI AND TEL AVIV | From the print edition 譯者：kate2016 IF THE world’s education systems have a common focus, it is to turn out school-leavers who are proficient in mathematics. Governments are impressed by evidence from the World Bank and others that better maths results raises GDP and incomes. That, together with the soul-searching provoked by the cross-country PISA comparisons of 15-year-olds’ mathematical attainment produced by the OECD, a club of mostly rich countries, is prompting educators in many places to look afresh at what maths to teach, and how to teach it. 如果說(shuō)全球的教育系統(tǒng)都有一個(gè)共同焦點(diǎn)的話，那就是培養(yǎng)出精通數(shù)學(xué)的畢業(yè)生。世界銀行和其他的數(shù)據(jù)表明，數(shù)學(xué)更好可以提高GDP和收入，這給政府留下了深刻印象。國(guó)際學(xué)生評(píng)估項(xiàng)目（PISA）對(duì)多國(guó)15歲學(xué)生的數(shù)學(xué)素養(yǎng)進(jìn)行了比較，引人深思。這些因素促進(jìn)了很多地方的教育者重新思考數(shù)學(xué)教什么，怎么教數(shù)學(xué)。國(guó)際學(xué)生評(píng)估項(xiàng)目（PISA）由經(jīng)濟(jì)合作與發(fā)展組織（OECD）制作發(fā)起，該組織的成員國(guó)大多是富裕國(guó)家。 Those countries languishing in the league tables fret about how to catch up without turning students off the subject with boring drill. Top performers, most of them Asian (see chart), fear that their focus on technical proficiency does not translate into an enthusiasm for maths after leaving school. And everyone worries about how to prepare pupils for a jobs market that will reward creative thinking ever more highly. 在排名表上日漸衰弱的國(guó)家擔(dān)憂如何才能讓學(xué)生趕上其他國(guó)家，而又不需要用枯燥的訓(xùn)練讓學(xué)生對(duì)課程失去興趣。表現(xiàn)最優(yōu)者大多數(shù)是亞洲人（見(jiàn)圖表），他們擔(dān)心在離開(kāi)學(xué)校后，對(duì)技術(shù)熟練能力的關(guān)注并不能轉(zhuǎn)化為對(duì)數(shù)學(xué)的熱愛(ài)。而且人人都在擔(dān)心怎樣讓學(xué)生們?yōu)榫蜆I(yè)市場(chǎng)做準(zhǔn)備。就業(yè)市場(chǎng)獎(jiǎng)勵(lì)創(chuàng)造性思維的程度將會(huì)越來(lái)越大。 Maths education has been a battlefield before: the American “math wars” of the 1980s pitted traditionalists, who emphasised fluency in pen-and-paper calculations, against reformers led by the country’s biggest teaching lobby, who put real-world problem-solving, often with the help of calculators, at the centre of the curriculum. A backlash followed as parents and academics worried that the “new math” left pupils ill-prepared for university courses in mathematics and the sciences. But as many countries have since found, training pupils to ace exams is not the same as equipping them to use their hard-won knowledge in work and life. 數(shù)學(xué)教育曾是一片戰(zhàn)場(chǎng)：20世紀(jì)80年代美國(guó)的“數(shù)學(xué)之戰(zhàn)”使傳統(tǒng)主義者和改革者對(duì)立。改革派由全國(guó)最大的教學(xué)游說(shuō)團(tuán)體領(lǐng)導(dǎo)。傳統(tǒng)主義者強(qiáng)調(diào)紙筆計(jì)算的熟練。改革者主張課程的中心是常在計(jì)算器的幫助下，來(lái)解決實(shí)際問(wèn)題。強(qiáng)烈的抵制隨之而來(lái)，家長(zhǎng)和學(xué)者擔(dān)心這個(gè)“新數(shù)學(xué)”會(huì)讓學(xué)生在大學(xué)課程中的數(shù)學(xué)和其他科學(xué)準(zhǔn)備不足。但是正如很多國(guó)家已經(jīng)發(fā)現(xiàn)的那樣，訓(xùn)練學(xué)生考試考得好，并不等于訓(xùn)練他們?cè)诠ぷ骱蜕钪?，運(yùn)用來(lái)之不易的知識(shí)。 Today’s reformers think new technology renders this old argument redundant. They include Conrad Wolfram, who worked on Mathematica, a program which allows users to solve equations, visualise mathematical functions and much more. He argues that computers make rote procedures, such as long division, obsolete. “If it is high-level problem-solving and critical thinking we’re after, there’s not much in evidence in a lot of curriculums,” he says. 如今的改革者認(rèn)為新技術(shù)使得這個(gè)舊爭(zhēng)論變得多余。改革者包括康拉德·沃爾夫拉姆（Conrad Wolfram），他在Mathematica工作。Mathematica是一款數(shù)學(xué)軟件，能幫用戶解方程，使數(shù)學(xué)函數(shù)可視化以及其他功能。他認(rèn)為電腦使機(jī)械過(guò)程過(guò)時(shí)了， 比如長(zhǎng)除法。他說(shuō)：“如果我們追求的是高水準(zhǔn)的解決問(wèn)題和批判性思維，然而在大多數(shù)課程中關(guān)于這方面的證據(jù)真的不多?！?div> Estonia’s government has commissioned Mr Wolfram’s consultancy in Oxfordshire to modernise maths courses for secondary-school pupils. Starting this month, it will pilot lessons built around open-ended problems which have no single solution. One example: “What’s the best algorithm for picking a romantic date?” (Possible answer: go on more dates with a lower quality threshold to maximise the chance of success.) Another: “Am I drunk?”, which leads into quantitative analysis involving body masses, rates of alcohol absorption and other variables. 愛(ài)沙尼亞政府委任沃爾夫拉姆（Mr Wolfram）在牛津郡做顧問(wèn)，為中學(xué)生改革數(shù)學(xué)課程。本月開(kāi)始，它將會(huì)試開(kāi)課程。這些課程設(shè)立開(kāi)放式問(wèn)題，沒(méi)有唯一的答案。舉一個(gè)例子：“選到一個(gè)浪漫的約會(huì)對(duì)象的最佳算法是什么？”（可能的答案：降低門檻、增加約會(huì)次數(shù)，使成功幾率最大化。）另一個(gè)例子：“我喝醉了嗎？”，這就會(huì)引向定量分析包括身體質(zhì)量，酒精攝入率以及其他可變因素。 Some PISA stars are also seeking a new approach. Singapore’s government commissioned David Hogan, an Australian maths educator, to assess its syllabus. It wants pupils to be able to explain what they know to their classmates, and to apply it in unfamiliar situations. 一些國(guó)際學(xué)生評(píng)估項(xiàng)目（PISA）之星也在尋求新的方法。新加坡政府委任澳大利亞數(shù)學(xué)教育家大衛(wèi)·霍根（David Hogan）來(lái)評(píng)估他們的教學(xué)大綱。他們想要學(xué)生能夠向他們的同學(xué)解釋他們所知道的的東西，并且能在不熟悉的情況中運(yùn)用這些知識(shí)。 Israel is also experimenting, aware of the imminent end of the windfall provided by the arrival of many mathematicians from the former Soviet Union. Some Israeli maths lessons from primary school onwards will soon be taught using inexpensive tablets, an approach inspired by Shimon Schocken. An academic at Herzliya University, from 2005 he has created a large online group of “self-learners” who build computers and programs from scratch. Pupils are encouraged to visualise their calculations, not just to get the right answers. (Your correspondent, raised on algebra worksheets, found an iPad exercise that involved grouping sheep in holding pens embarrassingly hard.) 以色列的數(shù)學(xué)也在進(jìn)行改革實(shí)驗(yàn)。他們意識(shí)到，很多前蘇聯(lián)數(shù)學(xué)家?guī)?lái)的額外收益即將結(jié)束。一些以色列數(shù)學(xué)課程，從小學(xué)開(kāi)始將很快使用便宜的平板電腦教學(xué)。這個(gè)方法受到了希蒙· 肖肯（Shimon Schocken）的啟發(fā)。他是荷茲利亞大學(xué)的一位學(xué)者，2005年他創(chuàng)立了一個(gè)大型在線自學(xué)者組織，這些自學(xué)者從零開(kāi)始學(xué)習(xí)組建計(jì)算機(jī)和編寫程序。學(xué)生被鼓勵(lì)使他們的計(jì)算可視化，而不僅僅就是得到正確答案。（本文作者當(dāng)年靠做習(xí)題學(xué)代數(shù)，發(fā)現(xiàn)一個(gè)在臨時(shí)圈養(yǎng)圈分羊的iPad練習(xí)令人尷尬的難。） Andreas Schleicher, the statistician in charge of the PISA studies, agrees that many of the algorithms—rote procedures for solving familiar problems—that children have traditionally spent years mastering are now easy to “digitise, automate and outsource”. But he cautions against concluding that they are therefore no longer worth teaching. A third of pupils tested in Shanghai, which tops the PISA maths tables—and where pupils drill in mental arithmetic, start algebra young and learn formulae by heart—can apply their knowledge to novel and difficult problems, compared with 2-3% in America and Europe. 安德烈斯·施萊歇（Andreas Schleicher）是一位統(tǒng)計(jì)學(xué)家，負(fù)責(zé)國(guó)際學(xué)生評(píng)估項(xiàng)目（PISA）的研究。他認(rèn)同過(guò)去很多算法（解決相似問(wèn)題的機(jī)械過(guò)程）孩子傳統(tǒng)上要花數(shù)年掌握，現(xiàn)在這些算法很容易實(shí)現(xiàn)“數(shù)字化、自動(dòng)化和外包化”。但是他反對(duì)那種認(rèn)為這些算法不再值得教的結(jié)論。在上海被測(cè)試的學(xué)生中，有1/3的人能夠把他們的知識(shí)運(yùn)用到新穎、困難的問(wèn)題中。而在美國(guó)和歐洲這一比例為2-3%。上海在國(guó)際學(xué)生評(píng)估項(xiàng)目（PISA）數(shù)學(xué)排名第一。上海的學(xué)生訓(xùn)練掌握心算，年幼開(kāi)始學(xué)代數(shù)、背誦學(xué)習(xí)數(shù)學(xué)方程。 Youngsters, in China and elsewhere, need to have the grounding to assess whether a computer-generated answer is close to right before they start to rely on whizzy technology. Then they can happily abandon the drill in long division and quadratic equations. For the majority of maths moderates, the good news is that both sides in the fruitless trench warfare between progressives and traditionalists look ready for a ceasefire. 中國(guó)或者其他地方的青少年，在他們開(kāi)始依賴高新技術(shù)之前，需要基礎(chǔ)訓(xùn)練來(lái)評(píng)估一個(gè)計(jì)算機(jī)產(chǎn)生的答案是否接近正確。然后他們才會(huì)高興地?cái)[脫長(zhǎng)除法和二次方程的練習(xí)。對(duì)于大多的數(shù)學(xué)改良派來(lái)說(shuō)有個(gè)好消息，那就是數(shù)學(xué)改革者和傳統(tǒng)主義者之間毫無(wú)結(jié)果的壕溝戰(zhàn)，使得雙方考慮暫時(shí)?；?。 From the print edition: International
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