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Teaching mathematics 教数学 Time for a ceasefire 暂时停火 Technology and fresh ideas are replacing classroom drill—and helping pupils to learn 科技和新鲜的想法正代替课堂训练，帮助学生们去学习 Feb 1st 2014 | SHANGHAI AND TEL AVIV | From the print edition 译者：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. 如果说全球的教育系统都有一个共同焦点的话，那就是培养出精通数学的毕业生。世界银行和其他的数据表明，数学更好可以提高GDP和收入，这给政府留下了深刻印象。国际学生评估项目（PISA）对多国15岁学生的数学素养进行了比较，引人深思。这些因素促进了很多地方的教育者重新思考数学教什么，怎么教数学。国际学生评估项目（PISA）由经济合作与发展组织（OECD）制作发起，该组织的成员国大多是富裕国家。 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. 在排名表上日渐衰弱的国家担忧如何才能让学生赶上其他国家，而又不需要用枯燥的训练让学生对课程失去兴趣。表现最优者大多数是亚洲人（见图表），他们担心在离开学校后，对技术熟练能力的关注并不能转化为对数学的热爱。而且人人都在担心怎样让学生们为就业市场做准备。就业市场奖励创造性思维的程度将会越来越大。 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. 数学教育曾是一片战场：20世纪80年代美国的“数学之战”使传统主义者和改革者对立。改革派由全国最大的教学游说团体领导。传统主义者强调纸笔计算的熟练。改革者主张课程的中心是常在计算器的帮助下，来解决实际问题。强烈的抵制随之而来，家长和学者担心这个“新数学”会让学生在大学课程中的数学和其他科学准备不足。但是正如很多国家已经发现的那样，训练学生考试考得好，并不等于训练他们在工作和生活中，运用来之不易的知识。 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. 如今的改革者认为新技术使得这个旧争论变得多余。改革者包括康拉德·沃尔夫拉姆（Conrad Wolfram），他在Mathematica工作。Mathematica是一款数学软件，能帮用户解方程，使数学函数可视化以及其他功能。他认为电脑使机械过程过时了， 比如长除法。他说：“如果我们追求的是高水准的解决问题和批判性思维，然而在大多数课程中关于这方面的证据真的不多。” 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. 爱沙尼亚政府委任沃尔夫拉姆（Mr Wolfram）在牛津郡做顾问，为中学生改革数学课程。本月开始，它将会试开课程。这些课程设立开放式问题，没有唯一的答案。举一个例子：“选到一个浪漫的约会对象的最佳算法是什么？”（可能的答案：降低门槛、增加约会次数，使成功几率最大化。）另一个例子：“我喝醉了吗？”，这就会引向定量分析包括身体质量，酒精摄入率以及其他可变因素。 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. 一些国际学生评估项目（PISA）之星也在寻求新的方法。新加坡政府委任澳大利亚数学教育家大卫·霍根（David Hogan）来评估他们的教学大纲。他们想要学生能够向他们的同学解释他们所知道的的东西，并且能在不熟悉的情况中运用这些知识。 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.) 以色列的数学也在进行改革实验。他们意识到，很多前苏联数学家带来的额外收益即将结束。一些以色列数学课程，从小学开始将很快使用便宜的平板电脑教学。这个方法受到了希蒙· 肖肯（Shimon Schocken）的启发。他是荷兹利亚大学的一位学者，2005年他创立了一个大型在线自学者组织，这些自学者从零开始学习组建计算机和编写程序。学生被鼓励使他们的计算可视化，而不仅仅就是得到正确答案。（本文作者当年靠做习题学代数，发现一个在临时圈养圈分羊的iPad练习令人尴尬的难。） 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·施莱歇（Andreas Schleicher）是一位统计学家，负责国际学生评估项目（PISA）的研究。他认同过去很多算法（解决相似问题的机械过程）孩子传统上要花数年掌握，现在这些算法很容易实现“数字化、自动化和外包化”。但是他反对那种认为这些算法不再值得教的结论。在上海被测试的学生中，有1/3的人能够把他们的知识运用到新颖、困难的问题中。而在美国和欧洲这一比例为2-3%。上海在国际学生评估项目（PISA）数学排名第一。上海的学生训练掌握心算，年幼开始学代数、背诵学习数学方程。 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. 中国或者其他地方的青少年，在他们开始依赖高新技术之前，需要基础训练来评估一个计算机产生的答案是否接近正确。然后他们才会高兴地摆脱长除法和二次方程的练习。对于大多的数学改良派来说有个好消息，那就是数学改革者和传统主义者之间毫无结果的壕沟战，使得双方考虑暂时停火。 From the print edition: International
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