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ConradWolfram_2010G-_用计算机教导孩子真正的数学_
We've got a real problem with math education right now. 目前我们的数学教育面临着实际的问题。
Basically , no one's very happy. 基本上,没有人感到很满意。
Basically:adv.主要地,基本上;
Those learning it think it's disconnected , uninteresting and hard. 那些学习数学的学生 觉得所学的知识与实际无关, 无趣以及困难重重。
disconnected:adj.分离的; v.切断(煤气、水或电的供应); (disconnect的过去式和过去分词) uninteresting:adj.无趣味的,乏味的;令人厌倦的;
Those trying to employ them think they don't know enough. 那些运用数学的人们 又觉得他们所学的知识不够。
employ:vt.使用,采用;雇用;使忙于,使从事于;n.使用;雇用;
Governments realize that it's a big deal for our economies , but don't know how to fix it. 政府觉察到这是一个涉及到我们经济发展的大问题, 但他们却无从下手。
a big deal:na.要人;重要的事; economies:n.经济;经济结构;节约;(economy的复数)
And teachers are also frustrated . 而老师们也感到很沮丧。
frustrated:adj.失意的,挫败的;泄气的;v.挫败;阻挠;(frustrate的过去式和过去分词)
Yet math is more important to the world than at any point in human history. 但是现在的数学 比人类历史上任何时间都更为重要。
So at one end we've got falling interest in education in math, and at the other end we've got a more mathematical world, a more quantitative world than we ever have had. 一方面我们逐渐丧失 对数学教学的兴趣, 另一方面,我们的世界却比以前 更加趋向于数学化、数量化的数形世界中。
mathematical:adj.数学的,数学上的;精确的; quantitative:adj.定量的;量的,数量的;
So what's the problem, why has this chasm opened up, and what can we do to fix it? 那么问题到底出在哪里,为什么产生了这样的分歧？ 我们该怎么解决这个问题？
chasm:n.峡谷;裂口;分歧;深坑;
Well actually, I think the answer is staring us right in the face: 我想,答案就在 我们眼前。
staring:adj.凝视的,目不转睛的;显眼的;v.凝视;盯着看;显眼;(stare的现在分词)
Use computers. 使用计算机。
I believe that correctly using computers is the silver bullet for making math education work. 我相信 恰当地使用计算机 是使数学教育变得有效的 一剂良方。
silver:n.银; v.给…镀(或包)银; adj.银色的; bullet:n.子弹;只选某党全部候选人的投票;豆子;vi.射出;迅速行进;
So to explain that, let me first talk a bit about what math looks like in the real world and what it looks like in education. 在解释之前, 让我简要对比一下我们现实世界中的数学 和教学中的数学。
See, in the real world math isn't necessarily done by mathematicians . 现实世界中 数学并不是数学家的专用品。
necessarily:adv.必要地;必定地,必然地; mathematicians:n.[数]数学家(mathematician的复数形式);
It's done by geologists , engineers, biologists , all sorts of different people -- modeling and simulation . 地质学家、 工程师、生物学家 等各行各业 都可能运用数学进行数模和模拟。
geologists:n.[地质]地质学家(geologist的复数形式); biologists:n.生物学家(biologist的复数); simulation:n.仿真;模拟;模仿;假装;
It's actually very popular. 数学的应用实际上是非常广泛。
But in education it looks very different -- dumbed-down problems, lots of calculating, mostly by hand. 然而,教学中的数学则完全不同-- 它带有很多愚蠢的问题,大量的运算-- 还多数是人工笔头运算。
Lots of things that seem simple and not difficult like in the real world, except if you're learning it. 许多看起来很简单的, 不及现实世界中复杂的问题, 在学习的过程中都变得很困难。
And another thing about math: math sometimes looks like math -- like in this example here -- and sometimes it doesn't -- like "Am I drunk?" 关于数学的另一点就在于 数学有时看起来很“数学” 就像这个例子-- 不过有时又不像-- 比如说“我醉了吗？”
And then you get an answer that's quantitative in the modern world. 你会得到一个在现实世界中量化的答案。
You wouldn't have expected that a few years back. 几年前,你并不能预期你会得到这样的答案。
But now you can find out all about -- unfortunately , my weight is a little higher than that, but -- all about what happens. 但是现在你可以了解到... 抱歉其实我的体重比这个更重一点,但是-- 了解到所发生的这一切。
unfortunately:adv.不幸地;
So let's zoom out a bit and ask, why are we teaching people math? 我们退一步再来看看, 我们为什么要教数学呢？
What's the point of teaching people math? 教数学有什么意义呢？
And in particular , why are we teaching them math in general ? 尤其是,为什么我们要将数学作为通识基础教育呢？
in particular:尤其,特别; in general:总之,通常;一般而言;
Why is it such an important part of education as a sort of compulsory subject? 为什么数学是教育的一个重要组成部分呢？ 为什么要作为必修课呢？
compulsory:adj.义务的;必修的;被强制的;n.(花样滑冰、竞技体操等的)规定动作;
Well, I think there are about three reasons: technical jobs so critical to the development of our economies, what I call "everyday living" -- to function in the world today, you've got to be pretty quantitative, much more so than a few years ago: figure out your mortgages , 我想,这大概有以下三个理由: 一是完成技术工作。 这对我们经济的发展至关重要, 就是我所说的日常生活。 二,要在今天的世界运作, 你就必须比几年前掌握 更多数学知识。 你需要去算你的房贷,
technical:adj.工艺的,科技的;技术上的;专门的; critical:adj.鉴定的;[核]临界的;批评的,爱挑剔的;危险的;决定性的;评论的; mortgages:n.抵押贷款; v.[经]抵押(mortgage的三单形式);
being skeptical of government statistics , those kinds of things -- and thirdly, what I would call something like logical mind training, logical thinking. 对政府的数据等种种数字保持怀疑。 第三,我把它称作 逻辑思维训练,逻辑思维。
skeptical:adj.怀疑的;怀疑论的,不可知论的; statistics:n.统计数字;统计资料;统计学;(statistic的复数) logical:adj.合逻辑的,合理的;逻辑学的;
Over the years we've put so much in society into being able to process and think logically . It's part of human society. 这几年 我们努力使社会运作和思维 都变得符合逻辑,并将这做为人类社会的一部分。
process:v.处理;加工;列队行进;n.过程,进行;方法,adj.经过特殊加工(或处理)的; logically:adv.逻辑上;合乎逻辑;
It's very important to learn that math is a great way to do that. 必须要指出 数学正是实现这种构想的重要工具。
So let's ask another question. 现在让我们考虑另一个问题。
What is math? 数学是什么？
What do we mean when we say we're doing math, or educating people to do math? 当我们说我们在做数学, 或者说在教数学的时候我们指的是什么呢？
Well, I think it's about four steps, roughly speaking, starting with posing the right question. 总体来说,我觉得有四个步骤, 正确地提问是第一步。
roughly:adv.粗糙地;概略地; posing:v.造成;引起;产生;提问;摆好姿势;(pose的现在分词)
What is it that we want to ask? What is it we're trying to find out here? 我们想问的到底是什么问题？我们想了解的是什么？
And this is the thing most screwed up in the outside world, beyond virtually any other part of doing math. 这步是在外部世界相比做数学的其他任何步骤中 最容易搞砸的一步。
screwed:v.用螺丝固定;旋紧;拧紧;拧上去;(screw的过去分词和过去式) virtually:adv.事实上,几乎;实质上;
People ask the wrong question, and surprisingly enough, they get the wrong answer, for that reason, if not for others. 如果人们问错了问题, 毫无疑问,他们会因此而得出错误的结果, 如果运算没有错误的话。
surprisingly:adv.令人惊讶地;出乎意料地
So the next thing is take that problem and turn it from a real world problem into a math problem. 接下来就是根据正确的问题。 将它从现实世界的问题 转化成一个数学问题。
That's stage two. 这就是第二步。
Once you've done that, then there's the computation step. 这样做之后我们就进入了运算的第三步骤。
computation:n.估计,计算;
Turn it from that into some answer in a mathematical form. 求出某个 数学形式的答案。
And of course, math is very powerful at doing that. 当然,数学在这一方面非常有用。
And then finally , turn it back to the real world. 最后,将解答转换成现实世界的问题。
finally:adv.终于;最终;(用于列举)最后;彻底地;
Did it answer the question? 这个答案有没有解决现实社会的问题呢？
And also verify it -- crucial step. 当然事实证明这也是关键一步。
verify:vt.核实;查证; crucial:adj.重要的;决定性的;定局的;决断的;
Now here's the crazy thing right now. 现在,疯狂的事情就是
In math education, we're spending about perhaps 80 percent of the time teaching people to do step three by hand. 在数学教育中, 我们或许花了八成的时间 教人们用笔头计算解决第三个步骤。
Yet, that's the one step computers can do better than any human after years of practice. 然而,这恰恰是计算机 比任何经过多年运算训练的人都做得更好地一步。
Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four -- conceptualizing problems, applying them, getting the teacher to run them through how to do that. 换言之,我们本应该用计算机 去完成上述步骤三, 而让学生花更多的精力 去学好上述步骤一、二和四; 将问题概念化,并且运用它们, 让老师示范怎么去做。
conceptualizing:vt.使概念化;vi.概念化; applying:v.申请,请求;使用;应用;(apply的现在分词)
See, crucial point here: math is not equal to calculating. 看到了吗？这里的重点就是: 数学并不等同于计算。
Math is a much broader subject than calculating. 数学是比计算更广泛的一门学科。
Now it's understandable that this has all got intertwined over hundreds of years. 现在我们不难理解出现这种将计算和数学混淆在一起的原因了。 几百年来,
understandable:adj.可以理解的;可以了解的; intertwined:adj.缠绕的;错综复杂的;v.使缠结,缠绕(intertwine的过去式);
There was only one way to do calculating and that was by hand. 我们进行计算的唯一方法就是笔算。
But in the last few decades that has totally changed. 仅仅是近几十年 这种情况才出现根本性的变化。
We've had the biggest transformation of any ancient subject that I could ever imagine with computers. 计算机的运用使任何古老学科 产生最大的变革成为了可能。
transformation:n.转变;(用于南非)民主改革;
Calculating was typically the limiting step, and now often it isn't. 计算通常是有限步骤的, 无限步骤的并不常见。
typically:adv.代表性地;作为特色地;
So I think in terms of the fact that math has been liberated from calculating. 因此,我觉得,虽然就事实而言,数学 已经从计算中解放出来。
liberated:adj.不受传统思想束缚的,开放的; v.解放; (liberate的过去式和过去分词)
But that math liberation didn't get into education yet. 但是数学的解放却仍没有被引入到教学之中。
liberation:n.释放,解放;
See, I think of calculating, in a sense , as the machinery of math. 我认为,在某种程度上,计算 只是机械化的数学。
in a sense:在某种意义上; machinery:n.机械;机器;机构;机械装置;
It's the chore. 计算是一种杂活,
It's the thing you'd like to avoid if you can, like to get a machine to do. 就是那种你想尽量避免,可以的话让机器来完成的工作。
It's a means to an end , not an end in itself, and automation allows us to have that machinery. 仅仅作为得出目的的手段,而不是目的本身。 而自动化允许我们 享用这种计算机器。
means to an end:达到目的的手段;达到某一目的的手段;手段目的; automation:n.自动化;自动操作;
Computers allow us to do that -- and this is not a small problem by any means . 计算机允许我们这样做。 而这无论从何种角度看都不是一个小问题。
by any means:无论如何;
I estimated that, just today, across the world, we spent about 106 average world lifetimes teaching people how to calculate by hand. 我估计,仅仅是今天世界范围之内, 我们就平均用了大约106年生命时间 去教人们做人工运算。
estimated:adj.估计的;预计的;估算的;
That's an amazing amount of human endeavor . 这是巨大的人类劳动。
endeavor:n.v.努力;尽力;试图;
So we better be damn sure -- and by the way , they didn't even have fun doing it, most of them -- so we better be damn sure that we know why we're doing that and it has a real purpose. 因此我们最好可以非常确定—— 顺便提一下,绝大多数人做人工运算时没有感到任何乐趣。 因此,我们最好可以非常确定 我们知道这样做的原因, 这样做有一个真正的目的。
damn:v.谴责;该死;n.诅咒;adj.可恶的; by the way:顺便说一下;
I think we should be assuming computers for doing the calculating and only doing hand calculations where it really makes sense to teach people that. 我认为我们应该预计计算机 可以完成运算, 仅仅在必要时才教人们人工笔头运算。
assuming:conj.假设…为真; adj.傲慢的; v.假定; (assume的现在分词)
And I think there are some cases. 当然我觉得某些情况下
For example: mental arithmetic . 比如说,心算。
mental arithmetic:n.心算;
I still do a lot of that, mainly for estimating . 我现在仍然经常运用心算,通常是用来预估。
mainly:adv.大多;大部分;主要地;首要地; estimating:v.估价;估算;(estimate的现在分词)
People say, "Is such and such true?" 当别人说,答案是这样的,
And I'll say, "Hmm, not sure." I'll think about it roughly. 我会说,嗯,不确定。 我认为答案大概是这样吧。
It's still quicker to do that and more practical . 心算在这方面仍然是比较快速而且有用的。
practical:adj.实际的;真实的;客观存在的;n.实习课;实践课;
So I think practicality is one case where it's worth teaching people by hand. 因此我认为实用性是 其中一种值得教授人工运算的情况。
practicality:n.实用性,实际性;实际,实例;
And then there are certain conceptual things that can also benefit from hand calculating, but I think they're relatively small in number. 另外还有一些 同样得益于人工运算的概念, 但我认为这种情况相对来说比较少。
relatively:adv.相当程度上;相当地;相对地;
One thing I often ask about is ancient Greek and how this relates. 比如说,我经常会问起 以古希腊教学数学来做一个类比。
See, the thing we're doing right now is we're forcing people to learn mathematics . 我们现在的教学 总是强迫学生去学习数学。
mathematics:n.数学;数学运算;
It's a major subject. 它是一个主科。
I'm not for one minute suggesting that, if people are interested in hand calculating or in following their own interests in any subject however bizarre -- they should do that. 注意的是,我从来没有刻意暗示说如果有人对人工运算感兴趣 或者按照他们自己的意愿 学习任何哪怕是毫无意义的学科; 相反,他们应当这样做。
bizarre:adj.奇异的(指态度,容貌,款式等);
That's absolutely the right thing, for people to follow their self-interest . 因为让人们发展自己的兴趣爱好, 完全就是一件正确的事。
absolutely:adv.绝对地;完全地; self-interest:n.私利;利己主义;
I was somewhat interested in ancient Greek, but I don't think that we should force the entire population to learn a subject like ancient Greek. 我个人对古希腊怀有某种特殊的情结, 但我不认为我们应该逼迫全体人民 都像古希腊学习数学的方法。
somewhat:n.几分;某物;adv.有点;多少;几分;稍微;
I don't think it's warranted . 我不认为这样是值得的。
warranted:v.使有必要;使正当;使恰当;(warrant的过去分词和过去式)
So I have this distinction between what we're making people do and the subject that's sort of mainstream and the subject that, in a sense, people might follow with their own interest and perhaps even be spiked into doing that. 因此我知道我们应该区分迫使人们学习 或是作为某种主流的学科 与人们在某种程度上自觉按照自己意愿学习的 或甚至是钟情于此的学科。
distinction:n.区别;区分;差别;卓越; mainstream:n.主流; spiked:adj.尖的;有穗的;加入标准的;
So what are the issues people bring up with this? 那人们为什么提出这一点呢？
issues:n.重要议题;争论的问题;v.宣布;公布;发出;(issue的第三人称单数和复数)
Well one of them is, they say, you need to get the basics first. 其中一点,他们会说,你必须先打好基础。
You shouldn't use the machine until you get the basics of the subject. 你在完全掌握学科的基础之前 不许用机器。
So my usual question is, what do you mean by "basics?" 那我通常的疑问就是,你所说的“基础”指的什么？
what do you mean by:你说…是什么意思？;
Basics of what? 什么的基础？
Are the basics of driving a car learning how to service it, or design it for that matter? 这样说来,请问驾车的基础 是否是怎样修车或者设计车模？
Are the basics of writing learning how to sharpen a quill ? 而写作的基础是否又是要学习如何削鹅毛笔？
sharpen:vt.削尖;磨快;使敏捷;加重;vi.尖锐;变锋利; quill:n.大翎毛;羽茎;(豪猪,刺猬的)刚毛;vt.刺穿;
I don't think so. 我认为不是这样的。
I think you need to separate the basics of what you're trying to do from how it gets done and the machinery of how it gets done and automation allows you to make that separation . 我认为你需要将你所要做的事情的基础 理解成两部分如何得出结果 与用机器如何得出结果区分开来。 自动化允许你作出如此的理解。
separation:n.分离;分开;分割;隔离;
A hundred years ago, it's certainly true that to drive a car you kind of needed to know a lot about the mechanics of the car and how the ignition timing worked and all sorts of things. 一百多年前,如果要学车你无疑是需要 知道某些汽车机械原理的 以及定时打火器等各零部件的工作原理。
mechanics:n.机械师;机械修理工;技工;力学;机械学;(mechanic的复数) ignition:n.点火,点燃;着火,燃烧;点火开关,点火装置;
But automation in cars allowed that to separate, so driving is now a quite separate subject, so to speak , from engineering of the car or learning how to service it. 但是汽车的自动化 实现两类学问的划分, 也就是说驾车现在与汽车工程学 是相对独立的两个学科, 修车又是另外一回事。
so to speak:可以说;打个譬喻说; engineering:n.工程;工程学;v.密谋策划;设计制造;改变…的基因;(engineer的现在分词)
So automation allows this separation and also allows -- in the case of driving, and I believe also in the future case of maths -- a democratized way of doing that. 正是自动化实现了这种区分, 不仅在驾车的例子中是这样的自动化, 因为我相信自动计算化也会在未来的数学中起同样的作用, 我们可以自主地选择怎样去做。
maths:数学(课) democratized:vt.使民主化;使大众化;vi.民主化;大众化;
It can be spread across a much larger number of people who can really work with that. 数学可以向更多实际运用的人 去传播。
So there's another thing that comes up with basics. 关于基本技能又有另外一个观点。
People confuse , in my view, the order of the invention of the tools with the order in which they should use them for teaching. 我认为,人们混淆 发明工具的次序 与在教学中应当运用这些工具的次序。
confuse:v.混同;使迷惑;使糊涂;使更难于理解;
So just because paper was invented before computers, it doesn't necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics. 不能因为纸比计算机先被发明,而人们就应该先在教学中运用纸张, 用纸张教数学不一定代表 你会比用计算机 更了解数学的基础。
My daughter gave me a rather nice anecdote on this. 我女儿的一则轶事为这一点做了生动地说明。
anecdote:n.轶事;奇闻;秘史;
She enjoys making what she calls "paper laptops ." 她很喜欢制作她所说的“纸制笔记本电脑”。
laptops:n.便携式电脑;笔记本电脑;(laptop的复数)
(Laughter) （笑声）
So I asked her one day, "You know, when I was your age, 有一天,我问她:“我在你这个年纪的时候,
I didn't make these. 我都不会做这些。
Why do you think that was?" 你觉得是为什么呢？”
And after a second or two, carefully reflecting , she said, "No paper?" 她很仔细地想了一两秒, 然后她说,“没有纸吗？”
reflecting:v.反映;映出(影像);反射;显示,表明,表达;(reflect的现在分词)
(Laughter) （笑声）
If you were born after computers and paper, it doesn't really matter which order you're taught with them in, you just want to have the best tool. 如果你在计算机和纸张的发明之后出生, 你学习数学究竟是先用纸张学习还是先用计算机学习其实并不重要, 你仅仅需要用最佳的工具来学习数学。
So another one that comes up is "Computers dumb math down." 那另一个观点就是“计算机使数学变蠢。”
dumb:adj.哑的,无说话能力的;不说话的,无声音的;
That somehow , if you use a computer, it's all mindless button-pushing, but if you do it by hand, it's all intellectual . 在一定程度上的确如此,如果你用计算机, 这仅仅是无需思考的按键操作, 但如果你进行人工笔头运算, 这完全又是智力训练。
somehow:adv.以某种方法;莫名其妙地; mindless:adj.愚蠢的;不小心的;不需要动脑筋的;不顾虑的; intellectual:n.知识分子;脑力劳动者;adj.智力的;脑力的;理智的;有才智的;
This one kind of annoys me, I must say. 我要说,这一点很困扰我。
annoys:vt.骚扰;惹恼;打搅;vi.惹恼;令人讨厌;打搅;n.烦恼(等于annoyance);
Do we really believe that the math that most people are doing in school practically today is more than applying procedures to problems they don't really understand, for reasons they don't get? 我们难道真的以为 现在通常人们在学校里 学习的数学 真的比按照解题步骤做题学得多吗? 他们大都还在用不理解的原理去解答他们不明白的问题。
practically:adv.实际地;几乎;事实上; procedures:n.程序;规程(procedure的复数);
I don't think so. 至少我认为如此。
And what's worse, what they're learning there isn't even practically useful anymore. 更严重的是,他们在学校所学的知识根本已经不再实用。
Might have been 50 years ago, but it isn't anymore. 或许50年前还在用,但现在已经不再使用了。
When they're out of education, they do it on a computer. 除了教育之外,他们都用计算机来完成。
Just to be clear, I think computers can really help with this problem, actually make it more conceptual. 要清醒地知道,计算机已经完全解决了这一问题, 事实上使它更容易理解。
Now, of course, like any great tool, they can be used completely mindlessly , like turning everything into a multimedia show, like the example I was shown of solving an equation by hand, where the computer was the teacher -- show the student how to manipulate and solve it by hand. 而如今,像任何伟大的工具一样, 它们可以彻底无须动脑筋地操作, 就像将任何东西都变成一场多媒体的盛宴一样, 就像这个解方程的例子, 计算机就好比是老师-- 它给学生演示如何现在用笔头进行运算和解题。
mindlessly:adv.不费心思地;愚蠢地; multimedia:n.多媒体;adj.多媒体的; manipulate:vt.操纵;操作;巧妙地处理;篡改;
This is just nuts. 这其实很荒谬。
Why are we using computers to show a student how to solve a problem by hand that the computer should be doing anyway? 为什么我们要用计算机来演示如何用笔算解题呢？ 明明计算机可以自行计算来解题。
All backwards. 完全是倒退了。
Let me show you that you can also make problems harder to calculate. 下面让我为你演示 计算机如何运算出更复杂的题目。
See, normally in school, you do things like solve quadratic equations . 通常在学校里, 你会学习像解二次方程式这类题目。
normally:adv.正常地;通常地,一般地; quadratic:adj.[数]二次的;n.二次方程式; equations:n.方程式;等式;均等;均势(equation的复数形式);
But you see, when you're using a computer, you can just substitute . 但如果你用计算机来做, 你仅需要做代换。
substitute:v.取代;(以…)代替;n.代用品;代替物;代替者;替补(运动员);
You can make it a quartic equation. Make it kind of harder, calculating-wise. 就可以将它变成四次方程,运算上更复杂。
Same principles applied -- calculations, harder. 同样的原理仍然适用, 只是运算上更复杂。
principles:n.原则;主义;本质;政策;(principle的复数) applied:adj.应用的;实用的;v.应用;使用;申请,请求;(apply的过去分词和过去式)
And problems in the real world look nutty and horrible like this. 而现实世界里的问题 的确就像这样复杂和难懂的,
nutty:adj.多坚果的;产坚果的;古怪的; horrible:可怕的,极讨厌的,
They've got hair all over them. 甚至更加棘手的问题。
They're not just simple, dumbed-down things that we see in school math. 它们都不是我们学校数学中学的那些简单愚蠢的东西。
And think of the outside world. 让我们想想外面的世界。
Do we really believe that engineering and biology and all of these other things that have so benefited from computers and maths have somehow conceptually gotten reduced by using computers? 难道我们真的以为工程学和生物学 等其他 得益于计算机和数学的学问 会由于应用计算机而在某种程度上被明显削弱吗？
biology:n.(一个地区全部的)生物;生物学; conceptually:adv.概念地;
I don't think so -- quite the opposite. 我认为,恰恰相反。
So the problem we've really got in math education is not that computers might dumb it down, but that we have dumbed-down problems right now. 我们在数学教育中的真正问题 并不是计算机使数学变得愚蠢, 而是我们现在的教学使问题变得愚蠢。
Well, another issue people bring up is somehow that hand calculating procedures teach understanding. 好,那人们提到的另一点 就是笔头运算的步骤在一定程度上 可以教导学生理解数学。
So if you go through lots of examples, you can get the answer, you can understand how the basics of the system work better. 因此如果你做过很多例题, 你就可以算出答案, 你也可以更好地理解数学系统的基础。
I think there is one thing that I think very valid here, which is that I think understanding procedures and processes is important. 我认为,这个观点里有一点是我非常认同的, 那就是我认为步骤和运算过程很重要。
valid:adj.有效的;有根据的;合法的;正当的; processes:n.过程; v.处理(process的第三人称单数形式);
But there's a fantastic way to do that in the modern world. 但是在现代社会我们有一个绝佳的途径去这样做,
fantastic:奇异的,空想的
It's called programming. 那就是编程。
Programming is how most procedures and processes get written down these days, and it's also a great way to engage students much more and to check they really understand. 现在多数程序和编码过程 大都以编程的方式写下来, 而这也是一种很好的方式 去鼓励学生更多的参与, 检验他们是否真正理解数学问题。
engage:v.吸引,占用;使参加;雇佣;使订婚;预定;
If you really want to check you understand math then write a program to do it. 如果你真的想检验你是否理解数学 就尝试编写一条程序。
So programming is the way I think we should be doing that. 因此我认为编程就是我们应该做的方向。
So to be clear, what I really am suggesting here is we have a unique opportunity to make maths both more practical and more conceptual, simultaneously . 一言以蔽之,在此我想说的是 编程为我们提供了一个独特的机会: 一方面使数学更实用, 同时另一方面又更加理论化。
unique:adj.独特的,稀罕的;[数]唯一的;n.独一无二的人或物; simultaneously:adv.同时地;
I can't think of any other subject where that's recently been possible. 我想,这种在数学教学的新局面在其他任何学科都还没有出现。
recently:adv.最近;新近;
It's usually some kind of choice between the vocational and the intellectual. 这通常像是一种 在职业规划或者是智力培养之间的两难选择。
vocational:adj.职业的,行业的;
But I think we can do both at the same time here. 但我想运用计算机我们可以两方面同时实现。
at the same time:同时;另一方面;与此同时;
And we open up so many more possibilities. 还为我们又打开了许多可能性。
You can do so many more problems. 我们可以解决更多的实际问题。
What I really think we gain from this is students getting intuition and experience in far greater quantities than they've ever got before. 我真切地感觉到这种教育的改革 可以让学生获得很多的知识和经验 比之前多得多的直观的认识和经验的累积。
intuition:n.直觉;直觉力;直觉的知识; quantities:n.数量;数目;大量;大宗;(quantity的复数)
And experience of harder problems -- being able to play with the math, interact with it, feel it. 和接触更加复杂的题目—— 畅游于数学的乐园,与数学沟通, 感受数学之美。
interact:v.互相影响;互相作用;n.幕间剧;幕间休息;
We want people who can feel the math instinctively . 我们希望学生可以本能地感受数学。
instinctively:adv.本能地;
That's what computers allow us to do. 而这恰恰是计算机使这成为可能。
Another thing it allows us to do is reorder the curriculum . 另一点是计算机使我们得以对教学课程重新排序。
reorder:vt.再订购;重新整理;vi.再订购;n.再订购; curriculum:n.课程;总课程;
Traditionally it's been by how difficult it is to calculate, but now we can reorder it by how difficult it is to understand the concepts, however hard the calculating. 以前是按照计算的难度编排课程教学的顺序, 但现在我们可以重新排序, 按照概念理念的难度, 而不管计算的难度。
Traditionally:adv.传统上;习惯上;传说上;
So calculus has traditionally been taught very late. 因此微积分传统上是很晚才能教授的项目。
Why is this? 原因是什么呢？
Well, it's damn hard doing the calculations, that's the problem. 问题就在于微积分的计算相当困难。
But actually many of the concepts are amenable to a much younger age group . 但事实上许多概念 是可以向更低年级的学生传授的。
amenable:adj.有责任的,应服从的;有义务的;经得起检验的; age group:n.年龄组;年龄段;
This was an example I built for my daughter. 这就是一个我为女儿所建的模型例子。
And very, very simple. 而且非常简单易懂。
We were talking about what happens when you increase the number of sides of a polygon to a very large number. 这个模型是用于理解 当多边形的边数增加时,图形的变化随之改变。 当边数增加到非常大的时候,
polygon:n.多边形;多角形物体;
And of course, it turns into a circle. 当然,多边形变成一个圆。
And by the way, she was also very insistent on being able to change the color, an important feature for this demonstration . 顺便一提,她对可以转换颜色这一点 特别乐此不疲, 这也是这个展示的一个重要的特征。
insistent:adj.坚持的;迫切的;显著的;引人注目的;紧急的; demonstration:n.示范;演示;论证;证明;
You can see that this is a very early step into limits and differential calculus and what happens when you take things to an extreme -- and very small sides and a very large number of sides. 你可以看到这是一个 对极限和微积分非常初步的启蒙, 也可以让你直观地看到趋于极限时所发生的变化, 无限小边数的情况和无限多边数的情况。
differential calculus:n.微分学; extreme:adj.极端的;极度的;偏激的;尽头的;n.极端;末端;最大程度;极端的事物;
Very simple example. 非常浅显的例子。
That's a view of the world that we don't usually give people for many, many years after this. 这是对世界的一种观察方式, 我们通常是很久很久之后才会教给学生这些。
And yet, that's a really important practical view of the world. 但它却是对世界的一种非常重要且实用的观察。
So one of the roadblocks we have in moving this agenda forward is exams. 那现在我们要改革 推进这个进程的路障 就在于考试。
roadblocks:n.路障;障碍(roadblock的复数);v.设置路障(roadblock的单三形式); agenda:n.议事日程;
In the end, if we test everyone by hand in exams, it's kind of hard to get the curricula changed to a point where they can use computers during the semesters . 最后,假如我们考试的方式是人工笔算, 就难以将目前的课程设计 改革成 在教学中可以使用计算机。
curricula:n.课程(curriculum的复数形式); semesters:n.学期(semester的复数);
And one of the reasons it's so important -- so it's very important to get computers in exams. 而既然作为一个如此重要的原因 在考试中使用计算机也是非常重要的。
And then we can ask questions, real questions, questions like, what's the best life insurance policy to get? -- real questions that people have in their everyday lives. 如此以来,我们就可以设置实际的问题, 比如说,“哪一种人寿保险的险种更适合我们？” 人们实际生活中遇到诸如此类的问题。
insurance policy:n.保险单;
And you see, this isn't some dumbed-down model here. 你可以看到,这就不是那些愚蠢的问题模型。
This is an actual model where we can be asked to optimize what happens. 这是一个我们优化选择时可以用到的实际模型。
optimize:vt.使最优化,使完善;vi.优化;持乐观态度;
How many years of protection do I need? 我需要多少年的险期？
What does that do to the payments and to the interest rates and so forth? 这对保险费有什么影响？ 对利率等其他的影响呢？
payments:n.付款;支付;收款;款项;报答;(payment的复数)
Now I'm not for one minute suggesting it's the only kind of question that should be asked in exams, but I think it's a very important type that right now just gets completely ignored and is critical for people's real understanding. 这里我没有刻意在暗示 这是考试中应该考的唯一类型的问题, 但是我认为至少这是非常重要的一类型, 也是目前的教育完全忽视的, 这对人们的实际理解也是至关重要的。
ignored:v.忽视;对…不予理会;佯装未见;不予理睬;(ignore的过去分词和过去式)
So I believe [there is] critical reform we have to do in computer-based math. 因此我相信 我们需要一场以计算机作为数学教学工具的改革。
reform:v.改革;改进;改良;(使)悔改;n.改革;改良;改善;
We have got to make sure that we can move our economies forward, and also our societies, based on the idea that people can really feel mathematics. 我们务必确保 通过让人们可以切实感受数学, 我们才可以推进经济发展和 社会进步。
This isn't some optional extra . 这绝不是可有可无的一环。
optional:adj.可选择的,随意的;n.选修科目; extra:adj.额外的:n.额外的事物:adv.额外:另外:
And the country that does this first will, in my view, leapfrog others in achieving a new economy even, an improved economy, an improved outlook . 而我认为首先这样做的国家 一定会鹤立鸡群, 甚至取得更大的经济成就, 更完善的经济体系, 和更卓越的国家面貌。
leapfrog:n.跳背游戏;跳蛙游戏;交互跃进;vi.跳背;跳蛙;交替前进;vt.跃过; improved:adj.改良的:v.改进:改善(improve的过去分词和过去式) outlook:n.见解;前景;人生观;可能性;
In fact, I even talk about us moving from what we often call now the " knowledge economy " 事实上,我甚至会说我们正在 从我们现在经常提到的“知识经济”时代迈向
knowledge economy:知识经济;
to what we might call a " computational knowledge economy," 可能被称作“计算机知识经济”的时代,
computational:adj.计算的;
where high-level math is integral to what everyone does in the way that knowledge currently is. 高端的数学知识对每个人 应用现代知识来说都不可或缺。
high-level:adj.高级的;高阶层的;在高空的; integral:adj.积分的;完整的,整体的;必须的;n.积分;部分;完整; in the way:妨碍;挡道; currently:adv.当前;一般地;
We can engage so many more students with this, and they can have a better time doing it. 我们可以让更多学生加入到这个改革, 让他们在最佳的时机得到这样的教育。
And let's understand: this is not an incremental sort of change. 我们要明白 这不是一场渐进式的改革。
incremental:adj.增加的,增值的;
We're trying to cross the chasm here between school math and the real-world math. 我们要试图跨越 学校数学和实际数学之间的鸿沟。
real-world:adj.现实生活的;工作的;
And you know if you walk across a chasm, you end up making it worse than if you didn't start at all -- bigger disaster . 你可以看到如果我们是走过这个鸿沟的话, 我们一定会跌得粉身碎骨, 甚至比没有开始改革更糟糕。
disaster:n.灾难,灾祸;不幸;
No, what I'm suggesting is that we should leap off, we should increase our velocity so it's high, and we should leap off one side and go the other -- of course, having calculated our differential equation very carefully. 这不是我想要的局面, 我想说,我们应该跳跃, 我们必须加快我们的起跳速度, 这样我们就可以跳得高, 我们应该从鸿沟的一头跳到另外一头, 当然要准确地计算这里微分方程。
velocity:n.[物]速度; differential equation:n.微分方程;
(Laughter) （笑声）
So I want to see a completely renewed , changed math curriculum built from the ground up , based on computers being there, computers that are now ubiquitous almost. 因此我想看到的是 一套全新的、经过改良的数学教学课程 在应用计算机的基础上 架构而起, 目前计算机已经是在各地非常普遍的
renewed:adj.再次发生的; v.重新开始; (renew的过去分词和过去式) from the ground up:从头开始;彻底; ubiquitous:adj.普遍存在的;无所不在的;
Calculating machines are everywhere and will be completely everywhere in a small number of years. 运算工具, 相信在不久的将来它会彻底覆盖每一个地方。
Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future. 现在我还不确定我们是否应该仍然沿用“数学”作为这个学科的名字, 但我确信 这门课将是未来学科的主流。
brand:v.铭刻于,铭记;打烙印于;印…商标于;n.商标,牌子;烙印;
Let's go for it , and while we're about it, let's have a bit of fun, for us, for the students and for TED here. 让我们共同努力吧。 并且在学习它的过程中 让大家 所有人,广大的学生以及TED这里的观众享受一些乐趣！
go for it:努力争取;加油;
Thanks. 谢谢
(Applause) （掌声）