TEDGlobal 2010
Conrad Wolfram: Teaching kids real math with computers
康拉德·沃尔夫拉姆Conrad Wolfram：用计算机教导孩子真正的数学
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从火箭发射到股票市场，人类社会最惊人的创举大都由数学支持。那为何孩子会丧失学习数学的兴趣呢？康拉德·沃尔夫拉姆指出部分原因在于，我们注重笔算的教学方式，不但沉闷繁琐，还与我们实际生活毫无关联。在这个演讲中，他为我们呈现了他鲜明的观点：通过计算机编程教导孩子学习数学。
Conrad Wolfram - Mathematician
Conrad Wolfram runs the worldwide arm of Wolfram Research, the mathematical lab behind the cutting-edge knowledge engine Wolfram Alpha. Full bio
Conrad Wolfram runs the worldwide arm of Wolfram Research, the mathematical lab behind the cutting-edge knowledge engine Wolfram Alpha. Full bio
Double-click the English transcript below to play the video.
00:15
We've got a real problem with math education right now.
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目前我们的数学教育面临着实际的问题。
00:19
Basically, no one's very happy.
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基本上，没有人感到很满意。
00:22
Those learning it
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那些学习数学的学生
00:24
think it's disconnected,
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觉得所学的知识与实际无关，
00:26
uninteresting and hard.
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无趣以及困难重重。
00:28
Those trying to employ them
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那些运用数学的人们
00:30
think they don't know enough.
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又觉得他们所学的知识不够。
00:32
Governments realize that it's a big deal for our economies,
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政府觉察到这是一个涉及到我们经济发展的大问题，
00:35
but don't know how to fix it.
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但他们却无从下手。
00:38
And teachers are also frustrated.
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而老师们也感到很沮丧。
00:40
Yet math is more important to the world
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但是现在的数学
00:43
than at any point in human history.
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比人类历史上任何时间都更为重要。
00:45
So at one end we've got falling interest
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一方面我们逐渐丧失
00:47
in education in math,
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对数学教学的兴趣，
00:49
and at the other end we've got a more mathematical world,
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另一方面，我们的世界却比以前
00:52
a more quantitative world than we ever have had.
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更加趋向于数学化、数量化的数形世界中。
00:56
So what's the problem, why has this chasm opened up,
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那么问题到底出在哪里，为什么产生了这样的分歧？
00:58
and what can we do to fix it?
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我们该怎么解决这个问题？
01:01
Well actually, I think the answer
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我想，答案就在
01:03
is staring us right in the face:
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我们眼前。
01:05
Use computers.
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使用计算机。
01:07
I believe
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我相信
01:09
that correctly using computers
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恰当地使用计算机
01:11
is the silver bullet
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是使数学教育变得有效的
01:13
for making math education work.
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一剂良方。
01:16
So to explain that,
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在解释之前，
01:18
let me first talk a bit about what math looks like in the real world
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让我简要对比一下我们现实世界中的数学
01:21
and what it looks like in education.
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和教学中的数学。
01:23
See, in the real world
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现实世界中
01:25
math isn't necessarily done by mathematicians.
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数学并不是数学家的专用品。
01:28
It's done by geologists,
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地质学家、
01:30
engineers, biologists,
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工程师、生物学家
01:32
all sorts of different people --
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等各行各业
01:34
modeling and simulation.
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都可能运用数学进行数模和模拟。
01:36
It's actually very popular.
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数学的应用实际上是非常广泛。
01:38
But in education it looks very different --
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然而，教学中的数学则完全不同--
01:41
dumbed-down problems, lots of calculating,
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它带有很多愚蠢的问题，大量的运算--
01:43
mostly by hand.
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还多数是人工笔头运算。
01:46
Lots of things that seem simple
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许多看起来很简单的，
01:48
and not difficult like in the real world,
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不及现实世界中复杂的问题，
01:50
except if you're learning it.
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在学习的过程中都变得很困难。
01:53
And another thing about math:
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关于数学的另一点就在于
01:55
math sometimes looks like math --
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数学有时看起来很“数学”
01:57
like in this example here --
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就像这个例子--
02:00
and sometimes it doesn't --
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不过有时又不像--
02:02
like "Am I drunk?"
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比如说“我醉了吗？”
02:07
And then you get an answer that's quantitative in the modern world.
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你会得到一个在现实世界中量化的答案。
02:10
You wouldn't have expected that a few years back.
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几年前，你并不能预期你会得到这样的答案。
02:13
But now you can find out all about --
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但是现在你可以了解到...
02:16
unfortunately, my weight is a little higher than that, but --
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抱歉其实我的体重比这个更重一点，但是--
02:19
all about what happens.
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了解到所发生的这一切。
02:21
So let's zoom out a bit and ask,
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我们退一步再来看看，
02:23
why are we teaching people math?
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我们为什么要教数学呢？
02:25
What's the point of teaching people math?
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教数学有什么意义呢？
02:28
And in particular, why are we teaching them math in general?
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尤其是，为什么我们要将数学作为通识基础教育呢？
02:31
Why is it such an important part of education
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为什么数学是教育的一个重要组成部分呢？
02:34
as a sort of compulsory subject?
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为什么要作为必修课呢？
02:36
Well, I think there are about three reasons:
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我想，这大概有以下三个理由：
02:39
technical jobs
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一是完成技术工作。
02:41
so critical to the development of our economies,
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这对我们经济的发展至关重要，
02:44
what I call "everyday living" --
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就是我所说的日常生活。
02:48
to function in the world today,
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二，要在今天的世界运作，
02:50
you've got to be pretty quantitative,
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你就必须比几年前掌握
02:52
much more so than a few years ago:
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更多数学知识。
02:54
figure out your mortgages,
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你需要去算你的房贷，
02:56
being skeptical of government statistics, those kinds of things --
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对政府的数据等种种数字保持怀疑。
03:00
and thirdly, what I would call something like
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第三，我把它称作
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logical mind training, logical thinking.
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逻辑思维训练，逻辑思维。
03:06
Over the years
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这几年
03:08
we've put so much in society
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我们努力使社会运作和思维
03:10
into being able to process and think logically. It's part of human society.
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都变得符合逻辑，并将这做为人类社会的一部分。
03:13
It's very important to learn that
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必须要指出
03:15
math is a great way to do that.
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数学正是实现这种构想的重要工具。
03:17
So let's ask another question.
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现在让我们考虑另一个问题。
03:19
What is math?
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数学是什么？
03:21
What do we mean when we say we're doing math,
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当我们说我们在做数学，
03:23
or educating people to do math?
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或者说在教数学的时候我们指的是什么呢？
03:25
Well, I think it's about four steps, roughly speaking,
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总体来说，我觉得有四个步骤，
03:28
starting with posing the right question.
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正确地提问是第一步。
03:30
What is it that we want to ask? What is it we're trying to find out here?
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我们想问的到底是什么问题？我们想了解的是什么？
03:33
And this is the thing most screwed up in the outside world,
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这步是在外部世界相比做数学的其他任何步骤中
03:35
beyond virtually any other part of doing math.
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最容易搞砸的一步。
03:38
People ask the wrong question,
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如果人们问错了问题，
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and surprisingly enough, they get the wrong answer,
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毫无疑问，他们会因此而得出错误的结果，
03:42
for that reason, if not for others.
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如果运算没有错误的话。
03:44
So the next thing is take that problem
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接下来就是根据正确的问题。
03:46
and turn it from a real world problem
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将它从现实世界的问题
03:48
into a math problem.
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转化成一个数学问题。
03:50
That's stage two.
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这就是第二步。
03:52
Once you've done that, then there's the computation step.
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这样做之后我们就进入了运算的第三步骤。
03:55
Turn it from that into some answer
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求出某个
03:57
in a mathematical form.
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数学形式的答案。
04:00
And of course, math is very powerful at doing that.
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当然，数学在这一方面非常有用。
04:02
And then finally, turn it back to the real world.
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最后，将解答转换成现实世界的问题。
04:04
Did it answer the question?
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这个答案有没有解决现实社会的问题呢？
04:06
And also verify it -- crucial step.
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当然事实证明这也是关键一步。
04:10
Now here's the crazy thing right now.
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现在，疯狂的事情就是
04:12
In math education,
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在数学教育中，
04:14
we're spending about perhaps 80 percent of the time
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我们或许花了八成的时间
04:17
teaching people to do step three by hand.
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教人们用笔头计算解决第三个步骤。
04:20
Yet, that's the one step computers can do
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然而，这恰恰是计算机
04:22
better than any human after years of practice.
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比任何经过多年运算训练的人都做得更好地一步。
04:25
Instead, we ought to be using computers
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换言之，我们本应该用计算机
04:28
to do step three
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去完成上述步骤三，
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and using the students to spend much more effort
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而让学生花更多的精力
04:33
on learning how to do steps one, two and four --
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去学好上述步骤一、二和四；
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conceptualizing problems, applying them,
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将问题概念化，并且运用它们，
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getting the teacher to run them through how to do that.
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让老师示范怎么去做。
04:41
See, crucial point here:
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看到了吗？这里的重点就是：
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math is not equal to calculating.
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数学并不等同于计算。
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Math is a much broader subject than calculating.
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数学是比计算更广泛的一门学科。
04:48
Now it's understandable that this has all got intertwined
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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。
04:51
over hundreds of years.
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几百年来，
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There was only one way to do calculating and that was by hand.
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我们进行计算的唯一方法就是笔算。
04:56
But in the last few decades
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仅仅是近几十年
04:58
that has totally changed.
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这种情况才出现根本性的变化。
05:00
We've had the biggest transformation of any ancient subject
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计算机的运用使任何古老学科
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that I could ever imagine with computers.
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产生最大的变革成为了可能。
05:07
Calculating was typically the limiting step,
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计算通常是有限步骤的，
05:09
and now often it isn't.
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无限步骤的并不常见。
05:11
So I think in terms of the fact that math
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因此，我觉得，虽然就事实而言，数学
05:13
has been liberated from calculating.
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已经从计算中解放出来。
05:16
But that math liberation didn't get into education yet.
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但是数学的解放却仍没有被引入到教学之中。
05:19
See, I think of calculating, in a sense,
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我认为，在某种程度上，计算
05:21
as the machinery of math.
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只是机械化的数学。
05:23
It's the chore.
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计算是一种杂活，
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It's the thing you'd like to avoid if you can, like to get a machine to do.
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就是那种你想尽量避免，可以的话让机器来完成的工作。
05:29
It's a means to an end, not an end in itself,
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仅仅作为得出目的的手段，而不是目的本身。
05:34
and automation allows us
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而自动化允许我们
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to have that machinery.
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享用这种计算机器。
05:38
Computers allow us to do that --
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计算机允许我们这样做。
05:40
and this is not a small problem by any means.
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而这无论从何种角度看都不是一个小问题。
05:43
I estimated that, just today, across the world,
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我估计，仅仅是今天世界范围之内，
05:46
we spent about 106 average world lifetimes
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我们就平均用了大约106年生命时间
05:49
teaching people how to calculate by hand.
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去教人们做人工运算。
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That's an amazing amount of human endeavor.
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这是巨大的人类劳动。
05:55
So we better be damn sure --
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因此我们最好可以非常确定——
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and by the way, they didn't even have fun doing it, most of them --
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顺便提一下，绝大多数人做人工运算时没有感到任何乐趣。
06:00
so we better be damn sure
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因此，我们最好可以非常确定
06:02
that we know why we're doing that
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我们知道这样做的原因，
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and it has a real purpose.
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这样做有一个真正的目的。
06:06
I think we should be assuming computers
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我认为我们应该预计计算机
06:08
for doing the calculating
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可以完成运算，
06:10
and only doing hand calculations where it really makes sense to teach people that.
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仅仅在必要时才教人们人工笔头运算。
06:13
And I think there are some cases.
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当然我觉得某些情况下
06:15
For example: mental arithmetic.
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比如说，心算。
06:17
I still do a lot of that, mainly for estimating.
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我现在仍然经常运用心算，通常是用来预估。
06:20
People say, "Is such and such true?"
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当别人说，答案是这样的，
06:22
And I'll say, "Hmm, not sure." I'll think about it roughly.
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我会说，嗯，不确定。 我认为答案大概是这样吧。
06:24
It's still quicker to do that and more practical.
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心算在这方面仍然是比较快速而且有用的。
06:26
So I think practicality is one case
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因此我认为实用性是
06:28
where it's worth teaching people by hand.
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其中一种值得教授人工运算的情况。
06:30
And then there are certain conceptual things
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另外还有一些
06:32
that can also benefit from hand calculating,
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同样得益于人工运算的概念，
06:34
but I think they're relatively small in number.
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但我认为这种情况相对来说比较少。
06:36
One thing I often ask about
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比如说，我经常会问起
06:38
is ancient Greek and how this relates.
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以古希腊教学数学来做一个类比。
06:41
See, the thing we're doing right now
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我们现在的教学
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is we're forcing people to learn mathematics.
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总是强迫学生去学习数学。
06:45
It's a major subject.
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它是一个主科。
06:47
I'm not for one minute suggesting that, if people are interested in hand calculating
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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
06:50
or in following their own interests
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或者按照他们自己的意愿
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in any subject however bizarre --
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学习任何哪怕是毫无意义的学科；
06:54
they should do that.
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相反，他们应当这样做。
06:56
That's absolutely the right thing,
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因为让人们发展自己的兴趣爱好，
06:58
for people to follow their self-interest.
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完全就是一件正确的事。
07:00
I was somewhat interested in ancient Greek,
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我个人对古希腊怀有某种特殊的情结，
07:02
but I don't think that we should force the entire population
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但我不认为我们应该逼迫全体人民
07:05
to learn a subject like ancient Greek.
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都像古希腊学习数学的方法。
07:07
I don't think it's warranted.
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我不认为这样是值得的。
07:09
So I have this distinction between what we're making people do
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因此我知道我们应该区分迫使人们学习
07:12
and the subject that's sort of mainstream
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或是作为某种主流的学科
07:14
and the subject that, in a sense, people might follow with their own interest
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与人们在某种程度上自觉按照自己意愿学习的
07:17
and perhaps even be spiked into doing that.
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或甚至是钟情于此的学科。
07:19
So what are the issues people bring up with this?
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那人们为什么提出这一点呢？
07:22
Well one of them is, they say, you need to get the basics first.
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其中一点，他们会说，你必须先打好基础。
07:25
You shouldn't use the machine
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你在完全掌握学科的基础之前
07:27
until you get the basics of the subject.
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不许用机器。
07:29
So my usual question is, what do you mean by "basics?"
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那我通常的疑问就是，你所说的“基础”指的什么？
07:32
Basics of what?
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什么的基础？
07:34
Are the basics of driving a car
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这样说来，请问驾车的基础
07:36
learning how to service it, or design it for that matter?
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是否是怎样修车或者设计车模？
07:39
Are the basics of writing learning how to sharpen a quill?
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而写作的基础是否又是要学习如何削鹅毛笔？
07:43
I don't think so.
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我认为不是这样的。
07:45
I think you need to separate the basics of what you're trying to do
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我认为你需要将你所要做的事情的基础
07:48
from how it gets done
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理解成两部分如何得出结果
07:50
and the machinery of how it gets done
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与用机器如何得出结果区分开来。
07:54
and automation allows you to make that separation.
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自动化允许你作出如此的理解。
07:57
A hundred years ago, it's certainly true that to drive a car
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一百多年前，如果要学车你无疑是需要
08:00
you kind of needed to know a lot about the mechanics of the car
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知道某些汽车机械原理的
08:02
and how the ignition timing worked and all sorts of things.
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以及定时打火器等各零部件的工作原理。
08:06
But automation in cars
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但是汽车的自动化
08:08
allowed that to separate,
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实现两类学问的划分，
08:10
so driving is now a quite separate subject, so to speak,
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也就是说驾车现在与汽车工程学
08:13
from engineering of the car
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是相对独立的两个学科，
08:16
or learning how to service it.
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修车又是另外一回事。
08:20
So automation allows this separation
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正是自动化实现了这种区分，
08:22
and also allows -- in the case of driving,
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不仅在驾车的例子中是这样的自动化，
08:24
and I believe also in the future case of maths --
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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，
08:26
a democratized way of doing that.
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我们可以自主地选择怎样去做。
08:28
It can be spread across a much larger number of people
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数学可以向更多实际运用的人
08:30
who can really work with that.
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去传播。
08:33
So there's another thing that comes up with basics.
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关于基本技能又有另外一个观点。
08:35
People confuse, in my view,
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我认为，人们混淆
08:37
the order of the invention of the tools
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发明工具的次序
08:40
with the order in which they should use them for teaching.
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与在教学中应当运用这些工具的次序。
08:43
So just because paper was invented before computers,
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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，
08:46
it doesn't necessarily mean you get more to the basics of the subject
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用纸张教数学不一定代表
08:49
by using paper instead of a computer
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你会比用计算机
08:51
to teach mathematics.
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更了解数学的基础。
08:55
My daughter gave me a rather nice anecdote on this.
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我女儿的一则轶事为这一点做了生动地说明。
08:58
She enjoys making what she calls "paper laptops."
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她很喜欢制作她所说的“纸制笔记本电脑”。
09:01
(Laughter)
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（笑声）
09:03
So I asked her one day, "You know, when I was your age,
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有一天，我问她：“我在你这个年纪的时候，
09:05
I didn't make these.
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我都不会做这些。
09:07
Why do you think that was?"
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你觉得是为什么呢？”
09:09
And after a second or two, carefully reflecting,
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她很仔细地想了一两秒，
09:11
she said, "No paper?"
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然后她说，“没有纸吗？”
09:13
(Laughter)
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（笑声）
09:19
If you were born after computers and paper,
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如果你在计算机和纸张的发明之后出生，
09:21
it doesn't really matter which order you're taught with them in,
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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，
09:24
you just want to have the best tool.
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你仅仅需要用最佳的工具来学习数学。
09:26
So another one that comes up is "Computers dumb math down."
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那另一个观点就是“计算机使数学变蠢。”
09:29
That somehow, if you use a computer,
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在一定程度上的确如此，如果你用计算机，
09:31
it's all mindless button-pushing,
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这仅仅是无需思考的按键操作，
09:33
but if you do it by hand,
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但如果你进行人工笔头运算，
09:35
it's all intellectual.
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这完全又是智力训练。
09:37
This one kind of annoys me, I must say.
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我要说，这一点很困扰我。
09:40
Do we really believe
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我们难道真的以为
09:42
that the math that most people are doing in school
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现在通常人们在学校里
09:44
practically today
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学习的数学
09:46
is more than applying procedures
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真的比按照解题步骤做题学得多吗?
09:48
to problems they don't really understand, for reasons they don't get?
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他们大都还在用不理解的原理去解答他们不明白的问题。
09:51
I don't think so.
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至少我认为如此。
09:53
And what's worse, what they're learning there isn't even practically useful anymore.
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更严重的是，他们在学校所学的知识根本已经不再实用。
09:56
Might have been 50 years ago, but it isn't anymore.
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或许50年前还在用，但现在已经不再使用了。
09:59
When they're out of education, they do it on a computer.
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除了教育之外，他们都用计算机来完成。
10:02
Just to be clear, I think computers can really help with this problem,
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要清醒地知道，计算机已经完全解决了这一问题，
10:05
actually make it more conceptual.
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事实上使它更容易理解。
10:07
Now, of course, like any great tool,
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而如今，像任何伟大的工具一样，
10:09
they can be used completely mindlessly,
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它们可以彻底无须动脑筋地操作，
10:11
like turning everything into a multimedia show,
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就像将任何东西都变成一场多媒体的盛宴一样，
10:14
like the example I was shown of solving an equation by hand,
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就像这个解方程的例子，
10:17
where the computer was the teacher --
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计算机就好比是老师--
10:19
show the student how to manipulate and solve it by hand.
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它给学生演示如何现在用笔头进行运算和解题。
10:22
This is just nuts.
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这其实很荒谬。
10:24
Why are we using computers to show a student how to solve a problem by hand
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为什么我们要用计算机来演示如何用笔算解题呢？
10:27
that the computer should be doing anyway?
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明明计算机可以自行计算来解题。
10:29
All backwards.
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完全是倒退了。
10:31
Let me show you
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下面让我为你演示
10:33
that you can also make problems harder to calculate.
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计算机如何运算出更复杂的题目。
10:36
See, normally in school,
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通常在学校里，
10:38
you do things like solve quadratic equations.
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你会学习像解二次方程式这类题目。
10:41
But you see, when you're using a computer,
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但如果你用计算机来做，
10:44
you can just substitute.
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你仅需要做代换。
10:48
You can make it a quartic equation. Make it kind of harder, calculating-wise.
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就可以将它变成四次方程，运算上更复杂。
10:50
Same principles applied --
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同样的原理仍然适用，
10:52
calculations, harder.
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只是运算上更复杂。
10:54
And problems in the real world
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而现实世界里的问题
10:56
look nutty and horrible like this.
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的确就像这样复杂和难懂的，
10:58
They've got hair all over them.
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甚至更加棘手的问题。
11:00
They're not just simple, dumbed-down things that we see in school math.
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它们都不是我们学校数学中学的那些简单愚蠢的东西。
11:04
And think of the outside world.
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让我们想想外面的世界。
11:06
Do we really believe that engineering and biology
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难道我们真的以为工程学和生物学
11:08
and all of these other things
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等其他
11:10
that have so benefited from computers and maths
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得益于计算机和数学的学问
11:12
have somehow conceptually gotten reduced by using computers?
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会由于应用计算机而在某种程度上被明显削弱吗？
11:15
I don't think so -- quite the opposite.
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我认为，恰恰相反。
11:18
So the problem we've really got in math education
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我们在数学教育中的真正问题
11:21
is not that computers might dumb it down,
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并不是计算机使数学变得愚蠢，
11:24
but that we have dumbed-down problems right now.
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而是我们现在的教学使问题变得愚蠢。
11:27
Well, another issue people bring up
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好，那人们提到的另一点
11:29
is somehow that hand calculating procedures
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就是笔头运算的步骤在一定程度上
11:31
teach understanding.
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可以教导学生理解数学。
11:33
So if you go through lots of examples,
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因此如果你做过很多例题，
11:35
you can get the answer,
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你就可以算出答案，
11:37
you can understand how the basics of the system work better.
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你也可以更好地理解数学系统的基础。
11:40
I think there is one thing that I think very valid here,
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我认为，这个观点里有一点是我非常认同的，
11:43
which is that I think understanding procedures and processes is important.
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那就是我认为步骤和运算过程很重要。
11:47
But there's a fantastic way to do that in the modern world.
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但是在现代社会我们有一个绝佳的途径去这样做，
11:50
It's called programming.
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那就是编程。
11:53
Programming is how most procedures and processes
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现在多数程序和编码过程
11:55
get written down these days,
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大都以编程的方式写下来，
11:57
and it's also a great way
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而这也是一种很好的方式
11:59
to engage students much more
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去鼓励学生更多的参与，
12:01
and to check they really understand.
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检验他们是否真正理解数学问题。
12:03
If you really want to check you understand math
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如果你真的想检验你是否理解数学
12:05
then write a program to do it.
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就尝试编写一条程序。
12:08
So programming is the way I think we should be doing that.
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因此我认为编程就是我们应该做的方向。
12:11
So to be clear, what I really am suggesting here
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一言以蔽之，在此我想说的是
12:13
is we have a unique opportunity
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编程为我们提供了一个独特的机会：
12:15
to make maths both more practical
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一方面使数学更实用，
12:17
and more conceptual, simultaneously.
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同时另一方面又更加理论化。
12:20
I can't think of any other subject where that's recently been possible.
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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。
12:23
It's usually some kind of choice
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这通常像是一种
12:25
between the vocational and the intellectual.
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在职业规划或者是智力培养之间的两难选择。
12:27
But I think we can do both at the same time here.
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但我想运用计算机我们可以两方面同时实现。
12:32
And we open up so many more possibilities.
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还为我们又打开了许多可能性。
12:35
You can do so many more problems.
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我们可以解决更多的实际问题。
12:37
What I really think we gain from this
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我真切地感觉到这种教育的改革
12:39
is students getting intuition and experience
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可以让学生获得很多的知识和经验
12:42
in far greater quantities than they've ever got before.
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比之前多得多的直观的认识和经验的累积。
12:45
And experience of harder problems --
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和接触更加复杂的题目——
12:47
being able to play with the math, interact with it,
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畅游于数学的乐园，与数学沟通，
12:49
feel it.
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感受数学之美。
12:51
We want people who can feel the math instinctively.
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我们希望学生可以本能地感受数学。
12:54
That's what computers allow us to do.
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而这恰恰是计算机使这成为可能。
12:57
Another thing it allows us to do is reorder the curriculum.
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另一点是计算机使我们得以对教学课程重新排序。
13:00
Traditionally it's been by how difficult it is to calculate,
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以前是按照计算的难度编排课程教学的顺序，
13:02
but now we can reorder it
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但现在我们可以重新排序，
13:04
by how difficult it is to understand the concepts,
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按照概念理念的难度，
13:06
however hard the calculating.
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而不管计算的难度。
13:08
So calculus has traditionally been taught very late.
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因此微积分传统上是很晚才能教授的项目。
13:11
Why is this?
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原因是什么呢？
13:13
Well, it's damn hard doing the calculations, that's the problem.
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问题就在于微积分的计算相当困难。
13:17
But actually many of the concepts
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但事实上许多概念
13:19
are amenable to a much younger age group.
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是可以向更低年级的学生传授的。
13:22
This was an example I built for my daughter.
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这就是一个我为女儿所建的模型例子。
13:25
And very, very simple.
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而且非常简单易懂。
13:28
We were talking about what happens
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这个模型是用于理解
13:30
when you increase the number of sides of a polygon
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当多边形的边数增加时，图形的变化随之改变。
13:32
to a very large number.
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当边数增加到非常大的时候，
13:36
And of course, it turns into a circle.
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当然，多边形变成一个圆。
13:38
And by the way, she was also very insistent
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顺便一提，她对可以转换颜色这一点
13:40
on being able to change the color,
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特别乐此不疲，
13:42
an important feature for this demonstration.
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这也是这个展示的一个重要的特征。
13:46
You can see that this is a very early step
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你可以看到这是一个
13:49
into limits and differential calculus
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对极限和微积分非常初步的启蒙，
13:51
and what happens when you take things to an extreme --
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也可以让你直观地看到趋于极限时所发生的变化，
13:54
and very small sides and a very large number of sides.
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无限小边数的情况和无限多边数的情况。
13:56
Very simple example.
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非常浅显的例子。
13:58
That's a view of the world
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这是对世界的一种观察方式，
14:00
that we don't usually give people for many, many years after this.
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我们通常是很久很久之后才会教给学生这些。
14:03
And yet, that's a really important practical view of the world.
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但它却是对世界的一种非常重要且实用的观察。
14:06
So one of the roadblocks we have
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那现在我们要改革
14:09
in moving this agenda forward
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推进这个进程的路障
14:12
is exams.
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就在于考试。
14:14
In the end, if we test everyone by hand in exams,
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最后，假如我们考试的方式是人工笔算，
14:17
it's kind of hard to get the curricula changed
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就难以将目前的课程设计
14:20
to a point where they can use computers
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改革成
14:22
during the semesters.
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在教学中可以使用计算机。
14:25
And one of the reasons it's so important --
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而既然作为一个如此重要的原因
14:27
so it's very important to get computers in exams.
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在考试中使用计算机也是非常重要的。
14:30
And then we can ask questions, real questions,
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如此以来，我们就可以设置实际的问题，
14:33
questions like, what's the best life insurance policy to get? --
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比如说，“哪一种人寿保险的险种更适合我们？”
14:36
real questions that people have in their everyday lives.
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人们实际生活中遇到诸如此类的问题。
14:40
And you see, this isn't some dumbed-down model here.
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你可以看到，这就不是那些愚蠢的问题模型。
14:42
This is an actual model where we can be asked to optimize what happens.
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这是一个我们优化选择时可以用到的实际模型。
14:45
How many years of protection do I need?
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我需要多少年的险期？
14:47
What does that do to the payments
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这对保险费有什么影响？
14:49
and to the interest rates and so forth?
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对利率等其他的影响呢？
14:52
Now I'm not for one minute suggesting it's the only kind of question
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这里我没有刻意在暗示
14:55
that should be asked in exams,
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这是考试中应该考的唯一类型的问题，
14:57
but I think it's a very important type
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但是我认为至少这是非常重要的一类型，
14:59
that right now just gets completely ignored
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也是目前的教育完全忽视的，
15:02
and is critical for people's real understanding.
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这对人们的实际理解也是至关重要的。
15:05
So I believe [there is] critical reform
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因此我相信
15:08
we have to do in computer-based math.
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我们需要一场以计算机作为数学教学工具的改革。
15:10
We have got to make sure
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我们务必确保
15:12
that we can move our economies forward,
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通过让人们可以切实感受数学，
15:15
and also our societies,
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我们才可以推进经济发展和
15:17
based on the idea that people can really feel mathematics.
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社会进步。
15:22
This isn't some optional extra.
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这绝不是可有可无的一环。
15:25
And the country that does this first
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而我认为首先这样做的国家
15:27
will, in my view, leapfrog others
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一定会鹤立鸡群，
15:30
in achieving a new economy even,
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甚至取得更大的经济成就，
15:33
an improved economy,
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更完善的经济体系，
15:35
an improved outlook.
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和更卓越的国家面貌。
15:37
In fact, I even talk about us moving
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事实上，我甚至会说我们正在
15:39
from what we often call now the "knowledge economy"
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从我们现在经常提到的“知识经济”时代迈向
15:42
to what we might call a "computational knowledge economy,"
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可能被称作“计算机知识经济”的时代，
15:45
where high-level math is integral to what everyone does
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高端的数学知识对每个人
15:48
in the way that knowledge currently is.
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应用现代知识来说都不可或缺。
15:50
We can engage so many more students with this,
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我们可以让更多学生加入到这个改革，
15:53
and they can have a better time doing it.
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让他们在最佳的时机得到这样的教育。
15:56
And let's understand:
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我们要明白
15:58
this is not an incremental sort of change.
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这不是一场渐进式的改革。
16:02
We're trying to cross the chasm here
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我们要试图跨越
16:04
between school math and the real-world math.
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学校数学和实际数学之间的鸿沟。
16:06
And you know if you walk across a chasm,
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你可以看到如果我们是走过这个鸿沟的话，
16:08
you end up making it worse than if you didn't start at all --
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我们一定会跌得粉身碎骨，
16:11
bigger disaster.
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甚至比没有开始改革更糟糕。
16:13
No, what I'm suggesting
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这不是我想要的局面，
16:15
is that we should leap off,
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我想说，我们应该跳跃，
16:17
we should increase our velocity
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我们必须加快我们的起跳速度，
16:19
so it's high,
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这样我们就可以跳得高，
16:21
and we should leap off one side and go the other --
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我们应该从鸿沟的一头跳到另外一头，
16:24
of course, having calculated our differential equation very carefully.
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当然要准确地计算这里微分方程。
16:27
(Laughter)
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（笑声）
16:29
So I want to see
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因此我想看到的是
16:31
a completely renewed, changed math curriculum
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一套全新的、经过改良的数学教学课程
16:33
built from the ground up,
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在应用计算机的基础上
16:35
based on computers being there,
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架构而起，
16:37
computers that are now ubiquitous almost.
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目前计算机已经是在各地非常普遍的
16:39
Calculating machines are everywhere
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运算工具，
16:41
and will be completely everywhere in a small number of years.
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相信在不久的将来它会彻底覆盖每一个地方。
16:44
Now I'm not even sure if we should brand the subject as math,
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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，
16:48
but what I am sure is
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但我确信
16:50
it's the mainstream subject of the future.
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这门课将是未来学科的主流。
16:53
Let's go for it,
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让我们共同努力吧。
16:56
and while we're about it,
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并且在学习它的过程中
16:58
let's have a bit of fun,
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让大家
17:00
for us, for the students and for TED here.
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所有人，广大的学生以及TED这里的观众享受一些乐趣！
17:03
Thanks.
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谢谢
17:05
(Applause)
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（掌声）
ABOUT THE SPEAKER
Conrad Wolfram - MathematicianConrad Wolfram runs the worldwide arm of Wolfram Research, the mathematical lab behind the cutting-edge knowledge engine Wolfram Alpha.
Why you should listen
Conrad Wolfram is the strategic director of Wolfram Research, where his job, in a nutshell, is understanding and finding new uses for the Mathematica technology. Wolfram is especially passionate about finding uses for Mathematica outside of pure computation, using it as a development platform for products that help communicate big ideas. The Demonstrations tool, for instance, makes a compelling case for never writing out another equation -- instead displaying data in interactive, graphical form.
Wolfram's work points up the changing nature of math in the past 30 years, as we've moved from adding machines to calculators to sophisticated math software, allowing us to achieve ever more complex computational feats. But, Wolfram says, many schools are still focused on hand-calculating; using automation, such as a piece of software, to do math is sometimes seen as cheating. This keeps schools from spending the time they need on the new tools of science and mathematics. As they gain significance for everyday living, he suggests, we need to learn to take advantage of these tools and learn to use them young. Learn more at computerbasedmath.org.
Conrad Wolfram | Speaker | TED.com