TEDGlobal 2010
Conrad Wolfram: Teaching kids real math with computers
康拉德·沃夫朗：使用電腦教導小孩們真正的數學
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從飛彈到股市，人類的許多驚人創舉皆拜賜於數學的威力。那麼，為什麼小孩們對數學失去興趣？康拉德·沃夫朗說，我們教導數學的那個用手求算的部分，不僅只是麻煩，而且與真正的數學以及真實的世界毫無關係。他解說了他的激進想法：透過電腦程式設計來教導小孩們學習數學。
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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當今的數學教育有個真正的問題
00:19
Basically, no one's very happy.
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基本上大家都不滿意
00:22
Those learning it
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學習數學的人認為
00:24
think it's disconnected,
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數學與生活無干
00:26
uninteresting and hard.
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既無趣又困難
00:28
Those trying to employ them
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利用數學的人則認為
00:30
think they don't know enough.
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還學得不夠
00:32
Governments realize that it's a big deal for our economies,
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政府明白數學對經濟至為重要
00:35
but don't know how to fix it.
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卻不知從何下手改善
00:38
And teachers are also frustrated.
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教師也感到沮喪
00:40
Yet math is more important to the world
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然而數學對今日世界的重要性
00:43
than at any point in human history.
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遠勝於人類歷史上的任何時點
00:45
So at one end we've got falling interest
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我們一面有數學學習
00:47
in education in math,
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興趣低落的問題
00:49
and at the other end we've got a more mathematical world,
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另一面卻要面對更加需要數學的世界
00:52
a more quantitative world than we ever have had.
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這個世界比以往更加需要量化
00:56
So what's the problem, why has this chasm opened up,
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到底出了什麼問題？
00:58
and what can we do to fix it?
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怎會裂出這條鴻溝？要如何修補？
01:01
Well actually, I think the answer
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其實我認為答案
01:03
is staring us right in the face:
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就在眼前：
01:05
Use computers.
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多多使用電腦
01:07
I believe
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我相信
01:09
that correctly using computers
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正確地使用電腦
01:11
is the silver bullet
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是讓數學教育
01:13
for making math education work.
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起死回生的妙藥
01:16
So to explain that,
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要清楚解釋之前
01:18
let me first talk a bit about what math looks like in the real world
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先談數學在真實世界中是怎個樣態
01:21
and what it looks like in education.
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它在教學中又是怎個樣態
01:23
See, in the real world
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瞧，在現實世界中
01:25
math isn't necessarily done by mathematicians.
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數學並不一定是數學家的專用品
01:28
It's done by geologists,
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地質學家用到它
01:30
engineers, biologists,
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工程師、生物學家
01:32
all sorts of different people --
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各行各業都用到數學
01:34
modeling and simulation.
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用來建模和模擬
01:36
It's actually very popular.
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實際上這是很通行的
01:38
But in education it looks very different --
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但在教育中情況則大大不同
01:41
dumbed-down problems, lots of calculating,
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無聊的問題－有許多計算題
01:43
mostly by hand.
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大多要動手計算
01:46
Lots of things that seem simple
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許多東西原本簡單
01:48
and not difficult like in the real world,
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如同在現實世界中那麼容易
01:50
except if you're learning it.
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不學沒事、一學就煩
01:53
And another thing about math:
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還有一件關於數學的事：
01:55
math sometimes looks like math --
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數學有時貌似數學
01:57
like in this example here --
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比方說這裡這個例子
02:00
and sometimes it doesn't --
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有時不像數學
02:02
like "Am I drunk?"
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例如「我喝醉了嗎？」
02:07
And then you get an answer that's quantitative in the modern world.
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這問題在今日世界得到的是一個量化的答案
02:10
You wouldn't have expected that a few years back.
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幾年前你想都想不到會是這樣
02:13
But now you can find out all about --
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但現在你可以發現一切－
02:16
unfortunately, my weight is a little higher than that, but --
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不幸我的體重比那還重了些，不過－
02:19
all about what happens.
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一切有關世上發生的事情
02:21
So let's zoom out a bit and ask,
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我們縮小這個來問個問題
02:23
why are we teaching people math?
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為什麼要教數學？
02:25
What's the point of teaching people math?
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教數學有什麼意義？
02:28
And in particular, why are we teaching them math in general?
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更確切地問，為何要廣泛進行數學教育？
02:31
Why is it such an important part of education
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為何數學是教育的重要一環
02:34
as a sort of compulsory subject?
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因何成為必修的課程？
02:36
Well, I think there are about three reasons:
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我想這大概有三個理由：
02:39
technical jobs
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技術方面的工作
02:41
so critical to the development of our economies,
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對我們的經濟發展具有關鍵性
02:44
what I call "everyday living" --
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我稱之為我們每日的生活
02:48
to function in the world today,
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在今日世界生活
02:50
you've got to be pretty quantitative,
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你必須很懂得量化
02:52
much more so than a few years ago:
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比幾年前還必須懂得
02:54
figure out your mortgages,
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例如弄清楚你的房貸
02:56
being skeptical of government statistics, those kinds of things --
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懷疑政府提出的統計數據等類的東西
03:00
and thirdly, what I would call something like
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還有我稱之為
03:03
logical mind training, logical thinking.
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邏輯推理訓練、邏輯思考的東西
03:06
Over the years
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多年來我們投入許多資源
03:08
we've put so much in society
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教導社會上的人
03:10
into being able to process and think logically. It's part of human society.
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如何邏輯思考－這是人類社會生活的一部分
03:13
It's very important to learn that
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學習邏輯思考是很重要的
03:15
math is a great way to do that.
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數學是學習邏輯思考的好方法
03:17
So let's ask another question.
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那麼我們來問另一個問題
03:19
What is math?
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數學到底是什麼？
03:21
What do we mean when we say we're doing math,
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我們說利用數學或教導數學
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or educating people to do math?
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到底什麼意思？
03:25
Well, I think it's about four steps, roughly speaking,
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我認為這可以粗分為四個步驟
03:28
starting with posing the right question.
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首先是要提出正確的問題
03:30
What is it that we want to ask? What is it we're trying to find out here?
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問的是什麼問題？找到的是怎樣的答案？
03:33
And this is the thing most screwed up in the outside world,
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在利用數學的所有部分之外
03:35
beyond virtually any other part of doing math.
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一般最常搞砸的就是
03:38
People ask the wrong question,
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問錯問題
03:40
and surprisingly enough, they get the wrong answer,
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得到錯誤的答案
03:42
for that reason, if not for others.
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原因常常就是問錯問題
03:44
So the next thing is take that problem
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接下來就是
03:46
and turn it from a real world problem
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把現實世界的問題
03:48
into a math problem.
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轉換成數學的問題
03:50
That's stage two.
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這是第二步
03:52
Once you've done that, then there's the computation step.
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完成這個步驟後就是求算
03:55
Turn it from that into some answer
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從這裡推演求算出
03:57
in a mathematical form.
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數學形式的答案
04:00
And of course, math is very powerful at doing that.
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這方面數學當然有其威力
04:02
And then finally, turn it back to the real world.
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然後再把答案轉換到現實世界
04:04
Did it answer the question?
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答案是否解答了問題？
04:06
And also verify it -- crucial step.
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同時也驗證答案－這一步很關鍵
04:10
Now here's the crazy thing right now.
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但最瘋狂的是：
04:12
In math education,
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我們在數學教育中
04:14
we're spending about perhaps 80 percent of the time
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花費80%的時間
04:17
teaching people to do step three by hand.
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教導學生用手做第三步
04:20
Yet, that's the one step computers can do
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可這正是電腦能做的一步
04:22
better than any human after years of practice.
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比學習多年的人都要做得好
04:25
Instead, we ought to be using computers
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所以我們應該把這一步
04:28
to do step three
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交給電腦來做
04:30
and using the students to spend much more effort
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讓學生多花點功夫學習如何做
04:33
on learning how to do steps one, two and four --
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第一、第二和第四個步－
04:35
conceptualizing problems, applying them,
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將問題概念化、應用那些概念
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getting the teacher to run them through how to do that.
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讓教師示範如何進行這些步驟
04:41
See, crucial point here:
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請看，關鍵點就在這裡：
04:43
math is not equal to calculating.
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數學並不等於求算
04:45
Math is a much broader subject than calculating.
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數學是比求算還要廣延的課題
04:48
Now it's understandable that this has all got intertwined
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我們都明白這經過幾百年
04:51
over hundreds of years.
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都混雜在一起了
04:53
There was only one way to do calculating and that was by hand.
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以往只有用手求算這麼個辦法
04:56
But in the last few decades
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不過在過去這幾十年來
04:58
that has totally changed.
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局面完全改觀了
05:00
We've had the biggest transformation of any ancient subject
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我們經歷了電腦這個最大的變革
05:03
that I could ever imagine with computers.
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比任何古往的科目都要巨大
05:07
Calculating was typically the limiting step,
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求算向來是讓人受到限制的步驟
05:09
and now often it isn't.
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很少例外
05:11
So I think in terms of the fact that math
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因此我認為事實上數學
05:13
has been liberated from calculating.
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已經逐漸從求算的步驟解放出來了
05:16
But that math liberation didn't get into education yet.
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可是數學的解放卻還沒有進入教育裡頭
05:19
See, I think of calculating, in a sense,
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這麼說吧，我把求算看做
05:21
as the machinery of math.
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數學的機械部分
05:23
It's the chore.
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是勞煩的差事
05:25
It's the thing you'd like to avoid if you can, like to get a machine to do.
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若能讓機器來做，這是你會盡可能避免的事
05:29
It's a means to an end, not an end in itself,
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是達到目的的工具，並不是目的本身
05:34
and automation allows us
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自動化使得我們
05:36
to have that machinery.
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有了那樣的機械
05:38
Computers allow us to do that --
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電腦使我們可以避免煩差事
05:40
and this is not a small problem by any means.
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這可是非同小可的一件事
05:43
I estimated that, just today, across the world,
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我估計在當今全世界上
05:46
we spent about 106 average world lifetimes
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我們平均花費大約106個生命時間
05:49
teaching people how to calculate by hand.
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教導人們如何用手求算
05:52
That's an amazing amount of human endeavor.
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那可是驚人的人力耗費
05:55
So we better be damn sure --
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那麼我們最好能確定－
05:57
and by the way, they didn't even have fun doing it, most of them --
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可大多數人都學得很不愉快
06:00
so we better be damn sure
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那麼我們最好能確定
06:02
that we know why we're doing that
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我們為什麼那麼做
06:04
and it has a real purpose.
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而且那麼做有個好理由
06:06
I think we should be assuming computers
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我認為我們應該讓電腦來
06:08
for doing the calculating
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承擔求算的任務
06:10
and only doing hand calculations where it really makes sense to teach people that.
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只有在確實有意義教導求算時才用手求算
06:13
And I think there are some cases.
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我還認為有些情況下
06:15
For example: mental arithmetic.
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可以使用心算
06:17
I still do a lot of that, mainly for estimating.
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我還常心算，主要是用來做估計
06:20
People say, "Is such and such true?"
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有人說那樣、那樣是真的
06:22
And I'll say, "Hmm, not sure." I'll think about it roughly.
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我說：哼，不確定，我來想一下
06:24
It's still quicker to do that and more practical.
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求算還是快些，也實際些
06:26
So I think practicality is one case
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因此我想講求實際的是
06:28
where it's worth teaching people by hand.
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教導人們求算的一個好理由
06:30
And then there are certain conceptual things
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然後還有些計算方面的事
06:32
that can also benefit from hand calculating,
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動手求算還是有益處的
06:34
but I think they're relatively small in number.
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不過我認為那些情況畢竟少數
06:36
One thing I often ask about
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我常常思考一件事
06:38
is ancient Greek and how this relates.
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是關於古希臘及與其相關的事物
06:41
See, the thing we're doing right now
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瞧，我們現在做的是
06:43
is we're forcing people to learn mathematics.
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強迫人們學習數學
06:45
It's a major subject.
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這是重大的科目
06:47
I'm not for one minute suggesting that, if people are interested in hand calculating
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我並不暗示說如果有人喜歡
06:50
or in following their own interests
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用手求算或出於個人興趣
06:52
in any subject however bizarre --
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無論如何詭異的科目－
06:54
they should do that.
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他們都應該去做
06:56
That's absolutely the right thing,
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隨著自己的興趣做事
06:58
for people to follow their self-interest.
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那絕對是沒有錯的事
07:00
I was somewhat interested in ancient Greek,
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我以往對古希臘很有興趣
07:02
but I don't think that we should force the entire population
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但我不認為我們應該強迫所有的人
07:05
to learn a subject like ancient Greek.
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學習像古希臘這樣的科目
07:07
I don't think it's warranted.
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我不認為這會有什麼成功的保證
07:09
So I have this distinction between what we're making people do
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我在我們強迫別人做的事和所謂
07:12
and the subject that's sort of mainstream
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主流科目之間做了區分
07:14
and the subject that, in a sense, people might follow with their own interest
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重點是人們有自己的興趣
07:17
and perhaps even be spiked into doing that.
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而且可能受到激勵而去做
07:19
So what are the issues people bring up with this?
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那麼為什麼要教導別人某樣東西呢？
07:22
Well one of them is, they say, you need to get the basics first.
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其中一項就是人家說的要學會基礎
07:25
You shouldn't use the machine
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還沒學會某科目的基礎之前
07:27
until you get the basics of the subject.
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不要使用機器
07:29
So my usual question is, what do you mean by "basics?"
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那我通常會問：你說的基礎是什麼？
07:32
Basics of what?
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什麼的基礎呢？
07:34
Are the basics of driving a car
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開車的基礎是學會
07:36
learning how to service it, or design it for that matter?
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如何保養汽車或設計汽車那回事嗎？
07:39
Are the basics of writing learning how to sharpen a quill?
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寫作的基礎是學會削羽毛管嗎？
07:43
I don't think so.
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我可不這麼認為
07:45
I think you need to separate the basics of what you're trying to do
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我想你得把你嘗試要做的事的基礎
07:48
from how it gets done
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從如何把它做成分開來
07:50
and the machinery of how it gets done
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也從把它做成的機械分開來
07:54
and automation allows you to make that separation.
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自動化讓你能做這樣的區分
07:57
A hundred years ago, it's certainly true that to drive a car
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也許沒錯，一百年前你要開車
08:00
you kind of needed to know a lot about the mechanics of the car
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那你得多了解汽車的機械原理
08:02
and how the ignition timing worked and all sorts of things.
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以及內燃機點火的時點等等
08:06
But automation in cars
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但是汽車上的自動化
08:08
allowed that to separate,
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使得那些都可以分開了
08:10
so driving is now a quite separate subject, so to speak,
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因此開車可說已經變成另外一個科目
08:13
from engineering of the car
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跟汽車設計
08:16
or learning how to service it.
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或學習如何保養已經毫無關係
08:20
So automation allows this separation
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自動化造成這樣的區分
08:22
and also allows -- in the case of driving,
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而且不僅在開車這回事上
08:24
and I believe also in the future case of maths --
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我相信將來在數學上
08:26
a democratized way of doing that.
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也會容許民主化的利用方式
08:28
It can be spread across a much larger number of people
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可以讓很大多數的人
08:30
who can really work with that.
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都能真正用來做事
08:33
So there's another thing that comes up with basics.
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那麼關於基礎還有一件事
08:35
People confuse, in my view,
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在我看來就是
08:37
the order of the invention of the tools
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人們對於發明工具的順序和他們應該
08:40
with the order in which they should use them for teaching.
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利用工具來進行教學的順序感到困惑
08:43
So just because paper was invented before computers,
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只因為紙張比電腦早先發明
08:46
it doesn't necessarily mean you get more to the basics of the subject
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並不表示你要獲得事物的基礎
08:49
by using paper instead of a computer
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用紙張來教數學
08:51
to teach mathematics.
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比用電腦來得根本
08:55
My daughter gave me a rather nice anecdote on this.
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我的女兒有個小插曲很好玩
08:58
She enjoys making what she calls "paper laptops."
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她喜歡用紙張做她所謂的紙張膝上型電腦
09:01
(Laughter)
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（笑聲）
09:03
So I asked her one day, "You know, when I was your age,
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有天我就問她：「在你這個年紀時
09:05
I didn't make these.
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我並沒有做這些東西
09:07
Why do you think that was?"
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你怎麼會以為以前有這種東西？」
09:09
And after a second or two, carefully reflecting,
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她專心想了一兩秒鐘
09:11
she said, "No paper?"
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說：「那個時候沒有紙張嗎？」
09:13
(Laughter)
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（笑聲）
09:19
If you were born after computers and paper,
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如果你在有電腦和紙張之後才出生的
09:21
it doesn't really matter which order you're taught with them in,
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那麼先用什麼來教你做事都無妨了
09:24
you just want to have the best tool.
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你要的是最好的工具
09:26
So another one that comes up is "Computers dumb math down."
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還有另外一個說法是「電腦讓人數學無能」
09:29
That somehow, if you use a computer,
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說得好像你如果用電腦
09:31
it's all mindless button-pushing,
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那都是毫無意識的按鍵盤
09:33
but if you do it by hand,
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而如果你用手做同樣的事
09:35
it's all intellectual.
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那就是純然的讀書高了
09:37
This one kind of annoys me, I must say.
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這種說法聽了教人不爽
09:40
Do we really believe
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我們還真的相信
09:42
that the math that most people are doing in school
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當今之世大多數人在學校裡
09:44
practically today
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學的數學還真的是
09:46
is more than applying procedures
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不明所以就把某些程序
09:48
to problems they don't really understand, for reasons they don't get?
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應用到他們並不明白的問題上？
09:51
I don't think so.
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我可不這麼認為
09:53
And what's worse, what they're learning there isn't even practically useful anymore.
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而更糟的是他們所學的已不復實用
09:56
Might have been 50 years ago, but it isn't anymore.
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也許50年前還實用，現在不實用了
09:59
When they're out of education, they do it on a computer.
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他們出了社會用的是電腦
10:02
Just to be clear, I think computers can really help with this problem,
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再說清楚，我認為這個問題電腦真的能幫上忙
10:05
actually make it more conceptual.
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讓這件事更符合概念
10:07
Now, of course, like any great tool,
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現在呢，就像任何好工具一樣
10:09
they can be used completely mindlessly,
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電腦可能毫無意識地被使用著
10:11
like turning everything into a multimedia show,
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例如把所有的東西都弄成多媒體表演
10:14
like the example I was shown of solving an equation by hand,
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如同我指出的那個用手解決方程式的例子那樣
10:17
where the computer was the teacher --
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既然電腦可以是教師
10:19
show the student how to manipulate and solve it by hand.
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還教學生怎麼用手解決方程式不成？
10:22
This is just nuts.
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這有夠蠢的
10:24
Why are we using computers to show a student how to solve a problem by hand
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為什麼用電腦來教學生如何用手解答問題？
10:27
that the computer should be doing anyway?
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電腦明明可以解答問題的！
10:29
All backwards.
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全都倒反了
10:31
Let me show you
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我來告訴你
10:33
that you can also make problems harder to calculate.
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你還可以讓問題更難求算
10:36
See, normally in school,
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通常在學校裡
10:38
you do things like solve quadratic equations.
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你做的是解答二次方程式
10:41
But you see, when you're using a computer,
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但是你若利用電腦
10:44
you can just substitute.
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你用替代的方法就行了
10:48
You can make it a quartic equation. Make it kind of harder, calculating-wise.
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弄個四次方程式，把它弄得較難求算
10:50
Same principles applied --
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同樣的原則還是適用－
10:52
calculations, harder.
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求算困難些
10:54
And problems in the real world
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而真實世界裡的問題
10:56
look nutty and horrible like this.
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看起來像這樣困難、恐怖
10:58
They've got hair all over them.
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生毛帶角的，恐怖得要死
11:00
They're not just simple, dumbed-down things that we see in school math.
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看來不像學校數學裡那樣簡單、無聊的東西
11:04
And think of the outside world.
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想一想外面的世界
11:06
Do we really believe that engineering and biology
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我們真的相信工程和生物
11:08
and all of these other things
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還有所有那些受惠於
11:10
that have so benefited from computers and maths
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電腦和數學良多的東西
11:12
have somehow conceptually gotten reduced by using computers?
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由於使用了電腦在概念上就扣了分嗎？
11:15
I don't think so -- quite the opposite.
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我可不這麼認為，應該反而加了分
11:18
So the problem we've really got in math education
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因此我們在數學教育裡遇到的問題
11:21
is not that computers might dumb it down,
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並不是電腦可能使數學變得無聊
11:24
but that we have dumbed-down problems right now.
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而是我們現在就已經把問題弄得無聊
11:27
Well, another issue people bring up
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好，另外還有個說法
11:29
is somehow that hand calculating procedures
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是說用手求算的程序
11:31
teach understanding.
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多少有助於教導理解
11:33
So if you go through lots of examples,
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你做了許多範例題
11:35
you can get the answer,
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你就能得到答案－
11:37
you can understand how the basics of the system work better.
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你就更能理解系統的基礎是如何運作的
11:40
I think there is one thing that I think very valid here,
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我想這說法裡有個地方是有根據的
11:43
which is that I think understanding procedures and processes is important.
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我是說理解程序和過程有其重要性
11:47
But there's a fantastic way to do that in the modern world.
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但現代世界裡有個新的方式來進行理解
11:50
It's called programming.
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它就叫做程式設計
11:53
Programming is how most procedures and processes
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程式設計是現今寫下
11:55
get written down these days,
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大多數程序和過程的方法
11:57
and it's also a great way
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而且這是很好的方法
11:59
to engage students much more
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讓學生能更深入檢驗
12:01
and to check they really understand.
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他們真的理解了些什麼
12:03
If you really want to check you understand math
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如果你真的要檢驗你理解了數學
12:05
then write a program to do it.
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那就寫個程式做做看
12:08
So programming is the way I think we should be doing that.
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因此我認為程式設計就是我們進行檢驗的方法
12:11
So to be clear, what I really am suggesting here
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再說清楚，我在此提議的是
12:13
is we have a unique opportunity
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我們有了獨特的機會
12:15
to make maths both more practical
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同時讓數學變得
12:17
and more conceptual, simultaneously.
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更實用也更符合概念
12:20
I can't think of any other subject where that's recently been possible.
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我想不到最近有其它科目像數學這樣有這種可能
12:23
It's usually some kind of choice
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這通常像是在天份發掘
12:25
between the vocational and the intellectual.
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和智識培養之間做選擇
12:27
But I think we can do both at the same time here.
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但我認為我們在這頂上可以兩者兼得
12:32
And we open up so many more possibilities.
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而且我們會因此開拓出許多可能性
12:35
You can do so many more problems.
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可以藉此同時處理許多問題
12:37
What I really think we gain from this
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我真的認為我們可以
12:39
is students getting intuition and experience
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由此達到讓比以往更多的學生
12:42
in far greater quantities than they've ever got before.
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獲得直覺和經驗的目的
12:45
And experience of harder problems --
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而且是得到處理更困難
12:47
being able to play with the math, interact with it,
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問題的經驗－能夠玩數學、
12:49
feel it.
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與數學互動、感受到數學
12:51
We want people who can feel the math instinctively.
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我們要人們能夠本能地感受到數學
12:54
That's what computers allow us to do.
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電腦讓我們能夠做到這點
12:57
Another thing it allows us to do is reorder the curriculum.
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電腦也讓我們能夠重新安排教學課程
13:00
Traditionally it's been by how difficult it is to calculate,
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傳統至今求算一直是困難的
13:02
but now we can reorder it
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現在我們可以依據數學概念
13:04
by how difficult it is to understand the concepts,
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理解的難易度重新安排課程
13:06
however hard the calculating.
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無論求算會是多麼困難
13:08
So calculus has traditionally been taught very late.
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比方說微積分傳統上都很晚才教
13:11
Why is this?
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為什麼這樣呢？
13:13
Well, it's damn hard doing the calculations, that's the problem.
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是呀，問題就在求算非常困難
13:17
But actually many of the concepts
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但事實上有許多概念
13:19
are amenable to a much younger age group.
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是許多年齡低很多的人所能理解的
13:22
This was an example I built for my daughter.
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這裡有個我給我女兒設計的例題
13:25
And very, very simple.
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非常非常簡單
13:28
We were talking about what happens
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我們談的是如果多邊形的邊
13:30
when you increase the number of sides of a polygon
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數量增加到很大很大
13:32
to a very large number.
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那個多邊形會變成怎樣
13:36
And of course, it turns into a circle.
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當然啦，那會變成一個圓形
13:38
And by the way, she was also very insistent
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順道說一下，她也很堅持
13:40
on being able to change the color,
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要能夠改變顏色
13:42
an important feature for this demonstration.
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這對這個示範是很重要的
13:46
You can see that this is a very early step
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這裡看到的是極限和微分
13:49
into limits and differential calculus
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概念的初步
13:51
and what happens when you take things to an extreme --
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那麼把多邊形的邊減少到極限會是怎樣
13:54
and very small sides and a very large number of sides.
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這樣：極少量的邊和極大量的邊
13:56
Very simple example.
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很簡單的範例題
13:58
That's a view of the world
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這種看世界的觀點
14:00
that we don't usually give people for many, many years after this.
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許多年前還沒有電腦時我們不會讓人們看到
14:03
And yet, that's a really important practical view of the world.
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但這是一種看世界的重要觀點
14:06
So one of the roadblocks we have
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那麼我們做這樣改變的路程上
14:09
in moving this agenda forward
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遇到的一個絆腳石
14:12
is exams.
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就是考試
14:14
In the end, if we test everyone by hand in exams,
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如果在考試裡還是測驗用手求算
14:17
it's kind of hard to get the curricula changed
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那麼在教學課程裡
14:20
to a point where they can use computers
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就很難改變成
14:22
during the semesters.
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學生們可以在學期裡使用電腦
14:25
And one of the reasons it's so important --
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很重要的一個理由－
14:27
so it's very important to get computers in exams.
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使用電腦考試因此是很重要的
14:30
And then we can ask questions, real questions,
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然後我們就可以問問題，問真正的問題
14:33
questions like, what's the best life insurance policy to get? --
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例如，最佳壽險會是怎樣的保單內容？－
14:36
real questions that people have in their everyday lives.
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大家在每日生活中會遇到的真正問題
14:40
And you see, this isn't some dumbed-down model here.
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瞧，這一點也不是無聊的模型
14:42
This is an actual model where we can be asked to optimize what happens.
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這種實際的模型可用來求問如何最佳化
14:45
How many years of protection do I need?
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我的壽險要擔保多少年呢？
14:47
What does that do to the payments
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那對保費會有怎樣的影響呢？
14:49
and to the interest rates and so forth?
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對利率等等會有怎樣的影響呢？
14:52
Now I'm not for one minute suggesting it's the only kind of question
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我可不是說這是在考試裡問的
14:55
that should be asked in exams,
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唯一類型的問題
14:57
but I think it's a very important type
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不過我認為這個類型很重要
14:59
that right now just gets completely ignored
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目前這方面完全被忽略了
15:02
and is critical for people's real understanding.
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但對於真正的理解卻是很關鍵的
15:05
So I believe [there is] critical reform
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因此我相信我們必須進行重大的
15:08
we have to do in computer-based math.
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以電腦為基的數學教學變革
15:10
We have got to make sure
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我們得確保
15:12
that we can move our economies forward,
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讓我們的經濟和社會
15:15
and also our societies,
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能夠向上提升
15:17
based on the idea that people can really feel mathematics.
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根據的想法是人人都真正能感受到數學
15:22
This isn't some optional extra.
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這可不是額外的選擇
15:25
And the country that does this first
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哪個國家先做
15:27
will, in my view, leapfrog others
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哪個國家就會領先一步
15:30
in achieving a new economy even,
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達到新經濟的平衡
15:33
an improved economy,
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達到經濟的改善
15:35
an improved outlook.
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達到前景的改善
15:37
In fact, I even talk about us moving
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其實我談的甚至牽涉到
15:39
from what we often call now the "knowledge economy"
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把我們常說的知識經濟推移到
15:42
to what we might call a "computational knowledge economy,"
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我們或可稱為計算知識經濟的地步
15:45
where high-level math is integral to what everyone does
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那時高端數學是人人從事工作的一部分
15:48
in the way that knowledge currently is.
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如同今日從事工作需要知識一般
15:50
We can engage so many more students with this,
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我們能夠讓更多學生深入參與
15:53
and they can have a better time doing it.
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讓他們的數學學習更加順暢
15:56
And let's understand:
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我們必須理解
15:58
this is not an incremental sort of change.
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這種改變不是漸進式的改變
16:02
We're trying to cross the chasm here
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我們正在嘗試的是要跨過
16:04
between school math and the real-world math.
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學校數學與真實世界之間的鴻溝
16:06
And you know if you walk across a chasm,
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大家都知道試圖跨過鴻溝時
16:08
you end up making it worse than if you didn't start at all --
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事態往往是比不嘗試之前還要嚴重
16:11
bigger disaster.
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災難變得更大
16:13
No, what I'm suggesting
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不，我提議的是
16:15
is that we should leap off,
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我們要跳越過去
16:17
we should increase our velocity
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我們應該加快速度
16:19
so it's high,
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讓速度變得很快
16:21
and we should leap off one side and go the other --
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我們應該從一邊飛躍起來跳到另一邊－
16:24
of course, having calculated our differential equation very carefully.
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當然得先很謹慎地求算我們的微分方程式
16:27
(Laughter)
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（笑聲）
16:29
So I want to see
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因此我想看到
16:31
a completely renewed, changed math curriculum
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一個完全更新改變的數學教程
16:33
built from the ground up,
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從根底開始建構
16:35
based on computers being there,
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以電腦的存在為基礎
16:37
computers that are now ubiquitous almost.
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電腦現今已經到處存在
16:39
Calculating machines are everywhere
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計算機器到處都是
16:41
and will be completely everywhere in a small number of years.
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而且在短短的幾年內會是無所不在
16:44
Now I'm not even sure if we should brand the subject as math,
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我現在還不確定這個科目是否應該稱為數學
16:48
but what I am sure is
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但我能確定的是
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it's the mainstream subject of the future.
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它會是將來的主流科目
16:53
Let's go for it,
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我們先來迎接吧
16:56
and while we're about it,
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既然我們要做
16:58
let's have a bit of fun,
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那就讓它對我們、對學生
17:00
for us, for the students and for TED here.
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和對TED這裡的人能提供一點樂趣
17:03
Thanks.
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謝謝各位
17:05
(Applause)
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（掌聲）
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