ABOUT THE SPEAKER
Conrad Wolfram - Mathematician
Conrad 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.

More profile about the speaker
Conrad Wolfram | Speaker | TED.com
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

Filmed:
1,742,493 views

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。
- Mathematician
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我們已經 got a real真實 problem問題 with math數學 education教育 right now.
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當今的數學教育有個真正的問題
00:19
Basically基本上, no one's那些 very happy快樂.
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基本上大家都不滿意
00:22
Those learning學習 it
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學習數學的人認為
00:24
think it's disconnected斷開的,
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數學與生活無干
00:26
uninteresting枯燥 and hard.
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既無趣又困難
00:28
Those trying to employ採用 them
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利用數學的人則認為
00:30
think they don't know enough足夠.
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還學得不夠
00:32
Governments政府 realize實現 that it's a big deal合同 for our economies經濟,
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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固定 it.
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卻不知從何下手改善
00:38
And teachers教師 are also frustrated受挫.
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教師也感到沮喪
00:40
Yet然而 math數學 is more important重要 to the world世界
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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人的 history歷史.
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遠勝於人類歷史上的任何時點
00:45
So at one end結束 we've我們已經 got falling落下 interest利益
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我們一面有數學學習
00:47
in education教育 in math數學,
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興趣低落的問題
00:49
and at the other end結束 we've我們已經 got a more mathematical數學的 world世界,
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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世界 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問題, why has this chasm裂口 opened打開 up,
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到底出了什麼問題?
00:58
and what can we do to fix固定 it?
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怎會裂出這條鴻溝?要如何修補?
01:01
Well actually其實, I think the answer回答
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其實我認為答案
01:03
is staring凝視 us right in the face面對:
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就在眼前:
01:05
Use computers電腦.
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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正確地 using運用 computers電腦
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正確地使用電腦
01:11
is the silver bullet子彈
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是讓數學教育
01:13
for making製造 math數學 education教育 work.
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起死回生的妙藥
01:16
So to explain說明 that,
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要清楚解釋之前
01:18
let me first talk a bit about what math數學 looks容貌 like in the real真實 world世界
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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容貌 like in education教育.
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它在教學中又是怎個樣態
01:23
See, in the real真實 world世界
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瞧,在現實世界中
01:25
math數學 isn't necessarily一定 doneDONE by mathematicians數學家.
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數學並不一定是數學家的專用品
01:28
It's doneDONE by geologists地質學家,
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地質學家用到它
01:30
engineers工程師, biologists生物學家,
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工程師、生物學家
01:32
all sorts排序 of different不同 people --
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各行各業都用到數學
01:34
modeling造型 and simulation模擬.
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用來建模和模擬
01:36
It's actually其實 very popular流行.
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實際上這是很通行的
01:38
But in education教育 it looks容貌 very different不同 --
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但在教育中情況則大大不同
01:41
dumbed-down簡單化 problems問題, lots of calculating計算,
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無聊的問題-有許多計算題
01:43
mostly大多 by hand.
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大多要動手計算
01:46
Lots of things that seem似乎 simple簡單
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許多東西原本簡單
01:48
and not difficult like in the real真實 world世界,
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如同在現實世界中那麼容易
01:50
except if you're learning學習 it.
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不學沒事、一學就煩
01:53
And another另一個 thing about math數學:
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還有一件關於數學的事:
01:55
math數學 sometimes有時 looks容貌 like math數學 --
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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有時 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回答 that's quantitative in the modern現代 world世界.
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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不會 have expected預期 that a few少數 years年份 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不幸, my weight重量 is a little higher更高 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發生.
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一切有關世上發生的事情
02:21
So let's zoom放大 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教學 people math數學?
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為什麼要教數學?
02:25
What's the point of teaching教學 people math數學?
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教數學有什麼意義?
02:28
And in particular特定, why are we teaching教學 them math數學 in general一般?
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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這樣 an important重要 part部分 of education教育
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為何數學是教育的重要一環
02:34
as a sort分類 of compulsory義務 subject學科?
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因何成為必修的課程?
02:36
Well, I think there are about three reasons原因:
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我想這大概有三個理由:
02:39
technical技術 jobs工作
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技術方面的工作
02:41
so critical危急 to the development發展 of our economies經濟,
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對我們的經濟發展具有關鍵性
02:44
what I call "everyday每天 living活的" --
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我稱之為我們每日的生活
02:48
to function功能 in the world世界 today今天,
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在今日世界生活
02:50
you've got to be pretty漂亮 quantitative,
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你必須很懂得量化
02:52
much more so than a few少數 years年份 ago:
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比幾年前還必須懂得
02:54
figure數字 out your mortgages抵押貸款,
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例如弄清楚你的房貸
02:56
being存在 skeptical懷疑的 of government政府 statistics統計, 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第三, what I would call something like
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還有我稱之為
03:03
logical合乎邏輯 mind心神 training訓練, logical合乎邏輯 thinking思維.
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邏輯推理訓練、邏輯思考的東西
03:06
Over the years年份
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多年來我們投入許多資源
03:08
we've我們已經 put so much in society社會
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教導社會上的人
03:10
into being存在 able能夠 to process處理 and think logically邏輯. It's part部分 of human人的 society社會.
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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重要 to learn學習 that
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學習邏輯思考是很重要的
03:15
math數學 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另一個 question.
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那麼我們來問另一個問題
03:19
What is math數學?
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數學到底是什麼?
03:21
What do we mean when we say we're doing math數學,
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我們說利用數學或教導數學
03:23
or educating教育 people to do math數學?
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到底什麼意思?
03:25
Well, I think it's about four steps腳步, roughly大致 speaking請講,
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我認為這可以粗分為四個步驟
03:28
starting開始 with posing冒充 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世界,
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在利用數學的所有部分之外
03:35
beyond virtually實質上 any other part部分 of doing math數學.
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一般最常搞砸的就是
03:38
People ask the wrong錯誤 question,
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問錯問題
03:40
and surprisingly出奇 enough足夠, they get the wrong錯誤 answer回答,
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得到錯誤的答案
03:42
for that reason原因, if not for others其他.
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原因常常就是問錯問題
03:44
So the next下一個 thing is take that problem問題
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接下來就是
03:46
and turn it from a real真實 world世界 problem問題
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把現實世界的問題
03:48
into a math數學 problem問題.
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轉換成數學的問題
03:50
That's stage階段 two.
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這是第二步
03:52
Once一旦 you've doneDONE that, then there's the computation計算 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回答
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從這裡推演求算出
03:57
in a mathematical數學的 form形成.
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數學形式的答案
04:00
And of course課程, math數學 is very powerful強大 at doing that.
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這方面數學當然有其威力
04:02
And then finally最後, turn it back to the real真實 world世界.
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然後再把答案轉換到現實世界
04:04
Did it answer回答 the question?
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答案是否解答了問題?
04:06
And also verify校驗 it -- crucial關鍵 step.
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同時也驗證答案-這一步很關鍵
04:10
Now here's這裡的 the crazy thing right now.
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但最瘋狂的是:
04:12
In math數學 education教育,
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我們在數學教育中
04:14
we're spending開支 about perhaps也許 80 percent百分 of the time
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花費80%的時間
04:17
teaching教學 people to do step three by hand.
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教導學生用手做第三步
04:20
Yet然而, that's the one step computers電腦 can do
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可這正是電腦能做的一步
04:22
better than any human人的 after years年份 of practice實踐.
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比學習多年的人都要做得好
04:25
Instead代替, we ought應該 to be using運用 computers電腦
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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運用 the students學生們 to spend much more effort功夫
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讓學生多花點功夫學習如何做
04:33
on learning學習 how to do steps腳步 one, two and four --
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第一、第二和第四個步-
04:35
conceptualizing概念化 problems問題, applying應用 them,
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將問題概念化、應用那些概念
04:38
getting得到 the teacher老師 to run them through通過 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關鍵 point here:
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請看,關鍵點就在這裡:
04:43
math數學 is not equal等於 to calculating計算.
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數學並不等於求算
04:45
Math數學 is a much broader更廣泛 subject學科 than calculating計算.
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數學是比求算還要廣延的課題
04:48
Now it's understandable可理解 that this has all got intertwined交織
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我們都明白這經過幾百年
04:51
over hundreds數以百計 of years年份.
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都混雜在一起了
04:53
There was only one way to do calculating計算 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少數 decades幾十年
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不過在過去這幾十年來
04:58
that has totally完全 changed.
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局面完全改觀了
05:00
We've我們已經 had the biggest最大 transformation轉型 of any ancient subject學科
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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想像 with computers電腦.
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比任何古往的科目都要巨大
05:07
Calculating計算 was typically一般 the limiting限制 step,
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求算向來是讓人受到限制的步驟
05:09
and now often經常 it isn't.
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很少例外
05:11
So I think in terms條款 of the fact事實 that math數學
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因此我認為事實上數學
05:13
has been liberated解放 from calculating計算.
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已經逐漸從求算的步驟解放出來了
05:16
But that math數學 liberation解放 didn't get into education教育 yet然而.
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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計算, in a sense,
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這麼說吧,我把求算看做
05:21
as the machinery機械 of math數學.
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數學的機械部分
05:23
It's the chore苦差事.
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是勞煩的差事
05:25
It's the thing you'd like to avoid避免 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手段 to an end結束, not an end結束 in itself本身,
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是達到目的的工具,並不是目的本身
05:34
and automation自動化 allows允許 us
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自動化使得我們
05:36
to have that machinery機械.
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有了那樣的機械
05:38
Computers電腦 allow允許 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問題 by any means手段.
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這可是非同小可的一件事
05:43
I estimated預計 that, just today今天, across橫過 the world世界,
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我估計在當今全世界上
05:46
we spent花費 about 106 average平均 world世界 lifetimes壽命
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我們平均花費大約106個生命時間
05:49
teaching教學 people how to calculate計算 by hand.
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教導人們如何用手求算
05:52
That's an amazing驚人 amount of human人的 endeavor努力.
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那可是驚人的人力耗費
05:55
So we better be damn該死的 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開玩笑 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該死的 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真實 purpose目的.
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而且那麼做有個好理由
06:06
I think we should be assuming假設 computers電腦
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我認為我們應該讓電腦來
06:08
for doing the calculating計算
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承擔求算的任務
06:10
and only doing hand calculations計算 where it really makes品牌 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心理 arithmetic算術.
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可以使用心算
06:17
I still do a lot of that, mainly主要 for estimating估計.
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我還常心算,主要是用來做估計
06:20
People say, "Is such這樣 and such這樣 true真正?"
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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大致.
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我說:哼,不確定,我來想一下
06:24
It's still quicker更快 to do that and more practical實際的.
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求算還是快些,也實際些
06:26
So I think practicality實際性 is one case案件
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因此我想講求實際的是
06:28
where it's worth價值 teaching教學 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某些 conceptual概念上的 things
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然後還有些計算方面的事
06:32
that can also benefit效益 from hand calculating計算,
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動手求算還是有益處的
06:34
but I think they're relatively相對 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經常 ask about
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我常常思考一件事
06:38
is ancient Greek希臘語 and how this relates涉及.
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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迫使 people to learn學習 mathematics數學.
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強迫人們學習數學
06:45
It's a major重大的 subject學科.
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這是重大的科目
06:47
I'm not for one minute分鐘 suggesting提示 that, if people are interested有興趣 in hand calculating計算
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我並不暗示說如果有人喜歡
06:50
or in following以下 their own擁有 interests利益
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用手求算或出於個人興趣
06:52
in any subject學科 however然而 bizarre奇異的 --
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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絕對 the right thing,
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隨著自己的興趣做事
06:58
for people to follow跟隨 their self-interest自我利益.
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那絕對是沒有錯的事
07:00
I was somewhat有些 interested有興趣 in ancient Greek希臘語,
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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整個 population人口
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但我不認為我們應該強迫所有的人
07:05
to learn學習 a subject學科 like ancient Greek希臘語.
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學習像古希臘這樣的科目
07:07
I don't think it's warranted必要.
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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分別 between之間 what we're making製造 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學科 that's sort分類 of mainstream主流
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主流科目之間做了區分
07:14
and the subject學科 that, in a sense, people might威力 follow跟隨 with their own擁有 interest利益
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重點是人們有自己的興趣
07:17
and perhaps也許 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問題 people bring帶來 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基本 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不能 use the machine
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還沒學會某科目的基礎之前
07:27
until直到 you get the basics基本 of the subject學科.
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不要使用機器
07:29
So my usual通常 question is, what do you mean by "basics基本?"
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那我通常會問:你說的基礎是什麼?
07:32
Basics基本 of what?
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什麼的基礎呢?
07:34
Are the basics基本 of driving主動 a car汽車
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開車的基礎是學會
07:36
learning學習 how to service服務 it, or design設計 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基本 of writing寫作 learning學習 how to sharpen削尖 a quill鵝毛筆?
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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分離 the basics基本 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得到 doneDONE
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從如何把它做成分開來
07:50
and the machinery機械 of how it gets得到 doneDONE
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也從把它做成的機械分開來
07:54
and automation自動化 allows允許 you to make that separation分割.
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自動化讓你能做這樣的區分
07:57
A hundred years年份 ago, it's certainly當然 true真正 that to drive駕駛 a car汽車
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也許沒錯,一百年前你要開車
08:00
you kind of needed需要 to know a lot about the mechanics機械學 of the car汽車
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那你得多了解汽車的機械原理
08:02
and how the ignition點火 timing定時 worked工作 and all sorts排序 of things.
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以及內燃機點火的時點等等
08:06
But automation自動化 in cars汽車
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但是汽車上的自動化
08:08
allowed允許 that to separate分離,
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使得那些都可以分開了
08:10
so driving主動 is now a quite相當 separate分離 subject學科, so to speak說話,
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因此開車可說已經變成另外一個科目
08:13
from engineering工程 of the car汽車
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跟汽車設計
08:16
or learning學習 how to service服務 it.
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或學習如何保養已經毫無關係
08:20
So automation自動化 allows允許 this separation分割
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自動化造成這樣的區分
08:22
and also allows允許 -- in the case案件 of driving主動,
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而且不僅在開車這回事上
08:24
and I believe also in the future未來 case案件 of maths數學 --
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我相信將來在數學上
08:26
a democratized民主化 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傳播 across橫過 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另一個 thing that comes up with basics基本.
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那麼關於基礎還有一件事
08:35
People confuse迷惑, in my view視圖,
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在我看來就是
08:37
the order訂購 of the invention發明 of the tools工具
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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訂購 in which哪一個 they should use them for teaching教學.
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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發明 before computers電腦,
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只因為紙張比電腦早先發明
08:46
it doesn't necessarily一定 mean you get more to the basics基本 of the subject學科
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並不表示你要獲得事物的基礎
08:49
by using運用 paper instead代替 of a computer電腦
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用紙張來教數學
08:51
to teach mathematics數學.
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比用電腦來得根本
08:55
My daughter女兒 gave me a rather nice不錯 anecdote軼事 on this.
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我的女兒有個小插曲很好玩
08:58
She enjoys享受 making製造 what she calls電話 "paper laptops筆記本電腦."
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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笑聲)
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(笑聲)
09:03
So I asked her one day, "You know, when I was your age年齡,
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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第二 or two, carefully小心 reflecting反映,
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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笑聲)
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(笑聲)
09:19
If you were born天生 after computers電腦 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哪一個 order訂購 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最好 tool工具.
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你要的是最好的工具
09:26
So another另一個 one that comes up is "Computers電腦 dumb math數學 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不知何故, if you use a computer電腦,
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說得好像你如果用電腦
09:31
it's all mindless沒頭腦 button-pushing鈕推,
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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知識分子.
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那就是純然的讀書高了
09:37
This one kind of annoys惹惱 me, I must必須 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數學 that most people are doing in school學校
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當今之世大多數人在學校裡
09:44
practically幾乎 today今天
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學的數學還真的是
09:46
is more than applying應用 procedures程序
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不明所以就把某些程序
09:48
to problems問題 they don't really understand理解, for reasons原因 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更差, what they're learning學習 there isn't even practically幾乎 useful有用 anymore.
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而更糟的是他們所學的已不復實用
09:56
Might威力 have been 50 years年份 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教育, they do it on a computer電腦.
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他們出了社會用的是電腦
10:02
Just to be clear明確, I think computers電腦 can really help with this problem問題,
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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其實 make it more conceptual概念上的.
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讓這件事更符合概念
10:07
Now, of course課程, like any great tool工具,
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現在呢,就像任何好工具一樣
10:09
they can be used completely全然 mindlessly盲目,
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電腦可能毫無意識地被使用著
10:11
like turning車削 everything into a multimedia多媒體 show顯示,
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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顯示 of solving an equation方程 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電腦 was the teacher老師 --
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既然電腦可以是教師
10:19
show顯示 the student學生 how to manipulate操作 and solve解決 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堅果.
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這有夠蠢的
10:24
Why are we using運用 computers電腦 to show顯示 a student學生 how to solve解決 a problem問題 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電腦 should be doing anyway無論如何?
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電腦明明可以解答問題的!
10:29
All backwards向後.
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全都倒反了
10:31
Let me show顯示 you
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我來告訴你
10:33
that you can also make problems問題 harder更難 to calculate計算.
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你還可以讓問題更難求算
10:36
See, normally一般 in school學校,
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通常在學校裡
10:38
you do things like solve解決 quadratic二次 equations方程.
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你做的是解答二次方程式
10:41
But you see, when you're using運用 a computer電腦,
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但是你若利用電腦
10:44
you can just substitute替代.
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你用替代的方法就行了
10:48
You can make it a quartic四次 equation方程. Make it kind of harder更難, calculating-wise計算明智.
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弄個四次方程式,把它弄得較難求算
10:50
Same相同 principles原則 applied應用的 --
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同樣的原則還是適用-
10:52
calculations計算, harder更難.
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求算困難些
10:54
And problems問題 in the real真實 world世界
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而真實世界裡的問題
10:56
look nutty瘋狂的 and horrible可怕 like this.
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看起來像這樣困難、恐怖
10:58
They've他們已經 got hair頭髮 all over them.
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生毛帶角的,恐怖得要死
11:00
They're not just simple簡單, dumbed-down簡單化 things that we see in school學校 math數學.
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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世界.
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想一想外面的世界
11:06
Do we really believe that engineering工程 and biology生物學
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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受益 from computers電腦 and maths數學
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電腦和數學良多的東西
11:12
have somehow不知何故 conceptually概念 gotten得到 reduced減少 by using運用 computers電腦?
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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相當 the opposite對面.
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我可不這麼認為,應該反而加了分
11:18
So the problem問題 we've我們已經 really got in math數學 education教育
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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電腦 might威力 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簡單化 problems問題 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另一個 issue問題 people bring帶來 up
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好,另外還有個說法
11:29
is somehow不知何故 that hand calculating計算 procedures程序
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是說用手求算的程序
11:31
teach understanding理解.
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多少有助於教導理解
11:33
So if you go through通過 lots of examples例子,
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你做了許多範例題
11:35
you can get the answer回答,
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你就能得到答案-
11:37
you can understand理解 how the basics基本 of the system系統 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有效 here,
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我想這說法裡有個地方是有根據的
11:43
which哪一個 is that I think understanding理解 procedures程序 and processes流程 is important重要.
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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奇妙 way to do that in the modern現代 world世界.
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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程序設計.
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它就叫做程式設計
11:53
Programming程序設計 is how most procedures程序 and processes流程
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程式設計是現今寫下
11:55
get written書面 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從事 students學生們 much more
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讓學生能更深入檢驗
12:01
and to check they really understand理解.
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他們真的理解了些什麼
12:03
If you really want to check you understand理解 math數學
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如果你真的要檢驗你理解了數學
12:05
then write a program程序 to do it.
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那就寫個程式做做看
12:08
So programming程序設計 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明確, what I really am suggesting提示 here
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再說清楚,我在此提議的是
12:13
is we have a unique獨特 opportunity機會
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我們有了獨特的機會
12:15
to make maths數學 both more practical實際的
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同時讓數學變得
12:17
and more conceptual概念上的, simultaneously同時.
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更實用也更符合概念
12:20
I can't think of any other subject學科 where that's recently最近 been possible可能.
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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平時 some kind of choice選擇
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這通常像是在天份發掘
12:25
between之間 the vocational專業 and the intellectual知識分子.
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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相同 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打開 up so many許多 more possibilities可能性.
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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許多 more problems問題.
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可以藉此同時處理許多問題
12:37
What I really think we gain獲得 from this
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我真的認為我們可以
12:39
is students學生們 getting得到 intuition直覺 and experience經驗
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由此達到讓比以往更多的學生
12:42
in far greater更大 quantities數量 than they've他們已經 ever got before.
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獲得直覺和經驗的目的
12:45
And experience經驗 of harder更難 problems問題 --
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而且是得到處理更困難
12:47
being存在 able能夠 to play with the math數學, interact相互作用 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數學 instinctively本能.
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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電腦 allow允許 us to do.
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電腦讓我們能夠做到這點
12:57
Another另一個 thing it allows允許 us to do is reorder重新排序 the curriculum課程.
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電腦也讓我們能夠重新安排教學課程
13:00
Traditionally傳統 it's been by how difficult it is to calculate計算,
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傳統至今求算一直是困難的
13:02
but now we can reorder重新排序 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理解 the concepts概念,
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理解的難易度重新安排課程
13:06
however然而 hard the calculating計算.
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無論求算會是多麼困難
13:08
So calculus結石 has traditionally傳統 been taught very late晚了.
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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該死的 hard doing the calculations計算, that's the problem問題.
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是呀,問題就在求算非常困難
13:17
But actually其實 many許多 of the concepts概念
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但事實上有許多概念
13:19
are amenable適合 to a much younger更年輕 age年齡 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內置 for my daughter女兒.
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這裡有個我給我女兒設計的例題
13:25
And very, very simple簡單.
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非常非常簡單
13:28
We were talking about what happens發生
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我們談的是如果多邊形的邊
13:30
when you increase增加 the number of sides雙方 of a polygon多邊形
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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課程, 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堅持
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順道說一下,她也很堅持
13:40
on being存在 able能夠 to change更改 the color顏色,
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要能夠改變顏色
13:42
an important重要 feature特徵 for this demonstration示範.
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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範圍 and differential微分 calculus結石
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概念的初步
13:51
and what happens發生 when you take things to an extreme極端 --
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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雙方 and a very large number of sides雙方.
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這樣:極少量的邊和極大量的邊
13:56
Very simple簡單 example.
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很簡單的範例題
13:58
That's a view視圖 of the world世界
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這種看世界的觀點
14:00
that we don't usually平時 give people for many許多, many許多 years年份 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然而, that's a really important重要 practical實際的 view視圖 of the world世界.
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但這是一種看世界的重要觀點
14:06
So one of the roadblocks路障 we have
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那麼我們做這樣改變的路程上
14:09
in moving移動 this agenda議程 forward前鋒
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遇到的一個絆腳石
14:12
is exams考試.
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就是考試
14:14
In the end結束, if we test測試 everyone大家 by hand in exams考試,
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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課程 changed
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那麼在教學課程裡
14:20
to a point where they can use computers電腦
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就很難改變成
14:22
during the semesters學期.
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學生們可以在學期裡使用電腦
14:25
And one of the reasons原因 it's so important重要 --
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很重要的一個理由-
14:27
so it's very important重要 to get computers電腦 in exams考試.
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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問題, real真實 questions問題,
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然後我們就可以問問題,問真正的問題
14:33
questions問題 like, what's the best最好 life insurance保險 policy政策 to get? --
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例如,最佳壽險會是怎樣的保單內容?-
14:36
real真實 questions問題 that people have in their everyday每天 lives生活.
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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簡單化 model模型 here.
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瞧,這一點也不是無聊的模型
14:42
This is an actual實際 model模型 where we can be asked to optimize優化 what happens發生.
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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許多 years年份 of protection保護 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支付
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那對保費會有怎樣的影響呢?
14:49
and to the interest利益 rates利率 and so forth向前?
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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分鐘 suggesting提示 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考試,
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唯一類型的問題
14:57
but I think it's a very important重要 type類型
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不過我認為這個類型很重要
14:59
that right now just gets得到 completely全然 ignored忽視
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目前這方面完全被忽略了
15:02
and is critical危急 for people's人們 real真實 understanding理解.
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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危急 reform改革
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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基於計算機的 math數學.
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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移動 our economies經濟 forward前鋒,
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讓我們的經濟和社會
15:15
and also our societies社會,
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能夠向上提升
15:17
based基於 on the idea理念 that people can really feel mathematics數學.
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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可選的 extra額外.
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這可不是額外的選擇
15:25
And the country國家 that does this first
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哪個國家先做
15:27
will, in my view視圖, leapfrog蛙跳 others其他
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哪個國家就會領先一步
15:30
in achieving實現 a new economy經濟 even,
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達到新經濟的平衡
15:33
an improved改善 economy經濟,
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達到經濟的改善
15:35
an improved改善 outlook外表.
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達到前景的改善
15:37
In fact事實, I even talk about us moving移動
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其實我談的甚至牽涉到
15:39
from what we often經常 call now the "knowledge知識 economy經濟"
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把我們常說的知識經濟推移到
15:42
to what we might威力 call a "computational計算 knowledge知識 economy經濟,"
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我們或可稱為計算知識經濟的地步
15:45
where high-level高水平 math數學 is integral積分 to what everyone大家 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知識 currently目前 is.
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如同今日從事工作需要知識一般
15:50
We can engage從事 so many許多 more students學生們 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理解:
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我們必須理解
15:58
this is not an incremental增加的 sort分類 of change更改.
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這種改變不是漸進式的改變
16:02
We're trying to cross交叉 the chasm裂口 here
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我們正在嘗試的是要跨過
16:04
between之間 school學校 math數學 and the real-world真實世界 math數學.
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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步行 across橫過 a chasm裂口,
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大家都知道試圖跨過鴻溝時
16:08
you end結束 up making製造 it worse更差 than if you didn't start開始 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災害.
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災難變得更大
16:13
No, what I'm suggesting提示
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不,我提議的是
16:15
is that we should leap飛躍 off,
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我們要跳越過去
16:17
we should increase增加 our velocity速度
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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飛躍 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課程, having calculated計算 our differential微分 equation方程 very carefully小心.
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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笑聲)
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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全然 renewed更新, changed math數學 curriculum課程
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一個完全更新改變的數學教程
16:33
built內置 from the ground地面 up,
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從根底開始建構
16:35
based基於 on computers電腦 being存在 there,
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以電腦的存在為基礎
16:37
computers電腦 that are now ubiquitous普及 almost幾乎.
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電腦現今已經到處存在
16:39
Calculating計算 machines are everywhere到處
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計算機器到處都是
16:41
and will be completely全然 everywhere到處 in a small number of years年份.
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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學科 as math數學,
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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
16:50
it's the mainstream主流 subject學科 of the future未來.
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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開玩笑,
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那就讓它對我們、對學生
17:00
for us, for the students學生們 and for TEDTED here.
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和對TED這裡的人能提供一點樂趣
17:03
Thanks謝謝.
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謝謝各位
17:05
(Applause掌聲)
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(掌聲)
Translated by Wenjer Leuschel
Reviewed by Shelley Krishna R. TSANG

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ABOUT THE SPEAKER
Conrad Wolfram - Mathematician
Conrad 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.

More profile about the speaker
Conrad Wolfram | Speaker | TED.com