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訓練學童用電腦學數學 (推薦觀看)

訓練學童用電腦學數學

1. 訓練學童用電腦學數學 (中英原稿)

Teaching kids real math with computers

( 英文演講)
https://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers#t-15684

(中文字幕)
https://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers#t-18636

(原文中譯)
當今的數學教育有個真正的問題 基本上大家都不滿意 學習數學的人認為 數學與生活無干 既無趣又困難 利用數學的人則認為 還學得不夠 政府明白數學對經濟至為重要 卻不知從何下手改善 教師也感到沮喪 然而數學對今日世界的重要性 遠勝於人類歷史上的任何時點 我們一面有數學學習 興趣低落的問題 另一面卻要面對更加需要數學的世界 這個世界比以往更加需要量化 


00:44
到底出了什麼問題? 怎會裂出這條鴻溝?要如何修補? 其實我認為答案 就在眼前: 多多使用電腦 我相信 正確地使用電腦 是讓數學教育 起死回生的妙藥 要清楚解釋之前 先談數學在真實世界中是怎個樣態 它在教學中又是怎個樣態 瞧,在現實世界中 數學並不一定是數學家的專用品 地質學家用到它 工程師、生物學家 各行各業都用到數學 用來建模和模擬 實際上這是很通行的 但在教育中情況則大大不同 無聊的問題-有許多計算題 大多要動手計算 許多東西原本簡單 如同在現實世界中那麼容易 不學沒事、一學就煩 還有一件關於數學的事: 數學有時貌似數學 比方說這裡這個例子 有時不像數學 例如「我喝醉了嗎?」 這問題在今日世界得到的是一個量化的答案 幾年前你想都想不到會是這樣 但現在你可以發現一切- 不幸我的體重比那還重了些,不過- 一切有關世上發生的事情 


02:09
我們縮小這個來問個問題 為什麼要教數學? 教數學有什麼意義? 更確切地問,為何要廣泛進行數學教育? 為何數學是教育的重要一環 因何成為必修的課程? 我想這大概有三個理由: 技術方面的工作 對我們的經濟發展具有關鍵性 我稱之為我們每日的生活 在今日世界生活 你必須很懂得量化 比幾年前還必須懂得 例如弄清楚你的房貸 懷疑政府提出的統計數據等類的東西 還有我稱之為 邏輯推理訓練、邏輯思考的東西 多年來我們投入許多資源 教導社會上的人 如何邏輯思考-這是人類社會生活的一部分 學習邏輯思考是很重要的 數學是學習邏輯思考的好方法 


03:05
那麼我們來問另一個問題 數學到底是什麼? 我們說利用數學或教導數學 到底什麼意思? 我認為這可以粗分為四個步驟 首先是要提出正確的問題 問的是什麼問題?找到的是怎樣的答案? 在利用數學的所有部分之外 一般最常搞砸的就是 問錯問題 得到錯誤的答案 原因常常就是問錯問題 接下來就是 把現實世界的問題 轉換成數學的問題 這是第二步 完成這個步驟後就是求算 從這裡推演求算出 數學形式的答案 這方面數學當然有其威力 然後再把答案轉換到現實世界 答案是否解答了問題? 同時也驗證答案-這一步很關鍵 但最瘋狂的是: 我們在數學教育中 花費80%的時間 教導學生用手做第三步 可這正是電腦能做的一步 比學習多年的人都要做得好 所以我們應該把這一步 交給電腦來做 讓學生多花點功夫學習如何做 第一、第二和第四個步- 將問題概念化、應用那些概念 讓教師示範如何進行這些步驟 


04:29
請看,關鍵點就在這裡: 數學並不等於求算 數學是比求算還要廣延的課題 我們都明白這經過幾百年 都混雜在一起了 以往只有用手求算這麼個辦法 不過在過去這幾十年來 局面完全改觀了 我們經歷了電腦這個最大的變革 比任何古往的科目都要巨大 求算向來是讓人受到限制的步驟 很少例外 因此我認為事實上數學 已經逐漸從求算的步驟解放出來了 可是數學的解放卻還沒有進入教育裡頭 這麼說吧,我把求算看做 數學的機械部分 是勞煩的差事 若能讓機器來做,這是你會盡可能避免的事 是達到目的的工具,並不是目的本身 自動化使得我們 有了那樣的機械 電腦使我們可以避免煩差事 這可是非同小可的一件事 我估計在當今全世界上 我們平均花費大約106個生命時間 教導人們如何用手求算 那可是驚人的人力耗費 那麼我們最好能確定- 可大多數人都學得很不愉快 那麼我們最好能確定 我們為什麼那麼做 而且那麼做有個好理由 


05:54
我認為我們應該讓電腦來 承擔求算的任務 只有在確實有意義教導求算時才用手求算 我還認為有些情況下 可以使用心算 我還常心算,主要是用來做估計 有人說那樣、那樣是真的 我說:哼,不確定,我來想一下 求算還是快些,也實際些 因此我想講求實際的是 教導人們求算的一個好理由 然後還有些計算方面的事 動手求算還是有益處的 不過我認為那些情況畢竟少數 我常常思考一件事 是關於古希臘及與其相關的事物 瞧,我們現在做的是 強迫人們學習數學 這是重大的科目 我並不暗示說如果有人喜歡 用手求算或出於個人興趣 無論如何詭異的科目- 他們都應該去做 隨著自己的興趣做事 那絕對是沒有錯的事 我以往對古希臘很有興趣 但我不認為我們應該強迫所有的人 學習像古希臘這樣的科目 我不認為這會有什麼成功的保證 我在我們強迫別人做的事和所謂 主流科目之間做了區分 重點是人們有自己的興趣 而且可能受到激勵而去做 


07:07
那麼為什麼要教導別人某樣東西呢? 其中一項就是人家說的要學會基礎 還沒學會某科目的基礎之前 不要使用機器 那我通常會問:你說的基礎是什麼? 什麼的基礎呢? 開車的基礎是學會 如何保養汽車或設計汽車那回事嗎? 寫作的基礎是學會削羽毛管嗎? 我可不這麼認為 我想你得把你嘗試要做的事的基礎 從如何把它做成分開來 也從把它做成的機械分開來 自動化讓你能做這樣的區分 也許沒錯,一百年前你要開車 那你得多了解汽車的機械原理 以及內燃機點火的時點等等 但是汽車上的自動化 使得那些都可以分開了 因此開車可說已經變成另外一個科目 跟汽車設計 或學習如何保養已經毫無關係 自動化造成這樣的區分 而且不僅在開車這回事上 我相信將來在數學上 也會容許民主化的利用方式 可以讓很大多數的人 都能真正用來做事 


08:21
那麼關於基礎還有一件事 在我看來就是 人們對於發明工具的順序和他們應該 利用工具來進行教學的順序感到困惑 只因為紙張比電腦早先發明 並不表示你要獲得事物的基礎 用紙張來教數學 比用電腦來得根本 我的女兒有個小插曲很好玩 她喜歡用紙張做她所謂的紙張膝上型電腦 (笑聲) 有天我就問她:「在你這個年紀時 我並沒有做這些東西 你怎麼會以為以前有這種東西?」 她專心想了一兩秒鐘 說:「那個時候沒有紙張嗎?」 (笑聲) 如果你在有電腦和紙張之後才出生的 那麼先用什麼來教你做事都無妨了 你要的是最好的工具 


09:14
還有另外一個說法是「電腦讓人數學無能」 說得好像你如果用電腦 那都是毫無意識的按鍵盤 而如果你用手做同樣的事 那就是純然的讀書高了 這種說法聽了教人不爽 我們還真的相信 當今之世大多數人在學校裡 學的數學還真的是 不明所以就把某些程序 應用到他們並不明白的問題上? 我可不這麼認為 而更糟的是他們所學的已不復實用 也許50年前還實用,現在不實用了 他們出了社會用的是電腦 再說清楚,我認為這個問題電腦真的能幫上忙 讓這件事更符合概念 現在呢,就像任何好工具一樣 電腦可能毫無意識地被使用著 例如把所有的東西都弄成多媒體表演 如同我指出的那個用手解決方程式的例子那樣 既然電腦可以是教師 還教學生怎麼用手解決方程式不成? 這有夠蠢的 為什麼用電腦來教學生如何用手解答問題? 電腦明明可以解答問題的! 全都倒反了 


10:19
我來告訴你 你還可以讓問題更難求算 通常在學校裡 你做的是解答二次方程式 但是你若利用電腦 你用替代的方法就行了 弄個四次方程式,把它弄得較難求算 同樣的原則還是適用- 求算困難些 而真實世界裡的問題 看起來像這樣困難、恐怖 生毛帶角的,恐怖得要死 看來不像學校數學裡那樣簡單、無聊的東西 想一想外面的世界 我們真的相信工程和生物 還有所有那些受惠於 電腦和數學良多的東西 由於使用了電腦在概念上就扣了分嗎? 我可不這麼認為,應該反而加了分 因此我們在數學教育裡遇到的問題 並不是電腦可能使數學變得無聊 而是我們現在就已經把問題弄得無聊 好,另外還有個說法 是說用手求算的程序 多少有助於教導理解 你做了許多範例題 你就能得到答案- 你就更能理解系統的基礎是如何運作的 我想這說法裡有個地方是有根據的 我是說理解程序和過程有其重要性 但現代世界裡有個新的方式來進行理解 它就叫做程式設計 


11:41
程式設計是現今寫下 大多數程序和過程的方法 而且這是很好的方法 讓學生能更深入檢驗 他們真的理解了些什麼 如果你真的要檢驗你理解了數學 那就寫個程式做做看 因此我認為程式設計就是我們進行檢驗的方法 再說清楚,我在此提議的是 我們有了獨特的機會 同時讓數學變得 更實用也更符合概念 我想不到最近有其它科目像數學這樣有這種可能 這通常像是在天份發掘 和智識培養之間做選擇 但我認為我們在這頂上可以兩者兼得 而且我們會因此開拓出許多可能性 可以藉此同時處理許多問題 我真的認為我們可以 由此達到讓比以往更多的學生 獲得直覺和經驗的目的 而且是得到處理更困難 問題的經驗-能夠玩數學、 與數學互動、感受到數學 我們要人們能夠本能地感受到數學 電腦讓我們能夠做到這點 


12:45
電腦也讓我們能夠重新安排教學課程 傳統至今求算一直是困難的 現在我們可以依據數學概念 理解的難易度重新安排課程 無論求算會是多麼困難 比方說微積分傳統上都很晚才教 為什麼這樣呢? 是呀,問題就在求算非常困難 但事實上有許多概念 是許多年齡低很多的人所能理解的 這裡有個我給我女兒設計的例題 非常非常簡單 我們談的是如果多邊形的邊 數量增加到很大很大 那個多邊形會變成怎樣 當然啦,那會變成一個圓形 順道說一下,她也很堅持 要能夠改變顏色 這對這個示範是很重要的 這裡看到的是極限和微分 概念的初步 那麼把多邊形的邊減少到極限會是怎樣 這樣:極少量的邊和極大量的邊 很簡單的範例題 這種看世界的觀點 許多年前還沒有電腦時我們不會讓人們看到 但這是一種看世界的重要觀點 那麼我們做這樣改變的路程上 遇到的一個絆腳石 就是考試 如果在考試裡還是測驗用手求算 那麼在教學課程裡 就很難改變成 學生們可以在學期裡使用電腦 


14:13
很重要的一個理由- 使用電腦考試因此是很重要的 然後我們就可以問問題,問真正的問題 例如,最佳壽險會是怎樣的保單內容?- 大家在每日生活中會遇到的真正問題 瞧,這一點也不是無聊的模型 這種實際的模型可用來求問如何最佳化 我的壽險要擔保多少年呢? 那對保費會有怎樣的影響呢? 對利率等等會有怎樣的影響呢? 我可不是說這是在考試裡問的 唯一類型的問題 不過我認為這個類型很重要 目前這方面完全被忽略了 但對於真正的理解卻是很關鍵的 


14:53
因此我相信我們必須進行重大的 以電腦為基的數學教學變革 我們得確保 讓我們的經濟和社會 能夠向上提升 根據的想法是人人都真正能感受到數學 這可不是額外的選擇 哪個國家先做 哪個國家就會領先一步 達到新經濟的平衡 達到經濟的改善 達到前景的改善 其實我談的甚至牽涉到 把我們常說的知識經濟推移到 我們或可稱為計算知識經濟的地步 那時高端數學是人人從事工作的一部分 如同今日從事工作需要知識一般 我們能夠讓更多學生深入參與 讓他們的數學學習更加順暢 我們必須理解 這種改變不是漸進式的改變 我們正在嘗試的是要跨過 學校數學與真實世界之間的鴻溝 大家都知道試圖跨過鴻溝時 事態往往是比不嘗試之前還要嚴重 災難變得更大 不,我提議的是 我們要跳越過去 我們應該加快速度 讓速度變得很快 我們應該從一邊飛躍起來跳到另一邊- 當然得先很謹慎地求算我們的微分方程式 


16:15
(笑聲) 


16:17
因此我想看到 一個完全更新改變的數學教程 從根底開始建構 以電腦的存在為基礎 電腦現今已經到處存在 計算機器到處都是 而且在短短的幾年內會是無所不在 我現在還不確定這個科目是否應該稱為數學 但我能確定的是 它會是將來的主流科目 我們先來迎接吧 既然我們要做 那就讓它對我們、對學生 和對TED這裡的人能提供一點樂趣 


16:51
謝謝各位 


16:53
(掌聲) 


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< 英文原稿> 

00:03
We've got a real problem with math education right now. Basically, no one's very happy. Those learning it think it's disconnected, uninteresting and hard. Those trying to employ them think they don't know enough. Governments realize that it's a big deal for our economies, but don't know how to fix it. And teachers are also frustrated. Yet math is more important to the world than at any point in human history. So at one end we've got falling interest in education in math, and at the other end we've got a more mathematical world, a more quantitative world than we ever have had. 


00:44
So what's the problem, why has this chasm opened up, and what can we do to fix it? Well actually, I think the answer is staring us right in the face: Use computers. I believe that correctly using computers is the silver bullet for making math education work. So to explain that, let me first talk a bit about what math looks like in the real world and what it looks like in education. See, in the real world math isn't necessarily done by mathematicians. It's done by geologists, engineers, biologists, all sorts of different people -- modeling and simulation. It's actually very popular. But in education it looks very different -- dumbed-down problems, lots of calculating, mostly by hand. Lots of things that seem simple and not difficult like in the real world, except if you're learning it. And another thing about math: math sometimes looks like math -- like in this example here -- and sometimes it doesn't -- like "Am I drunk?" And then you get an answer that's quantitative in the modern world. You wouldn't have expected that a few years back. But now you can find out all about -- unfortunately, my weight is a little higher than that, but -- all about what happens. 


02:09
So let's zoom out a bit and ask, why are we teaching people math? What's the point of teaching people math? And in particular, why are we teaching them math in general? Why is it such an important part of education as a sort of compulsory subject? Well, I think there are about three reasons: technical jobs so critical to the development of our economies, what I call "everyday living" -- to function in the world today, you've got to be pretty quantitative, much more so than a few years ago: figure out your mortgages, being skeptical of government statistics, those kinds of things -- and thirdly, what I would call something like logical mind training, logical thinking. Over the years we've put so much in society into being able to process and think logically. It's part of human society. It's very important to learn that math is a great way to do that. 


03:05
So let's ask another question. What is math? What do we mean when we say we're doing math, or educating people to do math? Well, I think it's about four steps, roughly speaking, starting with posing the right question. What is it that we want to ask? What is it we're trying to find out here? And this is the thing most screwed up in the outside world, beyond virtually any other part of doing math. People ask the wrong question, and surprisingly enough, they get the wrong answer, for that reason, if not for others. So the next thing is take that problem and turn it from a real world problem into a math problem. That's stage two. Once you've done that, then there's the computation step. Turn it from that into some answer in a mathematical form. And of course, math is very powerful at doing that. And then finally, turn it back to the real world. Did it answer the question? And also verify it -- crucial step. Now here's the crazy thing right now. In math education, we're spending about perhaps 80 percent of the time teaching people to do step three by hand. Yet, that's the one step computers can do better than any human after years of practice. Instead, we ought to be using computers to do step three and using the students to spend much more effort on learning how to do steps one, two and four -- conceptualizing problems, applying them, getting the teacher to run them through how to do that. 


04:29
See, crucial point here: math is not equal to calculating. Math is a much broader subject than calculating. Now it's understandable that this has all got intertwined over hundreds of years. There was only one way to do calculating and that was by hand. But in the last few decades that has totally changed. We've had the biggest transformation of any ancient subject that I could ever imagine with computers. Calculating was typically the limiting step, and now often it isn't. So I think in terms of the fact that math has been liberated from calculating. But that math liberation didn't get into education yet. See, I think of calculating, in a sense, as the machinery of math. It's the chore. It's the thing you'd like to avoid if you can, like to get a machine to do. It's a means to an end, not an end in itself, and automation allows us to have that machinery. Computers allow us to do that -- and this is not a small problem by any means. I estimated that, just today, across the world, we spent about 106 average world lifetimes teaching people how to calculate by hand. That's an amazing amount of human endeavor. So we better be damn sure -- and by the way, they didn't even have fun doing it, most of them -- so we better be damn sure that we know why we're doing that and it has a real purpose. 


05:54
I think we should be assuming computers for doing the calculating and only doing hand calculations where it really makes sense to teach people that. And I think there are some cases. For example: mental arithmetic. I still do a lot of that, mainly for estimating. People say, "Is such and such true?" And I'll say, "Hmm, not sure." I'll think about it roughly. It's still quicker to do that and more practical. So I think practicality is one case where it's worth teaching people by hand. And then there are certain conceptual things that can also benefit from hand calculating, but I think they're relatively small in number. One thing I often ask about is ancient Greek and how this relates. See, the thing we're doing right now is we're forcing people to learn mathematics. It's a major subject. I'm not for one minute suggesting that, if people are interested in hand calculating or in following their own interests in any subject however bizarre -- they should do that. That's absolutely the right thing, for people to follow their self-interest. I was somewhat interested in ancient Greek, but I don't think that we should force the entire population to learn a subject like ancient Greek. I don't think it's warranted. So I have this distinction between what we're making people do and the subject that's sort of mainstream and the subject that, in a sense, people might follow with their own interest and perhaps even be spiked into doing that. 


07:07
So what are the issues people bring up with this? Well one of them is, they say, you need to get the basics first. You shouldn't use the machine until you get the basics of the subject. So my usual question is, what do you mean by "basics?" Basics of what? Are the basics of driving a car learning how to service it, or design it for that matter? Are the basics of writing learning how to sharpen a quill? I don't think so. I think you need to separate the basics of what you're trying to do from how it gets done and the machinery of how it gets done and automation allows you to make that separation. A hundred years ago, it's certainly true that to drive a car you kind of needed to know a lot about the mechanics of the car and how the ignition timing worked and all sorts of things. But automation in cars allowed that to separate, so driving is now a quite separate subject, so to speak, from engineering of the car or learning how to service it. So automation allows this separation and also allows -- in the case of driving, and I believe also in the future case of maths -- a democratized way of doing that. It can be spread across a much larger number of people who can really work with that. 


08:21
So there's another thing that comes up with basics. People confuse, in my view, the order of the invention of the tools with the order in which they should use them for teaching. So just because paper was invented before computers, it doesn't necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics. My daughter gave me a rather nice anecdote on this. She enjoys making what she calls "paper laptops." (Laughter) So I asked her one day, "You know, when I was your age, I didn't make these. Why do you think that was?" And after a second or two, carefully reflecting, she said, "No paper?" (Laughter) If you were born after computers and paper, it doesn't really matter which order you're taught with them in, you just want to have the best tool. 


09:14
So another one that comes up is "Computers dumb math down." That somehow, if you use a computer, it's all mindless button-pushing, but if you do it by hand, it's all intellectual. This one kind of annoys me, I must say. Do we really believe that the math that most people are doing in school practically today is more than applying procedures to problems they don't really understand, for reasons they don't get? I don't think so. And what's worse, what they're learning there isn't even practically useful anymore. Might have been 50 years ago, but it isn't anymore. When they're out of education, they do it on a computer. Just to be clear, I think computers can really help with this problem, actually make it more conceptual. Now, of course, like any great tool, they can be used completely mindlessly, like turning everything into a multimedia show, like the example I was shown of solving an equation by hand, where the computer was the teacher -- show the student how to manipulate and solve it by hand. This is just nuts. Why are we using computers to show a student how to solve a problem by hand that the computer should be doing anyway? All backwards. 


10:19
Let me show you that you can also make problems harder to calculate. See, normally in school, you do things like solve quadratic equations. But you see, when you're using a computer, you can just substitute. You can make it a quartic equation. Make it kind of harder, calculating-wise. Same principles applied -- calculations, harder. And problems in the real world look nutty and horrible like this. They've got hair all over them. They're not just simple, dumbed-down things that we see in school math. And think of the outside world. Do we really believe that engineering and biology and all of these other things that have so benefited from computers and maths have somehow conceptually gotten reduced by using computers? I don't think so -- quite the opposite. So the problem we've really got in math education is not that computers might dumb it down, but that we have dumbed-down problems right now. Well, another issue people bring up is somehow that hand calculating procedures teach understanding. So if you go through lots of examples, you can get the answer, you can understand how the basics of the system work better. I think there is one thing that I think very valid here, which is that I think understanding procedures and processes is important. But there's a fantastic way to do that in the modern world. It's called programming. 


11:41
Programming is how most procedures and processes get written down these days, and it's also a great way to engage students much more and to check they really understand. If you really want to check you understand math then write a program to do it. So programming is the way I think we should be doing that. So to be clear, what I really am suggesting here is we have a unique opportunity to make maths both more practical and more conceptual, simultaneously. I can't think of any other subject where that's recently been possible. It's usually some kind of choice between the vocational and the intellectual. But I think we can do both at the same time here. And we open up so many more possibilities. You can do so many more problems. What I really think we gain from this is students getting intuition and experience in far greater quantities than they've ever got before. And experience of harder problems -- being able to play with the math, interact with it, feel it. We want people who can feel the math instinctively. That's what computers allow us to do. 


12:45
Another thing it allows us to do is reorder the curriculum. Traditionally it's been by how difficult it is to calculate, but now we can reorder it by how difficult it is to understand the concepts, however hard the calculating. So calculus has traditionally been taught very late. Why is this? Well, it's damn hard doing the calculations, that's the problem. But actually many of the concepts are amenable to a much younger age group. This was an example I built for my daughter. And very, very simple. We were talking about what happens when you increase the number of sides of a polygon to a very large number. And of course, it turns into a circle. And by the way, she was also very insistent on being able to change the color, an important feature for this demonstration. You can see that this is a very early step into limits and differential calculus and what happens when you take things to an extreme -- and very small sides and a very large number of sides. Very simple example. That's a view of the world that we don't usually give people for many, many years after this. And yet, that's a really important practical view of the world. So one of the roadblocks we have in moving this agenda forward is exams. In the end, if we test everyone by hand in exams, it's kind of hard to get the curricula changed to a point where they can use computers during the semesters. 


14:13
And one of the reasons it's so important -- so it's very important to get computers in exams. And then we can ask questions, real questions, questions like, what's the best life insurance policy to get? -- real questions that people have in their everyday lives. And you see, this isn't some dumbed-down model here. This is an actual model where we can be asked to optimize what happens. How many years of protection do I need? What does that do to the payments and to the interest rates and so forth? Now I'm not for one minute suggesting it's the only kind of question that should be asked in exams, but I think it's a very important type that right now just gets completely ignored and is critical for people's real understanding. 


14:53
So I believe [there is] critical reform we have to do in computer-based math. We have got to make sure that we can move our economies forward, and also our societies, based on the idea that people can really feel mathematics. This isn't some optional extra. And the country that does this first will, in my view, leapfrog others in achieving a new economy even, an improved economy, an improved outlook. In fact, I even talk about us moving from what we often call now the "knowledge economy" to what we might call a "computational knowledge economy," where high-level math is integral to what everyone does in the way that knowledge currently is. We can engage so many more students with this, and they can have a better time doing it. And let's understand: this is not an incremental sort of change. We're trying to cross the chasm here between school math and the real-world math. And you know if you walk across a chasm, you end up making it worse than if you didn't start at all -- bigger disaster. No, what I'm suggesting is that we should leap off, we should increase our velocity so it's high, and we should leap off one side and go the other -- of course, having calculated our differential equation very carefully. 


16:15
(Laughter) 


16:17
So I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being there, computers that are now ubiquitous almost. Calculating machines are everywhere and will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future. Let's go for it, and while we're about it, let's have a bit of fun, for us, for the students and for TED here. 


16:51
Thanks. 


16:53
(Applause) 

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