
請各位回想一下，是否曾非常喜歡某樣東西，例如一部電影、一張唱片、一首歌，或是一本書，你全心全意地向你所鍾愛的人推薦，你預期對方會有和你一樣的反應，等著等著，得到的答案卻是他恨死那東西了。在這場演講的介紹階段，我要告訴各位，這正是我過去六年每個工作天所面臨的情形。我在高中教數學，我販賣一種市場不想要，但法律規定必須購買的產品。我的意思是，這真是虧本生意。
我在學生學習數學過程中所見的刻板印象，對大家來說也都適用；如果我給各位做一個代數(II)的期末測驗，我預測及格率不會超過百分之二十五，這兩個事實並不是你或我學生的問題，而是當今美國所謂的數學教育面臨的問題。
首先，我將數學分成兩類，第一類是計算。這些內容你們大概都忘光了，例如將各項係數大於一的二次方程式做因式分解。這些東西很容易重新學習，假如你有相當堅實的推理基礎、數學推理基礎，我們稱之為數學程序在週遭世界的實際應用。這很難教，這是我們希望學生即使將來不進入數學領域工作，也能保有的能力；這也是以美國現今教學的方式，幾乎肯定無法使學生保有的能力。所以，我將告訴各位為什麼會這樣，為什麼這對社會來說是個不幸，我們能在這方面做些什麼，最後，為什麼現在是身為數學老師的大好時機。
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以下為系統擷取之英文原文
Can I ask you to please recall a time when you really loved something, a movie, an album, a song or a book, and you recommended it wholeheartedly to someone you also really liked, and you anticipated that reaction, you waited for it, and it came back, and the person hated it. So, by way of introduction, that is the exact same state in which I spent every working day of the last six years. I teach high school math. I sell a product to a market that doesn't want it, but is forced by law to buy it. I mean, that's kind of  it's just a losing proposition.
So there's a useful stereotype about students that I see, a useful stereotype about you all. I could give you guys an algebratwo final exam, and I would expect no higher than a 25 percent pass rate. And both of these facts say less about you or my students than they do about what we call math education in the U.S. today.
To start with, I'd like to break math down into two categories. One is computation. This is the stuff you've forgotten. For example, factoring quadratics with leading coefficients greater than one. This stuff is also really easy to relearn, provided you have a really strong grounding in reasoning, math reasoning. We'll call it the application of math processes to the world around us. This is hard to teach. This is what we would love students to retain, even if they don't go into mathematical fields. This is also something that, the way we teach it in the U.S. all but ensures they won't retain it. So, I'm going to talk about why that is, why that's such a calamity for society, what we can do about it, and, to close with, why this is an amazing time to be a math teacher.

首先，有五種徵兆，顯示你在課堂上所做的數學推理方式是錯誤的。第一是缺乏主動性，學生們不會自動自發的學習，當課程告一個段落，馬上有五隻手舉起來，要求你到他們課桌旁全部重新解釋一遍。學生缺乏持續力，記不得學過的東西；你會發現，三個月後，全部教過的觀念都得重新解釋一遍。我學生中有99%對解讀文字題很反感，另外1%的學生則熱衷於尋找應用在題目上的公式，這是相當沒有建設性的學習方式。
David Milch是《化外國度》（Deadwood）及其他一些精彩電視節目的創作者，對上述情況有個很棒的形容。他誓言停止創作當代戲劇及現今的單元連續劇，因為他意識到，當一個人每天花四小時沉溺在像《二個半男人》這種節目中時，沒有冒犯之意；他說，這會將神經傳導路徑形塑成期待簡單問題，他稱之為「一種無法做決定的焦慮」。人們對無法很快解決的問題沒有耐心，期望各種問題都像情境劇一樣，可以在22分鐘、三個廣告及一個罐頭笑聲中完成。我告訴大家，你們都知道，沒有一個值得解決的問題會那麼簡單。我非常擔心這一點，因為我將要在由我學生主宰的世界中退休，如果我用現在的方式教育他們，就是在跟我自己的未來及福祉過不去。我在這裡告訴大家，現今的教科書，尤其是那些已被大量採用的教科書，它們教導數學推理及耐心解決問題的方式，就像打開《二個半男人》節目，然後認為完成了一天的工作一樣。
(笑聲)
認真地看一下，這是一個來自物理教科書上的例子，在數學上也適用。注意看，首先它提供三個片段資訊，這些資訊最後都可套用到公式的某處，讓學生將結果計算出來。我相信在現實生活中，也請大家自問，哪些你解決過的、值得解決的問題，是事先就知道所有資訊的，或不需要由過多的資訊中過濾出有價值的部分，或資訊不足，你必須找出其他資訊補足。我想我們都同意，沒有任何值得解決的問題是像這樣的。我認為我們的教科書很瞭解該如何誤人子弟，因為，看看這個，這是一個練習題組，當實際做這個題組時，會看到像這樣的問題，我們只是換了數字並稍稍調整內容，如果學生仍不瞭解這類問題的模式，它會以解釋例題的方式幫助你回頭找到解題所需的公式，你事實上可以，我是說真的，將各個數字帶入公式，而不必懂物理，只要知道如何將教科書解碼即可，真令人汗顏。
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So first, five symptoms that you're doing math reasoning wrong in your classroom. One is a lack of initiative; your students don't selfstart. You finish your lecture block and immediately you have five hands going up asking you to reexplain the entire thing at their desks. Students lack perseverance. They lack retention; you find yourself reexplaining concepts three months later, wholesale. There's an aversion to word problems, which describes 99 percent of my students. And then the other one percent are eagerly looking for the formula to apply in that situation. This is really destructive.
David Milch, creator of "Deadwood" and other amazing TV shows, has a really good description for this. He swore off creating contemporary drama, shows set in the present day, because he saw that when people fill their mind with four hours a day of, for example, "Two and a Half Men," no disrespect, it shapes the neural pathways, he said, in such a way that they expect simple problems. He called it, "an impatience with irresolution." You're impatient with things that don't resolve quickly. You expect sitcomsized problems that wrap up in 22 minutes, three commercial breaks and a laugh track. And I'll put it to all of you, what you already know, that no problem worth solving is that simple. I am very concerned about this, because I'm going to retire in a world that my students will run. I'm doing bad things to my own future and wellbeing when I teach this way. I'm here to tell you that the way our textbooks, particularly, massadopted textbooks, teach math reasoning and patient problem solving, it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
(Laughter)
In all seriousness, here's an example from a physics textbook. It applies equally to math. Notice first of all here that you have exactly three pieces of information there, each of which will figure into a formula somewhere, eventually, which the student will then compute. I believe in real life. And ask yourself, what problem have you solved, ever, that was worth solving, where you knew all of the given information in advance, or where you didn't have a surplus of information and you had to filter it out, or where you didn't have sufficient information and you had to go find some. I'm sure we all agree that no problem worth solving is like that. And the textbook, I think, knows how it's hamstringing students. Because, watch this, this is the practice problem set. When it comes time to do the actually problem set, we have problems like this right here where we're just swapping out numbers and tweaking the context a little bit. And if the student still doesn't recognize the stamp this was molded from, it helpfully explains to you what sample problem you can return to to find the formula. You could literally, I mean this, pass this particular unit without knowing any physics, just knowing how to decode a textbook. That's a shame.

我可以在數學方面更精準地將這個問題做個診斷，這是個很酷的問題，我很喜歡，它是用滑雪纜車來定義坡度及斜率。圖片中事實上有四個不同的階層，我很好奇你們是否可以看出，這四個不同的階層，特別是當它們被壓縮在一起，同時給學生看的時候，如何造成對解題無耐心的問題。我在這裡詳細說明一下，這裡有個圖形，還有數學結構，說明了格線、標度、標示、點、軸這類的元素，解題前有些次步驟，引導我們進入真正想要討論的答案，即哪個部分坡度最陡。
我希望你們可以看得出來，我真的希望你們可以看出來，我們在此進行的，是採取一種強制性的問題及答案，但我們從這端舖了一條平順筆直的道路到另一端，並對學生跨過了途中一條小裂縫而讚不絕口，這就是我們現在的數學教育。所以我想告訴各位，如果我們可以用不同的方式將問題區分開來，並和學生們一起建立它們，就可以用耐心解決問題的方法，找到所有我們需要的資訊。
因此在這裡，我以一個圖像開始，並立即問以下問題：哪一個部份坡度最陡？這就開始了對話，因為圖像被設計成可用兩種答案進行辯論，所以學生們開始彼此爭辯，朋友和朋友，一對一，或分組爭論，不論什麼方式，因此最終我們知道，討論銀幕左下方或是中線以上的滑雪者有點惱人，我們瞭解到，如果我們標上ABCD四點來討論會簡單些，這個做法很棒。當我們開始給坡度下定義時，我們瞭解到，如果能用一些標度將其簡化，具體說明其含義，會是個不錯的主意，然後，一直到這時候，我們才將數學結構放上。是數學提供了這些對話，而不是由對話提供數學。這時候，我告訴各位，十分之九的班級都有能力繼續解決斜率、坡度之類的問題，但如果有需要，學生們可以共同發展這些解題的次步驟。
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So I can diagnose the problem a little more specifically in math. Here's a really cool problem. I like this. It's about defining steepness and slope using a ski lift. But what you have here is actually four separate layers. And I'm curious which of you can see the four separate layers, and, particularly, how when they're compressed together and presented to the student all at once, how that creates this impatient problem solving. I'll define them here. You have the visual. You also have the mathematical structure, talking about grids, measurements, labels, points, axes, that sort of thing. You have substeps, which all lead to what we really want to talk about, which section is the steepest.
So I hope you can see. I really hope you can see how, what we're doing here is taking a compelling question, a compelling answer, but we're paving a smooth, straight path from one to the other, and congratulating our students for how well they can step over the small cracks in the way. That's all we're doing here. So I want put to you, if we can separate these in a different way and build them up with students, we can have everything we're looking for in terms of patient problem solving.
So right here, I start with a visual, and I immediately ask the question: Which section is the steepest? And this starts conversation because the visual is created in such a way where you can defend two answers. So you get people arguing against each other, friend versus friend, in pairs, journaling, whatever. And then eventually we realize it's getting annoying to talk about the skier in the lower lefthand side of the screen or the skier just above the mid line. And we realize how great would it be if we just had some A, B, C, and D labels to talk about them more easily. And then as we start to define what does steepness mean, we realize it'd be nice to have some measurements to really narrow it down, specifically what that means. And then and only then, we throw down that mathematical structure. The math serves the conversation. The conversation doesn't serve the math. And at that point, I'll put it to you that nine out of 10 classes are good to go on the whole slope, steepness thing. But if you need to, your students can then develop those substeps together.

你們可以比較這邊這個和那邊那個，哪個可以產生耐心解決問題及數學推理的效果？以我過去的教學經驗，答案顯而易見。我以在此所說的觀念呼應愛因斯坦的話，我相信愛因斯坦此言是經驗的結晶，他說，規畫問題之能力的重要性無與倫比；但在我實際經驗中，在美國，我們只是給學生問題，並沒有讓學生參與問題的規畫。
所以我每星期五小時的課前準備工作中，有百分之九十的時間，是將這一類相當強制性的解題要素由教材中去除，然後重建成需要數學推理及耐心解題的形式，結果變成這樣。我喜歡這個跟水槽有關的題目，題目是：將水槽灌滿需要多久時間？瞭解嗎？我做的第一件事是，將教材中所有的次步驟刪除，學生必須自己發展這些，他們必須自己規劃解題的次步驟，然後注意到，所有寫在那裡的資訊都是解題所需，沒有一個是多餘的。我們先刪掉這些，學生必須自己決定，好，像是跟水槽高度有關嗎？跟尺寸有關嗎？跟閥的顏色有關嗎？哪些才是相關因素？以現今數學課程來說，這是個資訊相當不足的題目；這是個水槽，將它裝滿水需時多久，題目就是這樣。
因為現在是21世紀，所以我們喜歡用真實世界的元素來表達題目，而不是用教科書中常見的簡圖或剪貼圖。我們到外面去，照一張實物的相片，現在我們就有真實的物體了。需時多久才能將水裝滿呢？甚至更好些，我們拍攝影片，記錄人們裝水的過程，水灌得很慢，惱人的慢，過程冗長乏味，學生們不斷地看錶，翻著白眼，不時地想著，「天啊，到底要多久才會灌滿。」(笑聲)這樣各位就知道學生是怎麼被我騙上鉤的，對吧？
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Do you guys see how this, right here, compared to that  which one creates that patient problem solving, that math reasoning? It's been obvious in my practice, to me. And I'll yield the floor here for a second to Einstein, who, I believe, has paid his dues. He talked about the formulation of a problem being so incredibly important, and yet in my practice, in the U.S. here, we just give problems to students; we don't involve them in the formulation of the problem.
So 90 percent of what I do with my five hours of prep time per week is to take fairly compelling elements of problems like this from my textbook and rebuild them in a way that supports math reasoning and patient problem solving. And here's how it works. I like this question. It's about a water tank. The question is: How long will it take you to fill it up? Okay? First things first, we eliminate all the substeps. Students have to develop those. They have to formulate those. And then notice that all the information written on there is stuff you'll need. None of it's a distractor, so we lose that. Students need to decide, all right, well, does the height matter? Does the size of it matter? Does the color of the valve matter? What matters here? Such an underrepresented question in math curriculum. So now we have a water tank. How long will it take you to fill it up, and that's it.
And because this is the 21st century, and we would love to talk about the real world on its own terms, not in terms of line art or clip art that you so often see in textbooks, we go out, and we take a picture of it. So now we have the real deal. How long will it take it to fill it up? And, even better, is we take a video, a video of someone filling it up. And it's filling up slowly, agonizingly slowly. It's tedious. Students are looking at their watches, rolling their eyes, and they're all wondering at some point or another, "Man, how long is it going to take to fill up?" (Laughter) That's how you know you've baited the hook, right.

這個問題，就是這個，對我來說很有趣，因為就像我在介紹時所說的，因為沒經驗，所以我教導兒童，我教都是一些最需要輔導的孩子，有些孩子不願加入關於數學的討論，因為別人知道解題公式，別人比他更瞭解如何運用公式，所以他不願談論。但在這裡，每個人都位於直覺上的公平立足點，每個人都有用水裝過什麼的經驗，所以當我要孩子回答裝滿水需要多久時間這個問題時，就讓這些對數學或交談有恐懼感的孩子們加入了討論，我把學生姓名寫在黑板，連到他所猜的答案上，學生們就被帶入這個情境中了，然後依據我之前所述的程序進行。最棒的部份，或其中較佳的部份是，我們並非藉由教師版教科書後的解題提要來找答案，而只是將影片一直看到結束。(笑聲)那很可怕，對吧！因為教師版教科書後解題提要中的理論模式總是管用，那很棒，但當理論和實際情況不符合，談到錯誤的來源時，就令人驚慌了。但這些對話相當珍貴，是最有價值的精華。
我在此報告一些相當有趣的教學收穫，我學生第一天來上課時，已染上這些錯誤的學習方式病毒（1缺乏主動性2缺乏堅持性3缺乏持續力4對文字題反感5熱衷於公式）但一學期之後，他們變成這樣，我可以將一些全新的、他們完全不熟悉的教材寫在黑板上，然後他們會展開討論，時間比學期初長大約三或四分鐘。這真的很有趣，學生們不再對文字題反感，因為我們已經重新定義了文字題的做法；學生不再對數學產生恐懼，因為我們已逐漸重新定義數學的含義，這使數學變得有趣多了。
我鼓勵數學老師們多使用多媒體，因為它可將真實世界以高解析度、全彩的方式帶入教室；在公平的立足點上鼓勵學生運用直覺；盡可能詢問最簡短的問題，讓特定的問題在對話中出現；讓學生自己建構問題，因為愛因斯坦說應該這麼做；最後一點，總之，不要幫太多忙。因為教科書總是用錯誤的方式幫倒忙，它免除你培養學生耐心解決問題及數學推理的責任，所以別幫太多忙。
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And that question, off this right here, is really fun for me, because, like the intro, I teach kids, because of my inexperience, I teach the kids that are the most remedial, all right. And I've got kids who will not join a conversation about math because someone else has the formula, someone else knows how to work the formula better than me. So I won't talk about it. But here, everyone is on a level playing field of intuition. Everyone's filled something up with water before, so I get kids answering the question, how long will it take. I've got kids who are mathematically and conversationally intimidated joining the conversation. We put names on the board, attach them to guesses, and kids have bought in here. And then we follow the process I've described. And the best part here, or one of the better parts is that we don't get our answer from the answer key in the back of the teacher's edition. We, instead, just watch the end of the movie. (Laughter) And that's terrifying, all right. Because the theoretical models that always work out in the answer key in the back of a teacher's edition, that's great, but it's scary to talk about sources of error when the theoretical does not match up with the practical. But those conversations have been so valuable, among the most valuable.
So I'm here to report some really fun gains with students who come preinstalled with these viruses day one of the class. These are the kids who now, one semester in, I can put something on the board, totally new, totally foreign, and they'll have a conversation about it for three or four minutes more than they would have at the start of the year, which is just so fun. We're no longer averse to word problems, because we've redefined what a word problem is. We're no longer intimidated by math, because we're slowly redefining what math is. This has been a lot of fun.
I encourage math teachers I talk to to use multimedia, because it brings the real world into your classroom in high resolution and full color, to encourage student intuition for that level playing field, to ask the shortest question you possibly can and let those more specific questions come out in conversation, to let students build the problem, because Einstein said so, and to finally, in total, just be less helpful, because the textbook is helping you in all the wrong ways. It's buying you out of your obligation for patient problem solving and math reasoning, to be less helpful.

為什麼現在正是身為數學教師的好時機？因為我們口袋裡現在就有可以創造出這些高品質教材的工具，它相當普遍也相當便宜，而自由傳播教材的工具，在開放授權下，成了史上最便宜、最普及的方式。不久之前，我在我的部落格中放了一系列影片，兩星期內就有六千次點閱率，現在我仍會接到一些我從未去過的國家的老師來信，寫著，「哇！沒錯，我們確實藉此展開很棒的討論。喔！順便提一下，這是我將你的教材稍做改進的方法。」真的太棒了。我最近在部落格上貼了這個問題：在雜貨店裡，你會排哪一條櫃檯結賬？是一輛其中有19項物品的購物車後，還是排在四輛購物車後，其中分別有3、5、2、1項物品？這牽涉到線性規劃問題，是我課堂上的好教材，結果，這個問題讓我在幾星期前上了「早安美國」節目，真是太不可思議了，對吧？
從以上所談到的一切，我得到一個結論：不只是學生，一般人也很渴望得知這些，數學彰顯了真實世界的合理性，數學也是描述人類直覺的詞彙。所以我鼓勵大家，不論你們的教育程度如何，不論你是學生、家長、老師、政策制定者或其他身份，請堅持必須有更好的數學教學課程，我們需要更多耐心解決問題的人。謝謝。
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And why this is an amazing time to be a math teacher right now is because we have the tools to create this highquality curriculum in our front pocket. It's ubiquitous and fairly cheap. And the tools to distribute it freely, under open licenses has also never been cheaper or more ubiquitous. I put a video series on my blog not so long ago, and it got 6,000 views in two weeks. I get emails still from teachers in countries I've never visited saying, "Wow, yeah. We had a good conversation about that. Oh, and by the way, here's how I made your stuff better," which, wow. I put this problem on my blog recently. In a grocery store, which line do you get into, the one that has one cart and 19 items or the one with four carts and three, five, two and one items. And the linear modeling involved in that was some good stuff for my classroom, but it eventually got me on "Good Morning America" a few weeks later, which is just bizarre, right.
And from all of this, I can only conclude that people, not just students, are really hungry for this. Math makes sense of the world. Math is the vocabulary for your own intuition. So I just encourage you, whatever your stake is in education, whether you're a student, parent, teacher, policy maker, whatever, insist on better math curriculum. We need more patient problem solvers. Thank you.