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. (Laughter) 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, it's just a losing proposition.
請各位回想一下過去是否曾經 你非常喜歡一種東西 例如一部電影、一張唱片、一首歌或是一本書 在你全心全意的向 你所鍾愛的人推薦 你預期對方會有和你一樣的反應,你等著等著 得到的答案卻是,他恨死那東西了 在自我介紹階段, 我要告訴各位,這正是 我過去六年每個工作天所面臨情形的寫照 我在高中教數學 我生產一種 市場不想要,但法律規定必須買的產品。 我想這真的是賠錢貨喔。
So there's a useful stereotype about students that I see, a useful stereotype about you all. I could give you guys an algebra-two 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.
拿我學生對數學的刻板印象 用來預測你們的行為結果也管用 如果我給各位 作一個數學(II)的期末測驗 我不敢期望有超過 百分之二十五的人會通過。 這兩個事實並不是你或是我的學生特有的問題, 這反應出美國今天所面臨的 數學教育的通病。
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'd like 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,我將數學分成兩類。 第一類是計算,這些內容你大概都忘光了。 例如二次方係數大於一 的因式分解 這類數學是很容易學回來的。 假如你有堅實的推理, 數學推理的基礎, 我們把它稱作數學應用。 在生活週遭的實際應用 這很難教。 我們希望學生學習這些後 即使將來不進入數學的領域工作也要留下的能力 這也是我們美國數學教育之目標。 但幾乎可以保證最後這種能力不會被留下來 我會告訴各位為什麼會這樣 為什麼這種社會不幸會發生,我們能做些什麼 最後,為什麼現在是數學老師 改變現狀的大好機會
So first, five symptoms that you're doing math reasoning wrong in your classroom. One is a lack of initiative; your students don't self-start. You finish your lecture block and immediately you have five hands going up asking you to re-explain the entire thing at their desks. Students lack perseverance. They lack retention; you find yourself re-explaining 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 is eagerly looking for the formula to apply in that situation. This is really destructive.
首先 有五種症狀 顯示我們課堂上所做數學推理 的方式是錯誤的 第一 缺乏主動 學生們不會自動自發的學習 當你課程進行到一個段落 課椅上馬上有五隻手舉起來 要求你整個的重新解釋 第二 學生缺乏持久力 學過的東西不會記得,你會發現 教過的三個月之後,你得全部都要重來一遍 我的學生中有99%對於 解讀數學題意非常厭煩 另外1%的學生則 熱衷於找應用在 該題目的公式 這真的是非常沒有建設性
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 sitcom-sized 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 well-being when I teach this way. I'm here to tell you that the way our textbooks -- particularly mass-adopted 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.
David Milch,是“米蟲”影集及其他一些電視節目的製作人 對上述情況有一個很棒的形容 他誓言要停止製作 當代戲劇 及現今的的單元連續劇 因為他認為當一個人, 每天花四小時,沉溺在像”二個半男人”節目的劇情中時 人們神經的路徑會被定型為 期待簡單的問題 也就是他所謂的 ”對無法做決定的焦慮 “之中 人們對於無法立刻解決的問題沒有耐心 會期望各種問題都像情境劇一樣 可以在22分鐘 三個廣告橋段及同一個罐頭音樂中完成 而在真實情況中 如大家所知,沒有一個值得解決的問題會是那麼簡單 對這個現象我非常擔心 因為我要退休時的世界,會是由我的學生這一代的人來運作 如果我用現在的方式 教育他們, 那我就是在跟我自己的 未來及福利過不去 我在這裡要告訴大家,現今的教科書, 尤其是那些已被大量採用的教科書, 它們教數學推理及耐心解決問題的方式, 就像打開”二個半男人”電視節目,然後認為一天的工作完成了一樣
(Laughter)
(笑)
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; where you didn't have a surplus of information and you had to filter it out, or you didn't have sufficient information and 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 actual 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.
這問題要嚴肅以待,這裡是一個由物理教科書取出的例子 但在數學上也適用 注意看,首先它 提供三個片段資訊 而這些資訊剛好都最後套到公式 的某一個地方 學生就將它的結果計算出來 我相信 也請大家問問自己,在真實的生活中,有哪些你解決過的 有價值的問題 是你在事先就知道所有的資訊 是不需要你由過多的資訊中,過濾出有價值的部分 或是資訊不足, 你必須再找其他資訊補足的 我想我們都同意,沒有任何值得解決的問題是這樣處理的 而我認為我們的教科書很懂得讓我們的學生變成跛腳馬 看看這個就知道,這是一個練習題目組 當實際做這問題組時 問題像右邊這個 我們只要將數字切換進去, 做一些小處理答案就出來 如果學生仍然對於解這類型的題目的模式不懂 它會以解釋例題的方式 讓你回頭找到解題所需的公式 所以你可以逐一 將各數字帶入公式, 而不必懂物理 只是知道解碼 就得到答案 這實在太不應該了
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.
對於數學,我可以更精準的診斷它的問題點 這是我很喜歡舉出來的好問題 它是用滑雪纜車來定義 斜率及坡度 這圖上有四不同的個階層 我很好奇你們哪一個人,可以看可看出這是四個不同的階層 特別是當四張圖被壓縮在一起 同時給學生看的時候 這顯示出對解題無耐心是怎麼發生的 我將對其下一個定義 這裡有圖形 有數學結構 例如格子 尺寸 標示 點 軸 這類的元素 但解題前需要有次步驟, 用於決定前述各項結構中 哪些和斜率有關係 哪一個部分坡度最大
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 to put to you that 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.
我希望你可以看出 我們在此進行的 問題及答案都是強制性 但我們應該由問題的這端開始 舖一條筆直的道路,到解決問題那端 現在卻只因為學生知道 如何跨過路中的一個小裂縫而沾沾自喜 這就是我們現在的數學教育 所以我想像各位表達,我們是不是可以用不同的方法區分這些問題 把學生耐心解決問題 的整套的能力都建立起來
So right here I start with the 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 left-hand 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 would 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.
因此在這裡,我先展現一個圖像 並且立即問 哪一部份坡度最大 這就開始了對話 因為圖像是被設計成正反兩方都能找到論據的樣子 所以兩方開始爭辯 朋友和朋友 情侶對情侶 或是讀者和期刊 等等 因此最終我們知道 在討論銀幕 左下方的滑雪者 或是線以上的滑雪者有點惱人 另外我們也會體會 如果 我們就只討論ABCD四個標示點 事情會簡化一些 當我們給坡度下定義時 我們會了解如果能作一些量度,事情會比較好處理 可以把問題做歸納,了解問題的涵義。 只有在這時候,這個階段, 我們才將數學的結構鋪陳出來 數學提供的是對話 而不是用對話來服侍數學 我告訴各位 十分之九的課 先用這種方式 再進行斜率、坡度教學效果會很好 但假如你需要 你的學生們可以共同發展解題的次步驟
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.
你們可以比較這邊這個 和那邊那個 哪一個可以產生耐心解決問題及數學推理的效果呢 以我過去的教學經驗 答案很明顯的 我將我在這裡所說的,用來呼應愛因斯坦的話 我相信愛因斯坦此言是經驗累積的結晶 他認為規畫問題的能力是無與倫比的重要 但是在我實際的經驗,在美國 我們只給學生問題 我們沒有讓學生參與問題的形成及規畫
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? 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 side 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.
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, 是將這一類強制提供的解題要素 由我的教材中去除。 然後將其重建成需要數學推理及耐心解題的形式, 這是它運作的方式, 我喜歡這和水槽有關的題目 題目是:將水槽灌滿要多少時間? 重要的事情先做,我將教材中所有的次步驟刪除 學生必須自己去發展出來, 他們必須自己規劃解題的次步驟, 因此他們就注意到,所有寫在那裏的都是解題所需要的, 沒有一個是我們可以忽略分心的。 學生必須自己決定 , 水槽高度有關嗎?水槽尺寸有關嗎? 閥的顏色有關嗎?哪些才是真正的重要因素? 以現今的數學教材而言 這題目敘述不完整 我們有一個水槽, 要多久你才會將它填滿水,題目就這樣
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 then 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?
因為這是21世紀, 所以我們喜歡用真實世界的元素來敘述題目, 而不是用你常在教科書 看到的線圖或插圖, 我們出去外面對實物照一張像。 這樣我們就有真實世界, 要多久才會填滿水呢? 甚至可以作的更好一點,我們用影片 紀錄填加水的過程, 水灌得很慢、惱人的慢, 過程冗長 , 學生們一直看他們的錶 眼珠子不斷的轉, 臉上都有某程度的疑惑 “天啊 要多久才會灌滿啊” (笑聲) 各位就知道學生們是怎麼被我騙上鉤的
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, every student 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, 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.
由這個問題產生許多我認為很有趣的現象。 就像我在開頭時所說, 因為我沒有經驗,所以我教幼兒。 我所教都是哪些最需要矯正的幼兒, 他們有的不願意加入討論, 是因為別人有解題的公式 別人比他更了解解題的公式, 所以不願意談論。 但在這裡 每一個人的直覺立足點是公平的 因為每個人都有灌過水槽的經驗, 所以當我要幼兒回答,這要多久時間時, 就讓這些有數學或是交談壓迫感覺的小孩, 共同加入了討論。 我把學童姓名寫在黑板,指定他們來猜, 所以學生就被帶入情境。 然後依據我前面所敘述的程序去做。 這樣做最妙的地方,或是比較優的地方就是, 我們不由老師版教科書後面的 解題提要來找答案, 我們只要將影片一直看完。 (笑聲) 這種現象很可怕,對吧! 因為在老師版教科書後面的解題提要, 理論的公式總是管用的 那很棒呀,但是 但是當理論和實際情況不吻合, 產生了錯誤時就會引起驚慌, 但這正是這些對話中 最有價值的精華
So I'm here to report some really fun games with students who come pre-installed 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.
我來這裡報告一些有趣的教學心得, 我的學生第一天到教室時, 許多都是被錯誤數學教育所汙染的帶菌者 但一學期之後,他們變成這樣。 我可以將一些教材寫在黑板上, 針對新的、完全陌生的事務, 用比年初多約三或四分鐘。 作共同討論 這真的很有趣, 學生們不再對數學語義問題產生反感, 因為我們已經對數學語義重新定義; 學生不再對數學產生恐懼, 因為我們會慢慢的重新定義該情境的數學涵義是什麼 這樣就變的有趣許多了!
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.
我鼓勵數學老師們多多利用多媒體 將真實的世界情境 以高解析度 全彩的方式帶入教室 用公平方式鼓勵學生運用直覺 盡可能 問最簡短的問題 讓特定的問題在對話中出現 讓學生自己提出問題 就像愛因思坦所說哪樣 最後一點 不要幫太多忙 因為教科書用錯誤的方式在幫倒忙 它買斷你培養學生耐心解決問題 及數學推理的責任 所以不要幫太多忙
And why this is an amazing time to be a math teacher right now is because we have the tools to create this high-quality 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 line 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?
為什麼現在正是數學教師的大好時光 因為現在有工具可以創造 高品質且可攜帶於口袋的教材 它無所不在也相當便宜 而傳布教材的工具 因為大量供應及開放授權 而價格史上最低 普及性最高 不久之前我在我的部落格中放了一系列的影片 在兩個星期內就有六千個人來觀看 到現在我仍然接到一些我從未訪問過國家的老師來信 提到”哇!是的,我們確實用這個與學生有很好的對話 啊!順便告訴你,我有改進你教材的妙方” 效果真驚人 我最近在部落格放了這個問題, 在雜貨店裡,你要排哪一個結賬櫃檯 是要排在只有一輛購物車 但是上面有19項物品的後面 或是排在有四輛購物車 但是上面各有3、5、2、1項物品的後面 這是一個線性規劃的問題 是我課堂上的好教材 因為它,我幾星期前上了早安美國的節目 答案超乎一般想像 對吧
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 really 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. (Applause)
從前述影片受歡迎的程度 我得到一個結論就是 不只是學生 一般人也很渴望這些內容 數學彰顯真實世界的合理性 數學也是描述人類 直覺的詞彙 所以我鼓勵大家 不管你在教育上的角色是什麼 不管你是學生 家長老師或是政策制定者或其他 都請堅持要有一個更好的數學教材 我們需要更多有耐心解決問題的人,非常感謝