TEDxNYED
Dan Meyer: Math class needs a makeover
Dan Meyer:數學課程必須改頭換面
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現今的數學課程方向是教育學生善長於作數字運算作業，這剝奪了小孩學習一種比解題還重要的能力--規劃問題--的機會， 在TEDxNYED, Dan Meyer 展示一些會讓學生停止思考的課堂數學作業測驗
Double-click the English transcript below to play the video.
00:15
Can I ask you to please recall a time
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請各位回想一下過去是否曾經
00:18
when you really loved something --
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你非常喜歡一種東西
00:20
a movie, an album, a song or a book --
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例如一部電影、一張唱片、一首歌或是一本書
00:22
and you recommended it wholeheartedly
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在你全心全意的向
00:25
to someone you also really liked,
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你所鍾愛的人推薦
00:27
and you anticipated that reaction, you waited for it,
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你預期對方會有和你一樣的反應，你等著等著
00:29
and it came back, and the person hated it?
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得到的答案卻是，他恨死那東西了
00:32
So, by way of introduction,
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在自我介紹階段，
00:34
that is the exact same state
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我要告訴各位，這正是
00:36
in which I spent every working day of the last six years. (Laughter)
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我過去六年每個工作天所面臨情形的寫照
00:39
I teach high school math.
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我在高中教數學
00:41
I sell a product to a market
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我生產一種
00:44
that doesn't want it, but is forced by law to buy it.
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市場不想要，但法律規定必須買的產品。
00:47
I mean, it's just a losing proposition.
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我想這真的是賠錢貨喔。
00:50
So there's a useful stereotype about students that I see,
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拿我學生對數學的刻板印象
00:53
a useful stereotype about you all.
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用來預測你們的行為結果也管用
00:55
I could give you guys
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如果我給各位
00:57
an algebra-two final exam,
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作一個數學(II)的期末測驗
00:59
and I would expect no higher
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我不敢期望有超過
01:01
than a 25 percent pass rate.
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百分之二十五的人會通過。
01:03
And both of these facts say less about you or my students
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這兩個事實並不是你或是我的學生特有的問題，
01:06
than they do about what we call math education
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這反應出美國今天所面臨的
01:08
in the U.S. today.
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數學教育的通病。
01:10
To start with, I'd like to break math down into two categories.
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首先，我將數學分成兩類。
01:13
One is computation; this is the stuff you've forgotten.
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第一類是計算，這些內容你大概都忘光了。
01:16
For example, factoring quadratics with
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例如二次方係數大於一
01:18
leading coefficients greater than one.
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的因式分解
01:20
This stuff is also really easy to relearn,
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這類數學是很容易學回來的。
01:22
provided you have a really strong grounding
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假如你有堅實的推理，
01:24
in reasoning. Math reasoning --
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數學推理的基礎，
01:26
we'll call it the application
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我們把它稱作數學應用。
01:28
of math processes to the world around us --
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在生活週遭的實際應用
01:30
this is hard to teach.
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這很難教。
01:32
This is what we would love students to retain,
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我們希望學生學習這些後
01:34
even if they don't go into mathematical fields.
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即使將來不進入數學的領域工作也要留下的能力
01:36
This is also something that, the way we teach it in the U.S.
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這也是我們美國數學教育之目標。
01:38
all but ensures they won't retain it.
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但幾乎可以保證最後這種能力不會被留下來
01:41
So, I'd like to talk about why that is,
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我會告訴各位為什麼會這樣
01:42
why that's such a calamity for society, what we can do about it
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為什麼這種社會不幸會發生，我們能做些什麼
01:45
and, to close with, why this is an amazing time
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最後，為什麼現在是數學老師
01:47
to be a math teacher.
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改變現狀的大好機會
01:49
So first, five symptoms
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首先 有五種症狀
01:51
that you're doing math reasoning wrong
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顯示我們課堂上所做數學推理
01:53
in your classroom.
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的方式是錯誤的
01:55
One is a lack of initiative; your students don't self-start.
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第一 缺乏主動 學生們不會自動自發的學習
01:58
You finish your lecture block
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當你課程進行到一個段落
02:00
and immediately you have five hands going up
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課椅上馬上有五隻手舉起來
02:02
asking you to re-explain the entire thing at their desks.
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要求你整個的重新解釋
02:04
Students lack perseverance.
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第二 學生缺乏持久力
02:06
They lack retention; you find yourself
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學過的東西不會記得，你會發現
02:08
re-explaining concepts three months later, wholesale.
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教過的三個月之後，你得全部都要重來一遍
02:10
There's an aversion to word problems,
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我的學生中有99%對於
02:12
which describes 99 percent of my students.
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解讀數學題意非常厭煩
02:14
And then the other one percent
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另外1%的學生則
02:16
is eagerly looking for the formula
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熱衷於找應用在
02:18
to apply in that situation.
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該題目的公式
02:20
This is really destructive.
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這真的是非常沒有建設性
02:22
David Milch, creator of "Deadwood" and other amazing TV shows,
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David Milch，是“米蟲”影集及其他一些電視節目的製作人
02:25
has a really good description for this.
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對上述情況有一個很棒的形容
02:28
He swore off creating
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他誓言要停止製作
02:30
contemporary drama,
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當代戲劇
02:32
shows set in the present day,
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及現今的的單元連續劇
02:34
because he saw that when people fill their mind
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因為他認為當一個人，
02:36
with four hours a day of, for example, "Two and a Half Men," no disrespect,
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每天花四小時，沉溺在像”二個半男人”節目的劇情中時
02:39
it shapes the neural pathways, he said,
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人們神經的路徑會被定型為
02:41
in such a way that they expect simple problems.
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期待簡單的問題
02:44
He called it, "an impatience with irresolution."
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也就是他所謂的 ”對無法做決定的焦慮 “之中
02:47
You're impatient with things that don't resolve quickly.
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人們對於無法立刻解決的問題沒有耐心
02:50
You expect sitcom-sized problems that wrap up in 22 minutes,
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會期望各種問題都像情境劇一樣 可以在22分鐘
02:53
three commercial breaks and a laugh track.
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三個廣告橋段及同一個罐頭音樂中完成
02:56
And I'll put it to all of you,
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而在真實情況中
02:59
what you already know, that no problem worth solving is that simple.
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如大家所知，沒有一個值得解決的問題會是那麼簡單
03:02
I am very concerned about this
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對這個現象我非常擔心
03:04
because I'm going to retire in a world that my students will run.
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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
03:07
I'm doing bad things
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如果我用現在的方式
03:09
to my own future and well-being
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教育他們， 那我就是在跟我自己的
03:11
when I teach this way.
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未來及福利過不去
03:13
I'm here to tell you that the way our textbooks -- particularly
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我在這裡要告訴大家，現今的教科書，
03:16
mass-adopted textbooks -- teach math reasoning
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尤其是那些已被大量採用的教科書，
03:19
and patient problem solving,
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它們教數學推理及耐心解決問題的方式，
03:21
it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
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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
03:24
(Laughter)
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(笑)
03:26
In all seriousness. Here's an example from a physics textbook.
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這問題要嚴肅以待，這裡是一個由物理教科書取出的例子
03:29
It applies equally to math.
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但在數學上也適用
03:31
Notice, first of all here,
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注意看，首先它
03:33
that you have exactly three pieces of information there,
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提供三個片段資訊
03:35
each of which will figure into a formula
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而這些資訊剛好都最後套到公式
03:37
somewhere, eventually,
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的某一個地方
03:39
which the student will then compute.
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學生就將它的結果計算出來
03:41
I believe in real life.
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我相信
03:43
And ask yourself, what problem have you solved, ever,
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也請大家問問自己，在真實的生活中，有哪些你解決過的
03:45
that was worth solving
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有價值的問題
03:47
where you knew all of the given information in advance;
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是你在事先就知道所有的資訊
03:49
where you didn't have a surplus of information and you had to filter it out,
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是不需要你由過多的資訊中，過濾出有價值的部分
03:52
or you didn't have sufficient information
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或是資訊不足，
03:54
and had to go find some.
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你必須再找其他資訊補足的
03:56
I'm sure we all agree that no problem worth solving is like that.
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我想我們都同意，沒有任何值得解決的問題是這樣處理的
03:59
And the textbook, I think, knows how it's hamstringing students
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而我認為我們的教科書很懂得讓我們的學生變成跛腳馬
04:02
because, watch this, this is the practice problem set.
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看看這個就知道，這是一個練習題目組
04:05
When it comes time to do the actual problem set,
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當實際做這問題組時
04:07
we have problems like this right here
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問題像右邊這個
04:09
where we're just swapping out numbers and tweaking the context a little bit.
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我們只要將數字切換進去， 做一些小處理答案就出來
04:12
And if the student still doesn't recognize the stamp this was molded from,
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如果學生仍然對於解這類型的題目的模式不懂
04:15
it helpfully explains to you
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它會以解釋例題的方式
04:17
what sample problem you can return to to find the formula.
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讓你回頭找到解題所需的公式
04:20
You could literally, I mean this,
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所以你可以逐一
04:22
pass this particular unit without knowing any physics,
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將各數字帶入公式， 而不必懂物理
04:25
just knowing how to decode a textbook. That's a shame.
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只是知道解碼 就得到答案 這實在太不應該了
04:28
So I can diagnose the problem a little more specifically in math.
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對於數學，我可以更精準的診斷它的問題點
04:31
Here's a really cool problem. I like this.
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這是我很喜歡舉出來的好問題
04:33
It's about defining steepness and slope
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它是用滑雪纜車來定義
04:35
using a ski lift.
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斜率及坡度
04:37
But what you have here is actually four separate layers,
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這圖上有四不同的個階層
04:39
and I'm curious which of you can see the four separate layers
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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
04:42
and, particularly, how when they're compressed together
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特別是當四張圖被壓縮在一起
04:45
and presented to the student all at once,
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同時給學生看的時候
04:47
how that creates this impatient problem solving.
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這顯示出對解題無耐心是怎麼發生的
04:50
I'll define them here: You have the visual.
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我將對其下一個定義 這裡有圖形
04:52
You also have the mathematical structure,
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有數學結構
04:54
talking about grids, measurements, labels,
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例如格子 尺寸 標示
04:56
points, axes, that sort of thing.
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點 軸 這類的元素
04:58
You have substeps, which all lead to what we really want to talk about:
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但解題前需要有次步驟，
05:01
which section is the steepest.
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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
05:03
So I hope you can see.
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我希望你可以看出
05:05
I really hope you can see how what we're doing here
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我們在此進行的
05:07
is taking a compelling question, a compelling answer,
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問題及答案都是強制性
05:09
but we're paving a smooth, straight path
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但我們應該由問題的這端開始
05:11
from one to the other
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舖一條筆直的道路，到解決問題那端
05:13
and congratulating our students for how well
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現在卻只因為學生知道
05:15
they can step over the small cracks in the way.
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如何跨過路中的一個小裂縫而沾沾自喜
05:17
That's all we're doing here.
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這就是我們現在的數學教育
05:19
So I want to put to you that if we can separate these in a different way
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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
05:21
and build them up with students,
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把學生耐心解決問題
05:23
we can have everything we're looking for in terms of patient problem solving.
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的整套的能力都建立起來
05:26
So right here I start with the visual,
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因此在這裡，我先展現一個圖像
05:28
and I immediately ask the question:
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並且立即問
05:30
Which section is the steepest?
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哪一部份坡度最大
05:32
And this starts conversation
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這就開始了對話
05:34
because the visual is created in such a way where you can defend two answers.
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因為圖像是被設計成正反兩方都能找到論據的樣子
05:37
So you get people arguing against each other,
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所以兩方開始爭辯
05:39
friend versus friend,
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朋友和朋友
05:41
in pairs, journaling, whatever.
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情侶對情侶 或是讀者和期刊 等等
05:43
And then eventually we realize
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因此最終我們知道
05:45
it's getting annoying to talk about
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在討論銀幕
05:47
the skier in the lower left-hand side of the screen
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左下方的滑雪者
05:49
or the skier just above the mid line.
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或是線以上的滑雪者有點惱人
05:51
And we realize how great would it be
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另外我們也會體會 如果
05:53
if we just had some A, B, C and D labels
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我們就只討論ABCD四個標示點
05:55
to talk about them more easily.
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事情會簡化一些
05:57
And then as we start to define what does steepness mean,
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當我們給坡度下定義時
06:00
we realize it would be nice to have some measurements
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我們會了解如果能作一些量度，事情會比較好處理
06:02
to really narrow it down, specifically what that means.
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可以把問題做歸納，了解問題的涵義。
06:05
And then and only then,
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只有在這時候，這個階段，
06:07
we throw down that mathematical structure.
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我們才將數學的結構鋪陳出來
06:09
The math serves the conversation,
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數學提供的是對話
06:11
the conversation doesn't serve the math.
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而不是用對話來服侍數學
06:13
And at that point, I'll put it to you that nine out of 10 classes
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我告訴各位 十分之九的課 先用這種方式
06:16
are good to go on the whole slope, steepness thing.
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再進行斜率、坡度教學效果會很好
06:18
But if you need to,
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但假如你需要
06:20
your students can then develop those substeps together.
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你的學生們可以共同發展解題的次步驟
06:22
Do you guys see how this, right here, compared to that --
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你們可以比較這邊這個 和那邊那個
06:25
which one creates that patient problem solving, that math reasoning?
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哪一個可以產生耐心解決問題及數學推理的效果呢
06:28
It's been obvious in my practice, to me.
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以我過去的教學經驗 答案很明顯的
06:31
And I'll yield the floor here for a second to Einstein,
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我將我在這裡所說的，用來呼應愛因斯坦的話
06:33
who, I believe, has paid his dues.
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我相信愛因斯坦此言是經驗累積的結晶
06:35
He talked about the formulation of a problem being so incredibly important,
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他認為規畫問題的能力是無與倫比的重要
06:38
and yet in my practice, in the U.S. here,
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但是在我實際的經驗，在美國
06:40
we just give problems to students;
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我們只給學生問題
06:42
we don't involve them in the formulation of the problem.
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我們沒有讓學生參與問題的形成及規畫
06:46
So 90 percent of what I do
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所以我每星期五小時的
06:48
with my five hours of prep time per week
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課前準備工作中，有百分之九十的時間，
06:50
is to take fairly compelling elements
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是將這一類強制提供的解題要素
06:53
of problems like this from my textbook
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由我的教材中去除。
06:55
and rebuild them in a way that supports math reasoning and patient problem solving.
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然後將其重建成需要數學推理及耐心解題的形式，
06:58
And here's how it works.
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這是它運作的方式，
07:00
I like this question. It's about a water tank.
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我喜歡這和水槽有關的題目
07:02
The question is: How long will it take you to fill it up?
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題目是：將水槽灌滿要多少時間？
07:04
First things first, we eliminate all the substeps.
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重要的事情先做，我將教材中所有的次步驟刪除
07:06
Students have to develop those,
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學生必須自己去發展出來，
07:08
they have to formulate those.
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他們必須自己規劃解題的次步驟，
07:10
And then notice that all the information written on there is stuff you'll need.
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因此他們就注意到，所有寫在那裏的都是解題所需要的，
07:13
None of it's a distractor, so we lose that.
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沒有一個是我們可以忽略分心的。
07:15
Students need to decide, "All right, well,
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學生必須自己決定 ，
07:17
does the height matter? Does the side of it matter?
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水槽高度有關嗎？水槽尺寸有關嗎？
07:19
Does the color of the valve matter? What matters here?"
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閥的顏色有關嗎？哪些才是真正的重要因素？
07:22
Such an underrepresented question in math curriculum.
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以現今的數學教材而言 這題目敘述不完整
07:25
So now we have a water tank.
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我們有一個水槽，
07:27
How long will it take you to fill it up? And that's it.
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要多久你才會將它填滿水，題目就這樣
07:29
And because this is the 21st century
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因為這是21世紀，
07:31
and we would love to talk about the real world on its own terms,
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所以我們喜歡用真實世界的元素來敘述題目，
07:34
not in terms of line art or clip art
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而不是用你常在教科書
07:37
that you so often see in textbooks,
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看到的線圖或插圖，
07:39
we go out and we take a picture of it.
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我們出去外面對實物照一張像。
07:41
So now we have the real deal.
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這樣我們就有真實世界，
07:43
How long will it take it to fill it up?
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要多久才會填滿水呢？
07:45
And then even better is we take a video,
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甚至可以作的更好一點，我們用影片
07:47
a video of someone filling it up.
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紀錄填加水的過程，
07:50
And it's filling up slowly, agonizingly slowly.
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水灌得很慢、惱人的慢，
07:52
It's tedious.
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過程冗長 ，
07:54
Students are looking at their watches, rolling their eyes,
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學生們一直看他們的錶 眼珠子不斷的轉，
07:56
and they're all wondering at some point or another,
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臉上都有某程度的疑惑
07:59
"Man, how long is it going to take to fill up?"
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“天啊 要多久才會灌滿啊”
08:02
(Laughter)
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(笑聲)
08:07
That's how you know you've baited the hook, right?
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各位就知道學生們是怎麼被我騙上鉤的
08:11
And that question, off this right here, is really fun for me
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由這個問題產生許多我認為很有趣的現象。
08:14
because, like the intro,
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就像我在開頭時所說，
08:16
I teach kids -- because of my inexperience --
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因為我沒有經驗，所以我教幼兒。
08:19
I teach the kids that are the most remedial, all right?
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我所教都是哪些最需要矯正的幼兒，
08:21
And I've got kids who will not join a conversation about math
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他們有的不願意加入討論，
08:24
because someone else has the formula;
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是因為別人有解題的公式
08:26
someone else knows how to work the formula better than me,
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別人比他更了解解題的公式，
08:29
so I won't talk about it.
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所以不願意談論。
08:31
But here, every student is on a level playing field of intuition.
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但在這裡 每一個人的直覺立足點是公平的
08:34
Everyone's filled something up with water before,
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因為每個人都有灌過水槽的經驗，
08:37
so I get kids answering the question, "How long will it take?"
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所以當我要幼兒回答，這要多久時間時，
08:40
I've got kids who are mathematically and conversationally intimidated
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就讓這些有數學或是交談壓迫感覺的小孩，
08:43
joining the conversation.
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共同加入了討論。
08:45
We put names on the board, attach them to guesses,
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我把學童姓名寫在黑板，指定他們來猜，
08:48
and kids have bought in here.
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所以學生就被帶入情境。
08:50
And then we follow the process I've described.
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然後依據我前面所敘述的程序去做。
08:52
And the best part here, or one of the better parts
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這樣做最妙的地方，或是比較優的地方就是，
08:54
is that we don't get our answer from the answer key
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我們不由老師版教科書後面的
08:56
in the back of the teacher's edition.
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解題提要來找答案，
08:58
We, instead, just watch the end of the movie.
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我們只要將影片一直看完。
09:01
(Laughter)
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(笑聲)
09:03
And that's terrifying,
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這種現象很可怕，對吧！
09:05
because the theoretical models that always work out
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因為在老師版教科書後面的解題提要，
09:07
in the answer key in the back of a teacher's edition,
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理論的公式總是管用的
09:09
that's great, but
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那很棒呀，但是
09:11
it's scary to talk about sources of error
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但是當理論和實際情況不吻合，
09:13
when the theoretical does not match up with the practical.
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產生了錯誤時就會引起驚慌，
09:15
But those conversations have been so valuable,
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但這正是這些對話中
09:17
among the most valuable.
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最有價值的精華
09:19
So I'm here to report some really fun games
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我來這裡報告一些有趣的教學心得，
09:21
with students who come pre-installed
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我的學生第一天到教室時，
09:23
with these viruses day one of the class.
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許多都是被錯誤數學教育所汙染的帶菌者
09:25
These are the kids who now, one semester in,
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但一學期之後，他們變成這樣。
09:28
I can put something on the board,
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我可以將一些教材寫在黑板上，
09:30
totally new, totally foreign,
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針對新的、完全陌生的事務，
09:32
and they'll have a conversation about it for three or four minutes more
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用比年初多約三或四分鐘。
09:34
than they would have at the start of the year,
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作共同討論
09:36
which is just so fun.
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這真的很有趣，
09:38
We're no longer averse to word problems,
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學生們不再對數學語義問題產生反感，
09:41
because we've redefined what a word problem is.
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因為我們已經對數學語義重新定義；
09:44
We're no longer intimidated by math,
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學生不再對數學產生恐懼，
09:46
because we're slowly redefining what math is.
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因為我們會慢慢的重新定義該情境的數學涵義是什麼
09:48
This has been a lot of fun.
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這樣就變的有趣許多了！
09:50
I encourage math teachers I talk to to use multimedia,
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我鼓勵數學老師們多多利用多媒體
09:53
because it brings the real world into your classroom
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將真實的世界情境
09:55
in high resolution and full color;
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以高解析度 全彩的方式帶入教室
09:57
to encourage student intuition for that level playing field;
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用公平方式鼓勵學生運用直覺
10:00
to ask the shortest question you possibly can
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盡可能 問最簡短的問題
10:02
and let those more specific questions come out in conversation;
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讓特定的問題在對話中出現
10:05
to let students build the problem,
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讓學生自己提出問題
10:07
because Einstein said so;
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就像愛因思坦所說哪樣
10:09
and to finally, in total, just be less helpful,
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最後一點 不要幫太多忙
10:12
because the textbook is helping you in all the wrong ways:
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因為教科書用錯誤的方式在幫倒忙
10:14
It's buying you out of your obligation,
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它買斷你培養學生耐心解決問題
10:17
for patient problem solving and math reasoning, to be less helpful.
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及數學推理的責任 所以不要幫太多忙
10:20
And why this is an amazing time to be a math teacher right now
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為什麼現在正是數學教師的大好時光
10:23
is because we have the tools to create
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因為現在有工具可以創造
10:25
this high-quality curriculum in our front pocket.
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高品質且可攜帶於口袋的教材
10:27
It's ubiquitous and fairly cheap,
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它無所不在也相當便宜
10:29
and the tools to distribute it
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而傳布教材的工具
10:31
freely under open licenses
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因為大量供應及開放授權
10:33
has also never been cheaper or more ubiquitous.
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而價格史上最低 普及性最高
10:36
I put a video series on my blog not so long ago
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不久之前我在我的部落格中放了一系列的影片
10:38
and it got 6,000 views in two weeks.
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在兩個星期內就有六千個人來觀看
10:41
I get emails still from teachers in countries I've never visited
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到現在我仍然接到一些我從未訪問過國家的老師來信
10:44
saying, "Wow, yeah. We had a good conversation about that.
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提到”哇!是的，我們確實用這個與學生有很好的對話
10:47
Oh, and by the way, here's how I made your stuff better,"
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啊!順便告訴你，我有改進你教材的妙方”
10:50
which, wow.
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效果真驚人
10:52
I put this problem on my blog recently:
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我最近在部落格放了這個問題，
10:54
In a grocery store, which line do you get into,
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在雜貨店裡，你要排哪一個結賬櫃檯
10:56
the one that has one cart and 19 items
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是要排在只有一輛購物車 但是上面有19項物品的後面
10:58
or the line with four carts and three, five, two and one items.
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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
11:01
And the linear modeling involved in that was some good stuff for my classroom,
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這是一個線性規劃的問題 是我課堂上的好教材
11:04
but it eventually got me on "Good Morning America" a few weeks later,
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因為它，我幾星期前上了早安美國的節目
11:07
which is just bizarre, right?
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答案超乎一般想像 對吧
11:09
And from all of this, I can only conclude
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從前述影片受歡迎的程度 我得到一個結論就是
11:11
that people, not just students,
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不只是學生
11:13
are really hungry for this.
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一般人也很渴望這些內容
11:15
Math makes sense of the world.
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數學彰顯真實世界的合理性
11:17
Math is the vocabulary
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數學也是描述人類
11:19
for your own intuition.
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直覺的詞彙
11:21
So I just really encourage you, whatever your stake is in education --
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所以我鼓勵大家 不管你在教育上的角色是什麼
11:24
whether you're a student, parent, teacher, policy maker, whatever --
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不管你是學生 家長老師或是政策制定者或其他
11:27
insist on better math curriculum.
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都請堅持要有一個更好的數學教材
11:30
We need more patient problem solvers. Thank you. (Applause)
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我們需要更多有耐心解決問題的人，非常感謝
ABOUT THE SPEAKER
Dan Meyer - EducatorDan Meyer is exploring the way we teach teachers to teach kids.
Why you should listen
Dan Meyer asks, "How can we design the ideal learning experience for students?" As a part-time Googler, a provocative blogger and a full-time high-school math teacher, his perspective on curriculum design, teacher education and teacher retention is informed by tech trends and online discourse as much as front-line experience with students.
Meyer has spun off his enlightening message -- that teachers "be less helpful" and push their students to formulate the steps to solve math problems -- into a nationwide tour-of-duty on the speaking circuit.
Dan Meyer | Speaker | TED.com