TEDxNYED
Dan Meyer: Math class needs a makeover
达恩·迈尔Dan Meyer：数学课需要改革
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如今的数学教学是期望学生擅长于由数字堆砌的功课，这剥夺了孩子们更重要的解决问题的能力。在TEDxNYED，达恩·迈尔展示了基于课堂测试的数学练习，他鼓励学生停下来，并想一想。
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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我想请你回忆过去，
00:18
when you really loved something --
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当你真的热爱过一些东西，
00:20
a movie, an album, a song or a book --
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一部电影，一张专辑，一首歌曲或一本书，
00:22
and you recommended it wholeheartedly
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并且你全心全意地推荐它
00:25
to someone you also really liked,
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给你很喜欢的某个人，
00:27
and you anticipated that reaction, you waited for it,
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你期望他的反应，你一直等待着，
00:29
and it came back, and the person hated it?
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但反馈回来的却是他很讨厌它。
00:32
So, by way of introduction,
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因此，以这种方式介绍，
00:34
that is the exact same state
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在这种完全相同的状态下，
00:36
in which I spent every working day of the last six years. (Laughter)
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我度过了在过去六年工作中的每一天。
00:39
I teach high school math.
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我教授高中数学。
00:41
I sell a product to a market
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我是将一种产品卖到一个
00:44
that doesn't want it, but is forced by law to buy it.
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不受欢迎的市场，但是依据法律强制购买它。
00:47
I mean, it's just a losing proposition.
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我的意思是，这是一种 - 它只是一个亏本生意。
00:50
So there's a useful stereotype about students that I see,
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因此，关于学数学的学生和你们所有人，我看到一个很形象的
00:53
a useful stereotype about you all.
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刻板印象。
00:55
I could give you guys
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我可以给你们
00:57
an algebra-two final exam,
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一个代数Ⅱ的期末考试，
00:59
and I would expect no higher
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我不会期望
01:01
than a 25 percent pass rate.
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超过百分之25的合格率。
01:03
And both of these facts say less about you or my students
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这两个事实并不表示你们的问题或是我学生的问题，
01:06
than they do about what we call math education
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它表现的是在当今的美国，我们所谓的数学教育
01:08
in the U.S. today.
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所面临的问题。
01:10
To start with, I'd like to break math down into two categories.
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首先，我想把数学分成两类。
01:13
One is computation; this is the stuff you've forgotten.
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一个是计算。这是你们已经忘记的东西。
01:16
For example, factoring quadratics with
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例如，首项系数大于1的
01:18
leading coefficients greater than one.
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保理二次方程式。
01:20
This stuff is also really easy to relearn,
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这东西也很容易重新学习，
01:22
provided you have a really strong grounding
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只要你有一个关于推理，数学推理真正过硬
01:24
in reasoning. Math reasoning --
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的基础。
01:26
we'll call it the application
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我们将这称之为
01:28
of math processes to the world around us --
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我们周围世界里的数学程序应用。
01:30
this is hard to teach.
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这点是很难教授的。
01:32
This is what we would love students to retain,
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这也是我们希望学生能够掌握并应用，
01:34
even if they don't go into mathematical fields.
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即使他们不进入数学领域。
01:36
This is also something that, the way we teach it in the U.S.
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这也是在美国，由于我们教授它的方式
01:38
all but ensures they won't retain it.
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以致于学生不会应用它。
01:41
So, I'd like to talk about why that is,
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所以，我要谈谈这是为什么，
01:42
why that's such a calamity for society, what we can do about it
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为什么这是当今社会的灾难，我们可以为此做什么，
01:45
and, to close with, why this is an amazing time
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而且，更进一步，为什么现在成为一名数学老师，
01:47
to be a math teacher.
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这正是一个千载难逢的好时代。
01:49
So first, five symptoms
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首先，你在数学课堂上
01:51
that you're doing math reasoning wrong
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做的五个征兆
01:53
in your classroom.
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是在数学推理上错误的。
01:55
One is a lack of initiative; your students don't self-start.
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一个是缺乏主动性，你的学生没有自我行动。
01:58
You finish your lecture block
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你完成你的演讲，
02:00
and immediately you have five hands going up
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然后马上，你就会看到五个人举手
02:02
asking you to re-explain the entire thing at their desks.
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要求你到他们课桌前重新解释整件事。
02:04
Students lack perseverance.
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二学生缺乏坚持不懈。
02:06
They lack retention; you find yourself
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三他们缺少持续记忆力，你会发现自己
02:08
re-explaining concepts three months later, wholesale.
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在3个月后需要完完全全的重新解释这些概念。
02:10
There's an aversion to word problems,
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四还存在着一种对文字题的反感，
02:12
which describes 99 percent of my students.
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这占了我学生中的99%。
02:14
And then the other one percent
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五这种情况下，剩下的百分之一
02:16
is eagerly looking for the formula
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正在急切地寻找公式
02:18
to apply in that situation.
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来解决问题。
02:20
This is really destructive.
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这实在是毁灭性的。
02:22
David Milch, creator of "Deadwood" and other amazing TV shows,
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大卫·米尔希是“死木”美剧以及其他的精彩电视节目的创造者，
02:25
has a really good description for this.
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对此他有一个很好的描述。
02:28
He swore off creating
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他发誓要停止创造
02:30
contemporary drama,
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当代戏剧，
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shows set in the present day,
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及当今的单元连续剧，
02:34
because he saw that when people fill their mind
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因为他意识到，当人们的脑中
02:36
with four hours a day of, for example, "Two and a Half Men," no disrespect,
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一天四小时想到，例如，“两个半男人”，没有冒犯的意思，
02:39
it shapes the neural pathways, he said,
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这就形成了神经传导通路，他说，
02:41
in such a way that they expect simple problems.
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以这样一种方式，人们期望简单的问题。
02:44
He called it, "an impatience with irresolution."
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他称这是“一种无法做决定的焦虑。”
02:47
You're impatient with things that don't resolve quickly.
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人们对不能快速解决的事情不耐烦。
02:50
You expect sitcom-sized problems that wrap up in 22 minutes,
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人们期望情景喜剧一般可以在22分中结束，
02:53
three commercial breaks and a laugh track.
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3个广告中断和一个笑声背景。
02:56
And I'll put it to all of you,
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我告诉你们所有人，
02:59
what you already know, that no problem worth solving is that simple.
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如大家知道的，解决任何问题都不是那样简单。
03:02
I am very concerned about this
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我对此很担忧，
03:04
because I'm going to retire in a world that my students will run.
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因为我将要在我的学生主宰的世界中退休。
03:07
I'm doing bad things
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当我以这种方式教学，我正在做
03:09
to my own future and well-being
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对我自己的将来和幸福感
03:11
when I teach this way.
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不利的事情。
03:13
I'm here to tell you that the way our textbooks -- particularly
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我在这里告诉大家，我们的教科书，特别是
03:16
mass-adopted textbooks -- teach math reasoning
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大众常用的教科书，它们教授数学理论
03:19
and patient problem solving,
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和耐心解决的数学问题，
03:21
it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.
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功能上和打开“好汉两个半”电视系列剧一样，然后这就结束了（消磨时间）。
03:24
(Laughter)
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（笑声）
03:26
In all seriousness. Here's an example from a physics textbook.
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非常严肃地说，这是物理教科书上的一个例子，
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It applies equally to math.
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它同样适用于数学。
03:31
Notice, first of all here,
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首先注意到这里
03:33
that you have exactly three pieces of information there,
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大家有正好3个信息，
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each of which will figure into a formula
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每一个都可以表示成某个公式，
03:37
somewhere, eventually,
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最终，
03:39
which the student will then compute.
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学生可以以此计算。
03:41
I believe in real life.
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我相信在真实生活中。
03:43
And ask yourself, what problem have you solved, ever,
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问问自己，什么问题你们解决过的，
03:45
that was worth solving
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值得解决的，
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where you knew all of the given information in advance;
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在此问题中，你提前知道所有信息，
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where you didn't have a surplus of information and you had to filter it out,
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或者你没有多余信息，然后你需要过滤信息，
03:52
or you didn't have sufficient information
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或是你没有足够的信息，
03:54
and had to go find some.
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你需要去查找些信息。
03:56
I'm sure we all agree that no problem worth solving is like that.
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我肯定大家都同意解决问题都不是如上所述那么简单。
03:59
And the textbook, I think, knows how it's hamstringing students
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我认为，教科书实在是误人子弟。
04:02
because, watch this, this is the practice problem set.
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因为，看看这个，这是一个练习题，
04:05
When it comes time to do the actual problem set,
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当要解决实际问题时，
04:07
we have problems like this right here
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我们在这就有类似问题，
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where we're just swapping out numbers and tweaking the context a little bit.
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我们只是在交换数字和稍稍调整内容。
04:12
And if the student still doesn't recognize the stamp this was molded from,
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如果学生们不知道这类数学题模式，
04:15
it helpfully explains to you
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而它会帮助向你解释例题，
04:17
what sample problem you can return to to find the formula.
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什么样题你可以返回去找出公式。
04:20
You could literally, I mean this,
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你确实可以，我很认真的说，
04:22
pass this particular unit without knowing any physics,
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通过这个测试，没有任何物理知识，
04:25
just knowing how to decode a textbook. That's a shame.
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仅仅知道如何解译教科书。那是一种耻辱。
04:28
So I can diagnose the problem a little more specifically in math.
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我能在数学中更准确得解决这个问题。
04:31
Here's a really cool problem. I like this.
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这是一个很酷的问题。我喜欢这个。
04:33
It's about defining steepness and slope
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它是关于确定陡度和坡度
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using a ski lift.
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在一个滑雪缆车上.
04:37
But what you have here is actually four separate layers,
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但在这里有四个不同层面。
04:39
and I'm curious which of you can see the four separate layers
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我很好奇你们哪个可以看到这四个不同层面的，
04:42
and, particularly, how when they're compressed together
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特别是，它们如何压缩在一起，
04:45
and presented to the student all at once,
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一次性的展现给学生，
04:47
how that creates this impatient problem solving.
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又如何轻易地解决了问题。
04:50
I'll define them here: You have the visual.
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我将在这里详细说明它们。你有图像。
04:52
You also have the mathematical structure,
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你也有数学结构，
04:54
talking about grids, measurements, labels,
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谈到格子，测量，标注，
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points, axes, that sort of thing.
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点，轴，诸如此类。
04:58
You have substeps, which all lead to what we really want to talk about:
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你有子步骤，它们都引到我们真正想谈论的问题，
05:01
which section is the steepest.
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哪段是最陡的？
05:03
So I hope you can see.
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我希望你们能看到。
05:05
I really hope you can see how what we're doing here
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我真的希望你们能看到，我们在这里
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is taking a compelling question, a compelling answer,
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是在谈一个激发兴趣的问题，一个有意思的答案，
05:09
but we're paving a smooth, straight path
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但我们在开拓一个平整的，简单明了的路径，
05:11
from one to the other
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从一点到另一点，
05:13
and congratulating our students for how well
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然后祝贺我们的学生，为他们
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they can step over the small cracks in the way.
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能够很好的沿路踏过小缝隙。
05:17
That's all we're doing here.
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这就是我们做的。
05:19
So I want to put to you that if we can separate these in a different way
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所以我想展示给你们的是，如果我们能以不同方式分开这些，
05:21
and build them up with students,
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和学生们一起建造它们，
05:23
we can have everything we're looking for in terms of patient problem solving.
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我们可以迎刃处理所有需耐心解决的问题。
05:26
So right here I start with the visual,
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就在这里，我以图像开始，
05:28
and I immediately ask the question:
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我马上问个问题：
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Which section is the steepest?
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哪个部分最陡？
05:32
And this starts conversation
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这会引起交谈，
05:34
because the visual is created in such a way where you can defend two answers.
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因为这个图像创造的方式使你可以用两个答案辩解。
05:37
So you get people arguing against each other,
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这样你可以促进人们互相争论，
05:39
friend versus friend,
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朋友对朋友的争论，
05:41
in pairs, journaling, whatever.
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一对对，日志式，随便怎样。
05:43
And then eventually we realize
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然后我们最终认识到
05:45
it's getting annoying to talk about
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谈到
05:47
the skier in the lower left-hand side of the screen
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在屏幕左手边滑雪的人
05:49
or the skier just above the mid line.
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或是刚上中线的滑雪的人，这是挺讨厌的。
05:51
And we realize how great would it be
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我们认识到如果
05:53
if we just had some A, B, C and D labels
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我们更简单地谈论带有一些a,b,c,d的标志，
05:55
to talk about them more easily.
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这该是多么好。
05:57
And then as we start to define what does steepness mean,
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然后我们确定陡峭程度是指什么，
06:00
we realize it would be nice to have some measurements
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我们认识到这是一个好主意，如果有一些测量
06:02
to really narrow it down, specifically what that means.
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来真的缩小范围，特别是那意味着什么。
06:05
And then and only then,
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然后，在此之后，
06:07
we throw down that mathematical structure.
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我们放下那个数学架构。
06:09
The math serves the conversation,
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数学服务于对话，
06:11
the conversation doesn't serve the math.
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对话却不为数学服务。
06:13
And at that point, I'll put it to you that nine out of 10 classes
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在这一点，我将展示给你们，10个班里有9个班
06:16
are good to go on the whole slope, steepness thing.
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能很好的继续解决整个坡度，陡度的问题。
06:18
But if you need to,
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如果你需要
06:20
your students can then develop those substeps together.
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你的学生可以在一起展开那些子步骤。
06:22
Do you guys see how this, right here, compared to that --
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你们可以比较这边这个和那个－－
06:25
which one creates that patient problem solving, that math reasoning?
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哪个班处理了耐心解决的问题，以及数学推理呢？
06:28
It's been obvious in my practice, to me.
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在我的实践中，我一目了然。
06:31
And I'll yield the floor here for a second to Einstein,
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我这就要谈谈爱因斯坦，
06:33
who, I believe, has paid his dues.
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我认为他做了应有的贡献。
06:35
He talked about the formulation of a problem being so incredibly important,
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他谈到规划问题的能力是如此重要，
06:38
and yet in my practice, in the U.S. here,
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但在我的实践中， 在美国这儿，
06:40
we just give problems to students;
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我们仅仅把问题交给学生；
06:42
we don't involve them in the formulation of the problem.
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我们不把他们参与到问题的形成和规划过程中。
06:46
So 90 percent of what I do
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所以我做的90％的事
06:48
with my five hours of prep time per week
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和我每周5小时的准备时间
06:50
is to take fairly compelling elements
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是拿问题中非常吸引人的元素，
06:53
of problems like this from my textbook
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像在我的教科书里的这些问题，
06:55
and rebuild them in a way that supports math reasoning and patient problem solving.
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把它们重组成能支持数学推理和耐心解决问题的方法。
06:58
And here's how it works.
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这里是它如何运作的。
07:00
I like this question. It's about a water tank.
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我喜欢这个问题，这是关于一个水桶。
07:02
The question is: How long will it take you to fill it up?
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问题是：多长时间可以装满这个水桶呢？
07:04
First things first, we eliminate all the substeps.
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第一件事，我们简化子步骤。
07:06
Students have to develop those,
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学生们必须得展开这些子步骤，
07:08
they have to formulate those.
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用公式化表达它们。
07:10
And then notice that all the information written on there is stuff you'll need.
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然后认识到写在那里的东西是他们所需要的所有信息。
07:13
None of it's a distractor, so we lose that.
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没有一个是多余信息，用不着删除。
07:15
Students need to decide, "All right, well,
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学生们需要决定，好了，
07:17
does the height matter? Does the side of it matter?
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高度重要吗？它的尺码重要吗？
07:19
Does the color of the valve matter? What matters here?"
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开关的颜色重要吗？这里什么是重要的？
07:22
Such an underrepresented question in math curriculum.
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这在数学课里是不具代表性的问题。
07:25
So now we have a water tank.
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这样在这里我们有了一个水桶。
07:27
How long will it take you to fill it up? And that's it.
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多长时间会把它装满，就是这样。
07:29
And because this is the 21st century
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因为这是21世纪，
07:31
and we would love to talk about the real world on its own terms,
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我们喜欢以真实世界的表达方式来谈论真实的世界，
07:34
not in terms of line art or clip art
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不是以你常在教科书上看到的线条艺术
07:37
that you so often see in textbooks,
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或是插图艺术来表达。
07:39
we go out and we take a picture of it.
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我们出去应用，并给它照个相。
07:41
So now we have the real deal.
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现在我们有了真实的东西。
07:43
How long will it take it to fill it up?
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它要多长时间装满？
07:45
And then even better is we take a video,
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更好的，我们录像，
07:47
a video of someone filling it up.
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一个人在装满它的录像。
07:50
And it's filling up slowly, agonizingly slowly.
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它在慢慢的装满，令人不耐烦的等待。
07:52
It's tedious.
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这很乏味。
07:54
Students are looking at their watches, rolling their eyes,
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学生们在看他们的表，翻白眼，
07:56
and they're all wondering at some point or another,
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他们时不时地想，
07:59
"Man, how long is it going to take to fill up?"
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“伙计，这个要多长时间装满呐？”
08:02
(Laughter)
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（笑声）
08:07
That's how you know you've baited the hook, right?
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这就是你想要引起关注的地方，好。
08:11
And that question, off this right here, is really fun for me
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这个问题，就在这里，对我来说很有趣。
08:14
because, like the intro,
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因为，像个引言，
08:16
I teach kids -- because of my inexperience --
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我教小孩子，因为我的经验，
08:19
I teach the kids that are the most remedial, all right?
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我教那些需要辅导的孩子。
08:21
And I've got kids who will not join a conversation about math
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我不教那些不参与数学对话的孩子，
08:24
because someone else has the formula;
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因为他们懂公式，
08:26
someone else knows how to work the formula better than me,
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也知道如何更好地运用公式，甚至比我好。
08:29
so I won't talk about it.
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所以我不讨论这个。
08:31
But here, every student is on a level playing field of intuition.
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但在这里，每个人都在同一个水平上运用直觉。
08:34
Everyone's filled something up with water before,
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每个人都装过水，
08:37
so I get kids answering the question, "How long will it take?"
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所以我让孩子们自己回答这个问题：多长时间装满水。
08:40
I've got kids who are mathematically and conversationally intimidated
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我让这些在数学上和交谈上感到恐惧的孩子们
08:43
joining the conversation.
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也加入谈论。
08:45
We put names on the board, attach them to guesses,
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我们把名字写到黑板上，把它们连到猜测的答案上，
08:48
and kids have bought in here.
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孩子们把答案带到这里。
08:50
And then we follow the process I've described.
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然后孩子们遵照我描述的过程。
08:52
And the best part here, or one of the better parts
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最好的部分，或者说，一个更好的部分
08:54
is that we don't get our answer from the answer key
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是我们没有从
08:56
in the back of the teacher's edition.
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教师版教科书的后面得到答案。
08:58
We, instead, just watch the end of the movie.
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我们却看到了录像的结尾。
09:01
(Laughter)
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（笑声）
09:03
And that's terrifying,
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这很有意思，对吧。
09:05
because the theoretical models that always work out
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因为理论模型总是得出
09:07
in the answer key in the back of a teacher's edition,
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和教师版后面的答案一样的答案，
09:09
that's great, but
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这很好，
09:11
it's scary to talk about sources of error
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但谈到错误的来源，
09:13
when the theoretical does not match up with the practical.
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当理论不符合实际的时候，就让人害怕。
09:15
But those conversations have been so valuable,
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但那些谈论是如此珍贵，
09:17
among the most valuable.
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是最有价值的。
09:19
So I'm here to report some really fun games
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所以我在这里报导一些很有趣的收获，
09:21
with students who come pre-installed
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关于在数学课程刚开始的时候，学生带有这些预置的
09:23
with these viruses day one of the class.
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错误征兆。
09:25
These are the kids who now, one semester in,
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这些孩子现在已经学了一个学期，
09:28
I can put something on the board,
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我能在黑板上写下一些
09:30
totally new, totally foreign,
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全新的，完全陌生的问题，
09:32
and they'll have a conversation about it for three or four minutes more
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他们能就此展开讨论，
09:34
than they would have at the start of the year,
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这比他们在学年开始的时候长3，4分钟，
09:36
which is just so fun.
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这真有意思。
09:38
We're no longer averse to word problems,
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孩子们不再反感数学问答题，
09:41
because we've redefined what a word problem is.
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因为他们从新定义数学问答题。
09:44
We're no longer intimidated by math,
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孩子们不再恐惧数学，
09:46
because we're slowly redefining what math is.
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因为他们在慢慢从新定义数学。
09:48
This has been a lot of fun.
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这个过程很有意思。
09:50
I encourage math teachers I talk to to use multimedia,
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我鼓励数学教师多多应用多媒体，
09:53
because it brings the real world into your classroom
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因为它把真实世界以高清晰度的图片和全色彩图像带到
09:55
in high resolution and full color;
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你的教室，
09:57
to encourage student intuition for that level playing field;
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来鼓励学生对那个场景的直觉，
10:00
to ask the shortest question you possibly can
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来问你可能问的最短的问题，
10:02
and let those more specific questions come out in conversation;
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让那些更细，更具体的问题随着交谈显露出来，
10:05
to let students build the problem,
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让学生们着手解决问题，
10:07
because Einstein said so;
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正如爱因斯坦所说，
10:09
and to finally, in total, just be less helpful,
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总而言之，就是给予更少的帮助，
10:12
because the textbook is helping you in all the wrong ways:
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因为教科书在以非常错误的方式帮助大家。
10:14
It's buying you out of your obligation,
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它在免去大家
10:17
for patient problem solving and math reasoning, to be less helpful.
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对耐心问题的解决和数学推理的责任，更没有帮助性。
10:20
And why this is an amazing time to be a math teacher right now
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那为什么现在是作为一个数学教师的绝好时候，
10:23
is because we have the tools to create
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这是因为我们有工具来创造
10:25
this high-quality curriculum in our front pocket.
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这个高质量的数学课程，就在我们的口袋里的工具手机等等。
10:27
It's ubiquitous and fairly cheap,
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它无所不在，非常便宜。
10:29
and the tools to distribute it
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还有自由散播它的工具，
10:31
freely under open licenses
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在公开授权下，
10:33
has also never been cheaper or more ubiquitous.
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从没有比现在更便宜或更普遍的社交互联网工具。
10:36
I put a video series on my blog not so long ago
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不久之前，我把一系列的录像放在我的博客里，
10:38
and it got 6,000 views in two weeks.
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在两周之内它得到6000的观看率。
10:41
I get emails still from teachers in countries I've never visited
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我还收到很多我从来没有去过的国家的邮件，有的教师写邮件说：
10:44
saying, "Wow, yeah. We had a good conversation about that.
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“哦，是的。我们就此展开了很好的讨论。
10:47
Oh, and by the way, here's how I made your stuff better,"
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哦，顺便说一下，这是我如何改善你教学材料的例子。”
10:50
which, wow.
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这么多，喔。
10:52
I put this problem on my blog recently:
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我最近在我的博客上提出这个问题。
10:54
In a grocery store, which line do you get into,
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在一个杂物店，你排那个队，
10:56
the one that has one cart and 19 items
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一个是有一个手推车和19个商品
10:58
or the line with four carts and three, five, two and one items.
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或是另一个有4个手推车和3，5，2 和1个商品。
11:01
And the linear modeling involved in that was some good stuff for my classroom,
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这里牵涉到的线性模型，它成为了我的教学中很好的材料，
11:04
but it eventually got me on "Good Morning America" a few weeks later,
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但它最后让我上了在几周之前“早安，美国”的节目，
11:07
which is just bizarre, right?
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这的确是不可思议，对吧。
11:09
And from all of this, I can only conclude
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从所有这些，我只能下此结论
11:11
that people, not just students,
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人们，不仅仅是学生，
11:13
are really hungry for this.
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都非常渴望这个。
11:15
Math makes sense of the world.
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数学解读世界。
11:17
Math is the vocabulary
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数学也是描述人们
11:19
for your own intuition.
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直觉的词汇。
11:21
So I just really encourage you, whatever your stake is in education --
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所以我只是鼓励大家，无论你们是什么教育程度，
11:24
whether you're a student, parent, teacher, policy maker, whatever --
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无论你们是学生，还是家长，老师，决策者，不论是什么，
11:27
insist on better math curriculum.
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请坚持更好地应用数学教程。
11:30
We need more patient problem solvers. Thank you. (Applause)
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我们需要更多耐心的问题解决者。谢谢。
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