Math class needs a makeover | Dan Meyer

翻译人员: Limin Wang 校对人员: Alice2167 King

Can I ask you to please recall a time

我想请你回忆过去，

when you really loved something --

当你真的热爱过一些东西，

a movie, an album, a song or a book --

一部电影，一张专辑，一首歌曲或一本书，

and you recommended it wholeheartedly

并且你全心全意地推荐它

to someone you also really liked,

给你很喜欢的某个人，

and you anticipated that reaction, you waited for it,

你期望他的反应，你一直等待着，

and it came back, and the person hated it?

但反馈回来的却是他很讨厌它。

So, by way of introduction,

因此，以这种方式介绍，

that is the exact same state

在这种完全相同的状态下，

in which I spent every working day of the last six years. (Laughter)

我度过了在过去六年工作中的每一天。

I teach high school math.

我教授高中数学。

I sell a product to a market

我是将一种产品卖到一个

that doesn't want it, but is forced by law to buy it.

不受欢迎的市场，但是依据法律强制购买它。

I mean, it's just a losing proposition.

我的意思是，这是一种 - 它只是一个亏本生意。

So there's a useful stereotype about students that I see,

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

a useful stereotype about you all.

刻板印象。

I could give you guys

我可以给你们

an algebra-two final exam,

一个代数Ⅱ的期末考试，

and I would expect no higher

我不会期望

than a 25 percent pass rate.

超过百分之25的合格率。

And both of these facts say less about you or my students

这两个事实并不表示你们的问题或是我学生的问题，

than they do about what we call math education

它表现的是在当今的美国，我们所谓的数学教育

in the U.S. today.

所面临的问题。

To start with, I'd like to break math down into two categories.

首先，我想把数学分成两类。

One is computation; this is the stuff you've forgotten.

一个是计算。这是你们已经忘记的东西。

For example, factoring quadratics with

例如，首项系数大于1的

leading coefficients greater than one.

保理二次方程式。

This stuff is also really easy to relearn,

这东西也很容易重新学习，

provided you have a really strong grounding

只要你有一个关于推理，数学推理真正过硬

in reasoning. Math reasoning --

的基础。

we'll call it the application

我们将这称之为

of math processes to the world around us --

我们周围世界里的数学程序应用。

this is hard to teach.

这点是很难教授的。

This is what we would love students to retain,

这也是我们希望学生能够掌握并应用，

even if they don't go into mathematical fields.

即使他们不进入数学领域。

This is also something that, the way we teach it in the U.S.

这也是在美国，由于我们教授它的方式

all but ensures they won't retain it.

以致于学生不会应用它。

So, I'd like to talk about why that is,

所以，我要谈谈这是为什么，

why that's such a calamity for society, what we can do about it

为什么这是当今社会的灾难，我们可以为此做什么，

and, to close with, why this is an amazing time

而且，更进一步，为什么现在成为一名数学老师，

to be a math teacher.

这正是一个千载难逢的好时代。

So first, five symptoms

首先，你在数学课堂上

that you're doing math reasoning wrong

做的五个征兆

in your classroom.

是在数学推理上错误的。

One is a lack of initiative; your students don't self-start.

一个是缺乏主动性，你的学生没有自我行动。

You finish your lecture block

你完成你的演讲，

and immediately you have five hands going up

然后马上，你就会看到五个人举手

asking you to re-explain the entire thing at their desks.

要求你到他们课桌前重新解释整件事。

Students lack perseverance.

二学生缺乏坚持不懈。

They lack retention; you find yourself

三他们缺少持续记忆力，你会发现自己

re-explaining concepts three months later, wholesale.

在3个月后需要完完全全的重新解释这些概念。

There's an aversion to word problems,

四还存在着一种对文字题的反感，

which describes 99 percent of my students.

这占了我学生中的99%。

And then the other one percent

五这种情况下，剩下的百分之一

is eagerly looking for the formula

正在急切地寻找公式

to apply in that situation.

来解决问题。

This is really destructive.

这实在是毁灭性的。

David Milch, creator of "Deadwood" and other amazing TV shows,

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，

has a really good description for this.

对此他有一个很好的描述。

He swore off creating

他发誓要停止创造

contemporary drama,

当代戏剧，

shows set in the present day,

及当今的单元连续剧，

because he saw that when people fill their mind

因为他意识到，当人们的脑中

with four hours a day of, for example, "Two and a Half Men," no disrespect,

一天四小时想到，例如，“两个半男人”，没有冒犯的意思，

it shapes the neural pathways, he said,

这就形成了神经传导通路，他说，

in such a way that they expect simple problems.

以这样一种方式，人们期望简单的问题。

He called it, "an impatience with irresolution."

他称这是“一种无法做决定的焦虑。”

You're impatient with things that don't resolve quickly.

人们对不能快速解决的事情不耐烦。

You expect sitcom-sized problems that wrap up in 22 minutes,

人们期望情景喜剧一般可以在22分中结束，

three commercial breaks and a laugh track.

3个广告中断和一个笑声背景。

And I'll put it to all of you,

我告诉你们所有人，

what you already know, that no problem worth solving is that simple.

如大家知道的，解决任何问题都不是那样简单。

I am very concerned about this

我对此很担忧，

because I'm going to retire in a world that my students will run.

因为我将要在我的学生主宰的世界中退休。

I'm doing bad things

当我以这种方式教学，我正在做

to my own future and well-being

对我自己的将来和幸福感

when I teach this way.

不利的事情。

I'm here to tell you that the way our textbooks -- particularly

我在这里告诉大家，我们的教科书，特别是

mass-adopted textbooks -- teach math reasoning

大众常用的教科书，它们教授数学理论

and patient problem solving,

和耐心解决的数学问题，

it's functionally equivalent to turning on "Two and a Half Men" and calling it a day.

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）。

(Laughter)

（笑声）

In all seriousness. Here's an example from a physics textbook.

非常严肃地说，这是物理教科书上的一个例子，

It applies equally to math.

它同样适用于数学。

Notice, first of all here,

首先注意到这里

that you have exactly three pieces of information there,

大家有正好3个信息，

each of which will figure into a formula

每一个都可以表示成某个公式，

somewhere, eventually,

最终，

which the student will then compute.

学生可以以此计算。

I believe in real life.

我相信在真实生活中。

And ask yourself, what problem have you solved, ever,

问问自己，什么问题你们解决过的，

that was worth solving

值得解决的，

where you knew all of the given information in advance;

在此问题中，你提前知道所有信息，

where you didn't have a surplus of information and you had to filter it out,

或者你没有多余信息，然后你需要过滤信息，

or you didn't have sufficient information

或是你没有足够的信息，

and had to go find some.

你需要去查找些信息。

I'm sure we all agree that no problem worth solving is like that.

我肯定大家都同意解决问题都不是如上所述那么简单。

And the textbook, I think, knows how it's hamstringing students

我认为，教科书实在是误人子弟。

because, watch this, this is the practice problem set.

因为，看看这个，这是一个练习题，

When it comes time to do the actual problem set,

当要解决实际问题时，

we have problems like this right here

我们在这就有类似问题，

where we're just swapping out numbers and tweaking the context a little bit.

我们只是在交换数字和稍稍调整内容。

And if the student still doesn't recognize the stamp this was molded from,

如果学生们不知道这类数学题模式，

it helpfully explains to you

而它会帮助向你解释例题，

what sample problem you can return to to find the formula.

什么样题你可以返回去找出公式。

You could literally, I mean this,

你确实可以，我很认真的说，

pass this particular unit without knowing any physics,

通过这个测试，没有任何物理知识，

just knowing how to decode a textbook. That's a shame.

仅仅知道如何解译教科书。那是一种耻辱。

So I can diagnose the problem a little more specifically in math.

我能在数学中更准确得解决这个问题。

Here's a really cool problem. I like this.

这是一个很酷的问题。我喜欢这个。

It's about defining steepness and slope

它是关于确定陡度和坡度

using a ski lift.

在一个滑雪缆车上.

But what you have here is actually four separate layers,

但在这里有四个不同层面。

and I'm curious which of you can see the four separate layers

我很好奇你们哪个可以看到这四个不同层面的，

and, particularly, how when they're compressed together

特别是，它们如何压缩在一起，

and presented to the student all at once,

一次性的展现给学生，

how that creates this impatient problem solving.

又如何轻易地解决了问题。

I'll define them here: You have the visual.

我将在这里详细说明它们。你有图像。

You also have the mathematical structure,

你也有数学结构，

talking about grids, measurements, labels,

谈到格子，测量，标注，

points, axes, that sort of thing.

点，轴，诸如此类。

You have substeps, which all lead to what we really want to talk about:

你有子步骤，它们都引到我们真正想谈论的问题，

which section is the steepest.

哪段是最陡的？

So I hope you can see.

我希望你们能看到。

I really hope you can see how what we're doing here

我真的希望你们能看到，我们在这里

is taking a compelling question, a compelling answer,

是在谈一个激发兴趣的问题，一个有意思的答案，

but we're paving a smooth, straight path

但我们在开拓一个平整的，简单明了的路径，

from one to the other

从一点到另一点，

and congratulating our students for how well

然后祝贺我们的学生，为他们

they can step over the small cracks in the way.

能够很好的沿路踏过小缝隙。

That's all we're doing here.

这就是我们做的。

So I want to put to you that if we can separate these in a different way

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，

and build them up with students,

和学生们一起建造它们，

we can have everything we're looking for in terms of patient problem solving.

我们可以迎刃处理所有需耐心解决的问题。

So right here I start with the visual,

就在这里，我以图像开始，

and I immediately ask the question:

我马上问个问题：

Which section is the steepest?

哪个部分最陡？

And this starts conversation

这会引起交谈，

because the visual is created in such a way where you can defend two answers.

因为这个图像创造的方式使你可以用两个答案辩解。

So you get people arguing against each other,

这样你可以促进人们互相争论，

friend versus friend,

朋友对朋友的争论，

in pairs, journaling, whatever.

一对对，日志式，随便怎样。

And then eventually we realize

然后我们最终认识到

it's getting annoying to talk about

谈到

the skier in the lower left-hand side of the screen

在屏幕左手边滑雪的人

or the skier just above the mid line.

或是刚上中线的滑雪的人，这是挺讨厌的。

And we realize how great would it be

我们认识到如果

if we just had some A, B, C and D labels

我们更简单地谈论带有一些a,b,c,d的标志，

to talk about them more easily.

这该是多么好。

And then as we start to define what does steepness mean,

然后我们确定陡峭程度是指什么，

we realize it would be nice to have some measurements

我们认识到这是一个好主意，如果有一些测量

to really narrow it down, specifically what that means.

来真的缩小范围，特别是那意味着什么。

And then and only then,

然后，在此之后，

we throw down that mathematical structure.

我们放下那个数学架构。

The math serves the conversation,

数学服务于对话，

the conversation doesn't serve the math.

对话却不为数学服务。

And at that point, I'll put it to you that nine out of 10 classes

在这一点，我将展示给你们，10个班里有9个班

are good to go on the whole slope, steepness thing.

能很好的继续解决整个坡度，陡度的问题。

But if you need to,

如果你需要

your students can then develop those substeps together.

你的学生可以在一起展开那些子步骤。

Do you guys see how this, right here, compared to that --

你们可以比较这边这个和那个－－

which one creates that patient problem solving, that math reasoning?

哪个班处理了耐心解决的问题，以及数学推理呢？

It's been obvious in my practice, to me.

在我的实践中，我一目了然。

And I'll yield the floor here for a second to Einstein,

我这就要谈谈爱因斯坦，

who, I believe, has paid his dues.

我认为他做了应有的贡献。

He talked about the formulation of a problem being so incredibly important,

他谈到规划问题的能力是如此重要，

and yet in my practice, in the U.S. here,

但在我的实践中， 在美国这儿，

we just give problems to students;

我们仅仅把问题交给学生；

we don't involve them in the formulation of the problem.

我们不把他们参与到问题的形成和规划过程中。

So 90 percent of what I do

所以我做的90％的事

with my five hours of prep time per week

和我每周5小时的准备时间

is to take fairly compelling elements

是拿问题中非常吸引人的元素，

of problems like this from my textbook

像在我的教科书里的这些问题，

and rebuild them in a way that supports math reasoning and patient problem solving.

把它们重组成能支持数学推理和耐心解决问题的方法。

And here's how it works.

这里是它如何运作的。

I like this question. It's about a water tank.

我喜欢这个问题，这是关于一个水桶。

The question is: How long will it take you to fill it up?

问题是：多长时间可以装满这个水桶呢？

First things first, we eliminate all the substeps.

第一件事，我们简化子步骤。

Students have to develop those,

学生们必须得展开这些子步骤，

they have to formulate those.

用公式化表达它们。

And then notice that all the information written on there is stuff you'll need.

然后认识到写在那里的东西是他们所需要的所有信息。

None of it's a distractor, so we lose that.

没有一个是多余信息，用不着删除。

Students need to decide, "All right, well,

学生们需要决定，好了，

does the height matter? Does the side of it matter?

高度重要吗？它的尺码重要吗？

Does the color of the valve matter? What matters here?"

开关的颜色重要吗？这里什么是重要的？

Such an underrepresented question in math curriculum.

这在数学课里是不具代表性的问题。

So now we have a water tank.

这样在这里我们有了一个水桶。

How long will it take you to fill it up? And that's it.

多长时间会把它装满，就是这样。

And because this is the 21st century

因为这是21世纪，

and we would love to talk about the real world on its own terms,

我们喜欢以真实世界的表达方式来谈论真实的世界，

not in terms of line art or clip art

不是以你常在教科书上看到的线条艺术

that you so often see in textbooks,

或是插图艺术来表达。

we go out and we take a picture of it.

我们出去应用，并给它照个相。

So now we have the real deal.

现在我们有了真实的东西。

How long will it take it to fill it up?

它要多长时间装满？

And then even better is we take a video,

更好的，我们录像，

a video of someone filling it up.

一个人在装满它的录像。

And it's filling up slowly, agonizingly slowly.

它在慢慢的装满，令人不耐烦的等待。

It's tedious.

这很乏味。

Students are looking at their watches, rolling their eyes,

学生们在看他们的表，翻白眼，

and they're all wondering at some point or another,

他们时不时地想，

"Man, how long is it going to take to fill up?"

“伙计，这个要多长时间装满呐？”

(Laughter)

（笑声）

That's how you know you've baited the hook, right?

这就是你想要引起关注的地方，好。

And that question, off this right here, is really fun for me

这个问题，就在这里，对我来说很有趣。

because, like the intro,

因为，像个引言，

I teach kids -- because of my inexperience --

我教小孩子，因为我的经验，

I teach the kids that are the most remedial, all right?

我教那些需要辅导的孩子。

And I've got kids who will not join a conversation about math

我不教那些不参与数学对话的孩子，

because someone else has the formula;

因为他们懂公式，

someone else knows how to work the formula better than me,

也知道如何更好地运用公式，甚至比我好。

so I won't talk about it.

所以我不讨论这个。

But here, every student is on a level playing field of intuition.

但在这里，每个人都在同一个水平上运用直觉。

Everyone's filled something up with water before,

每个人都装过水，

so I get kids answering the question, "How long will it take?"

所以我让孩子们自己回答这个问题：多长时间装满水。

I've got kids who are mathematically and conversationally intimidated

我让这些在数学上和交谈上感到恐惧的孩子们

joining the conversation.

也加入谈论。

We put names on the board, attach them to guesses,

我们把名字写到黑板上，把它们连到猜测的答案上，

and kids have bought in here.

孩子们把答案带到这里。

And then we follow the process I've described.

然后孩子们遵照我描述的过程。

And the best part here, or one of the better parts

最好的部分，或者说，一个更好的部分

is that we don't get our answer from the answer key

是我们没有从

in the back of the teacher's edition.

教师版教科书的后面得到答案。

We, instead, just watch the end of the movie.

我们却看到了录像的结尾。

(Laughter)

（笑声）

And that's terrifying,

这很有意思，对吧。

because the theoretical models that always work out

因为理论模型总是得出

in the answer key in the back of a teacher's edition,

和教师版后面的答案一样的答案，

that's great, but

这很好，

it's scary to talk about sources of error

但谈到错误的来源，

when the theoretical does not match up with the practical.

当理论不符合实际的时候，就让人害怕。

But those conversations have been so valuable,

但那些谈论是如此珍贵，

among the most valuable.

是最有价值的。

So I'm here to report some really fun games

所以我在这里报导一些很有趣的收获，

with students who come pre-installed

关于在数学课程刚开始的时候，学生带有这些预置的

with these viruses day one of the class.

错误征兆。

These are the kids who now, one semester in,

这些孩子现在已经学了一个学期，

I can put something on the board,

我能在黑板上写下一些

totally new, totally foreign,

全新的，完全陌生的问题，

and they'll have a conversation about it for three or four minutes more

他们能就此展开讨论，

than they would have at the start of the year,

这比他们在学年开始的时候长3，4分钟，

which is just so fun.

这真有意思。

We're no longer averse to word problems,

孩子们不再反感数学问答题，

because we've redefined what a word problem is.

因为他们从新定义数学问答题。

We're no longer intimidated by math,

孩子们不再恐惧数学，

because we're slowly redefining what math is.

因为他们在慢慢从新定义数学。

This has been a lot of fun.

这个过程很有意思。

I encourage math teachers I talk to to use multimedia,

我鼓励数学教师多多应用多媒体，

because it brings the real world into your classroom

因为它把真实世界以高清晰度的图片和全色彩图像带到

in high resolution and full color;

你的教室，

to encourage student intuition for that level playing field;

来鼓励学生对那个场景的直觉，

to ask the shortest question you possibly can

来问你可能问的最短的问题，

and let those more specific questions come out in conversation;

让那些更细，更具体的问题随着交谈显露出来，

to let students build the problem,

让学生们着手解决问题，

because Einstein said so;

正如爱因斯坦所说，

and to finally, in total, just be less helpful,

总而言之，就是给予更少的帮助，

because the textbook is helping you in all the wrong ways:

因为教科书在以非常错误的方式帮助大家。

It's buying you out of your obligation,

它在免去大家

for patient problem solving and math reasoning, to be less helpful.

对耐心问题的解决和数学推理的责任，更没有帮助性。

And why this is an amazing time to be a math teacher right now

那为什么现在是作为一个数学教师的绝好时候，

is because we have the tools to create

这是因为我们有工具来创造

this high-quality curriculum in our front pocket.

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。

It's ubiquitous and fairly cheap,

它无所不在，非常便宜。

and the tools to distribute it

还有自由散播它的工具，

freely under open licenses

在公开授权下，

has also never been cheaper or more ubiquitous.

从没有比现在更便宜或更普遍的社交互联网工具。

I put a video series on my blog not so long ago

不久之前，我把一系列的录像放在我的博客里，

and it got 6,000 views in two weeks.

在两周之内它得到6000的观看率。

I get emails still from teachers in countries I've never visited

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：

saying, "Wow, yeah. We had a good conversation about that.

“哦，是的。我们就此展开了很好的讨论。

Oh, and by the way, here's how I made your stuff better,"

哦，顺便说一下，这是我如何改善你教学材料的例子。”

which, wow.

这么多，喔。

I put this problem on my blog recently:

我最近在我的博客上提出这个问题。

In a grocery store, which line do you get into,

在一个杂物店，你排那个队，

the one that has one cart and 19 items

一个是有一个手推车和19个商品

or the line with four carts and three, five, two and one items.

或是另一个有4个手推车和3，5，2 和1个商品。

And the linear modeling involved in that was some good stuff for my classroom,

这里牵涉到的线性模型，它成为了我的教学中很好的材料，

but it eventually got me on "Good Morning America" a few weeks later,

但它最后让我上了在几周之前“早安，美国”的节目，

which is just bizarre, right?

这的确是不可思议，对吧。

And from all of this, I can only conclude

从所有这些，我只能下此结论

that people, not just students,

人们，不仅仅是学生，

are really hungry for this.

都非常渴望这个。

Math makes sense of the world.

数学解读世界。

Math is the vocabulary

数学也是描述人们

for your own intuition.

直觉的词汇。

So I just really encourage you, whatever your stake is in education --

所以我只是鼓励大家，无论你们是什么教育程度，

whether you're a student, parent, teacher, policy maker, whatever --

无论你们是学生，还是家长，老师，决策者，不论是什么，

insist on better math curriculum.

请坚持更好地应用数学教程。

We need more patient problem solvers. Thank you. (Applause)

我们需要更多耐心的问题解决者。谢谢。