Math class needs a makeover | Dan Meyer

譯者: 張新永 Davidchang 審譯者: Lin Su-Wei(林書暐)

Can I ask you to please recall a time

請各位回想一下過去是否曾經

when you really loved something --

你非常喜歡一種東西

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

例如一部電影、一張唱片、一首歌或是一本書

and you recommended it wholeheartedly

在你全心全意的向

to someone you also really liked,

你所鍾愛的人推薦

and you anticipated that reaction, you waited for it,

你預期對方會有和你一樣的反應，你等著等著

and it came back, and the person hated it?

得到的答案卻是，他恨死那東西了

So, by way of introduction,

在自我介紹階段，

that is the exact same state

我要告訴各位，這正是

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

我過去六年每個工作天所面臨情形的寫照

I teach high school math.

我在高中教數學

I sell a product to a market

我生產一種

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

市場不想要，但法律規定必須買的產品。

I mean, it's just a losing proposition.

我想這真的是賠錢貨喔。

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

拿我學生對數學的刻板印象

a useful stereotype about you all.

用來預測你們的行為結果也管用

I could give you guys

如果我給各位

an algebra-two final exam,

作一個數學(II)的期末測驗

and I would expect no higher

我不敢期望有超過

than a 25 percent pass rate.

百分之二十五的人會通過。

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

這兩個事實並不是你或是我的學生特有的問題，

than they do about what we call math education

這反應出美國今天所面臨的

in the U.S. today.

數學教育的通病。

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

首先，我將數學分成兩類。

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

第一類是計算，這些內容你大概都忘光了。

For example, factoring quadratics with

例如二次方係數大於一

leading coefficients greater than one.

的因式分解

This stuff is also really easy to relearn,

這類數學是很容易學回來的。

provided you have a really strong grounding

假如你有堅實的推理，

in reasoning. Math reasoning --

數學推理的基礎，

we'll call it the application

我們把它稱作數學應用。

of math processes to the world around us --

在生活週遭的實際應用

this is hard to teach.

這很難教。

This is what we would love students to retain,

我們希望學生學習這些後

even if they don't go into mathematical fields.

即使將來不進入數學的領域工作也要留下的能力

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

這也是我們美國數學教育之目標。

all but ensures they won't retain it.

但幾乎可以保證最後這種能力不會被留下來

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

我會告訴各位為什麼會這樣

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

為什麼這種社會不幸會發生，我們能做些什麼

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

最後，為什麼現在是數學老師

to be a math teacher.

改變現狀的大好機會

So first, five symptoms

首先 有五種症狀

that you're doing math reasoning wrong

顯示我們課堂上所做數學推理

in your classroom.

的方式是錯誤的

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

第一 缺乏主動 學生們不會自動自發的學習

You finish your lecture block

當你課程進行到一個段落

and immediately you have five hands going up

課椅上馬上有五隻手舉起來

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

要求你整個的重新解釋

Students lack perseverance.

第二 學生缺乏持久力

They lack retention; you find yourself

學過的東西不會記得，你會發現

re-explaining concepts three months later, wholesale.

教過的三個月之後，你得全部都要重來一遍

There's an aversion to word problems,

我的學生中有99%對於

which describes 99 percent of my students.

解讀數學題意非常厭煩

And then the other one percent

另外1%的學生則

is eagerly looking for the formula

熱衷於找應用在

to apply in that situation.

該題目的公式

This is really destructive.

這真的是非常沒有建設性

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

David Milch，是“米蟲”影集及其他一些電視節目的製作人

has a really good description for this.

對上述情況有一個很棒的形容

He swore off creating

他誓言要停止製作

contemporary drama,

當代戲劇

shows set in the present day,

及現今的的單元連續劇

because he saw that when people fill their mind

因為他認為當一個人，

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

每天花四小時，沉溺在像”二個半男人”節目的劇情中時

it shapes the neural pathways, he said,

人們神經的路徑會被定型為

in such a way that they expect simple problems.

期待簡單的問題

He called it, "an impatience with irresolution."

也就是他所謂的 ”對無法做決定的焦慮 “之中

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

人們對於無法立刻解決的問題沒有耐心

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

會期望各種問題都像情境劇一樣 可以在22分鐘

three commercial breaks and a laugh track.

三個廣告橋段及同一個罐頭音樂中完成

And I'll put it to all of you,

而在真實情況中

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

如大家所知，沒有一個值得解決的問題會是那麼簡單

I am very concerned about this

對這個現象我非常擔心

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

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

I'm doing bad things

如果我用現在的方式

to my own future and well-being

教育他們， 那我就是在跟我自己的

when I teach this way.

未來及福利過不去

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

我在這裡要告訴大家，現今的教科書，

mass-adopted textbooks -- teach math reasoning

尤其是那些已被大量採用的教科書，

and patient problem solving,

它們教數學推理及耐心解決問題的方式，

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

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.

這問題要嚴肅以待，這裡是一個由物理教科書取出的例子

It applies equally to math.

但在數學上也適用

Notice, first of all here,

注意看，首先它

that you have exactly three pieces of information there,

提供三個片段資訊

each of which will figure into a formula

而這些資訊剛好都最後套到公式

somewhere, eventually,

的某一個地方

which the student will then compute.

學生就將它的結果計算出來

I believe in real life.

我相信

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

也請大家問問自己，在真實的生活中，有哪些你解決過的

that was worth solving

有價值的問題

where you knew all of the given information in advance;

是你在事先就知道所有的資訊

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

是不需要你由過多的資訊中，過濾出有價值的部分

or you didn't have sufficient information

或是資訊不足，

and had to go find some.

你必須再找其他資訊補足的

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

我想我們都同意，沒有任何值得解決的問題是這樣處理的

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

而我認為我們的教科書很懂得讓我們的學生變成跛腳馬

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

看看這個就知道，這是一個練習題目組

When it comes time to do the actual problem set,

當實際做這問題組時

we have problems like this right here

問題像右邊這個

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

我們只要將數字切換進去， 做一些小處理答案就出來

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

如果學生仍然對於解這類型的題目的模式不懂

it helpfully explains to you

它會以解釋例題的方式

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

讓你回頭找到解題所需的公式

You could literally, I mean this,

所以你可以逐一

pass this particular unit without knowing any physics,

將各數字帶入公式， 而不必懂物理

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

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.

對於數學，我可以更精準的診斷它的問題點

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

這是我很喜歡舉出來的好問題

It's about defining steepness and slope

它是用滑雪纜車來定義

using a ski lift.

斜率及坡度

But what you have here is actually four separate layers,

這圖上有四不同的個階層

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

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

and, particularly, how when they're compressed together

特別是當四張圖被壓縮在一起

and presented to the student all at once,

同時給學生看的時候

how that creates this impatient problem solving.

這顯示出對解題無耐心是怎麼發生的

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

我將對其下一個定義 這裡有圖形

You also have the mathematical structure,

有數學結構

talking about grids, measurements, labels,

例如格子 尺寸 標示

points, axes, that sort of thing.

點 軸 這類的元素

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

但解題前需要有次步驟，

which section is the steepest.

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.

我希望你可以看出

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

我們在此進行的

is taking a compelling question, a compelling answer,

問題及答案都是強制性

but we're paving a smooth, straight path

但我們應該由問題的這端開始

from one to the other

舖一條筆直的道路，到解決問題那端

and congratulating our students for how well

現在卻只因為學生知道

they can step over the small cracks in the way.

如何跨過路中的一個小裂縫而沾沾自喜

That's all we're doing here.

這就是我們現在的數學教育

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

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

and build them up with students,

把學生耐心解決問題

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

的整套的能力都建立起來

So right here I start with the visual,

因此在這裡，我先展現一個圖像

and I immediately ask the question:

並且立即問

Which section is the steepest?

哪一部份坡度最大

And this starts conversation

這就開始了對話

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

因為圖像是被設計成正反兩方都能找到論據的樣子

So you get people arguing against each other,

所以兩方開始爭辯

friend versus friend,

朋友和朋友

in pairs, journaling, whatever.

情侶對情侶 或是讀者和期刊 等等

And then eventually we realize

因此最終我們知道

it's getting annoying to talk about

在討論銀幕

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

左下方的滑雪者

or the skier just above the mid line.

或是線以上的滑雪者有點惱人

And we realize how great would it be

另外我們也會體會 如果

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

我們就只討論ABCD四個標示點

to talk about them more easily.

事情會簡化一些

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

當我們給坡度下定義時

we realize it would be nice to have some measurements

我們會了解如果能作一些量度，事情會比較好處理

to really narrow it down, specifically what that means.

可以把問題做歸納，了解問題的涵義。

And then and only then,

只有在這時候，這個階段，

we throw down that mathematical structure.

我們才將數學的結構鋪陳出來

The math serves the conversation,

數學提供的是對話

the conversation doesn't serve the math.

而不是用對話來服侍數學

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

我告訴各位 十分之九的課 先用這種方式

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

再進行斜率、坡度教學效果會很好

But if you need to,

但假如你需要

your students can then develop those substeps together.

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 --

你們可以比較這邊這個 和那邊那個

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

哪一個可以產生耐心解決問題及數學推理的效果呢

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

以我過去的教學經驗 答案很明顯的

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

我將我在這裡所說的，用來呼應愛因斯坦的話

who, I believe, has paid his dues.

我相信愛因斯坦此言是經驗累積的結晶

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

他認為規畫問題的能力是無與倫比的重要

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

但是在我實際的經驗，在美國

we just give problems to students;

我們只給學生問題

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

我們沒有讓學生參與問題的形成及規畫

So 90 percent of what I do

所以我每星期五小時的

with my five hours of prep time per week

課前準備工作中，有百分之九十的時間，

is to take fairly compelling elements

是將這一類強制提供的解題要素

of problems like this from my textbook

由我的教材中去除。

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

然後將其重建成需要數學推理及耐心解題的形式，

And here's how it works.

這是它運作的方式，

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

我喜歡這和水槽有關的題目

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

題目是：將水槽灌滿要多少時間？

First things first, we eliminate all the substeps.

重要的事情先做，我將教材中所有的次步驟刪除

Students have to develop those,

學生必須自己去發展出來，

they have to formulate those.

他們必須自己規劃解題的次步驟，

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

因此他們就注意到，所有寫在那裏的都是解題所需要的，

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

沒有一個是我們可以忽略分心的。

Students need to decide, "All right, well,

學生必須自己決定 ，

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

水槽高度有關嗎？水槽尺寸有關嗎？

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

閥的顏色有關嗎？哪些才是真正的重要因素？

Such an underrepresented question in math curriculum.

以現今的數學教材而言 這題目敘述不完整

So now we have a water tank.

我們有一個水槽，

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

要多久你才會將它填滿水，題目就這樣

And because this is the 21st century

因為這是21世紀，

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

所以我們喜歡用真實世界的元素來敘述題目，

not in terms of line art or clip art

而不是用你常在教科書

that you so often see in textbooks,

看到的線圖或插圖，

we go out and we take a picture of it.

我們出去外面對實物照一張像。

So now we have the real deal.

這樣我們就有真實世界，

How long will it take it to fill it up?

要多久才會填滿水呢？

And then even better is we take a video,

甚至可以作的更好一點，我們用影片

a video of someone filling it up.

紀錄填加水的過程，

And it's filling up slowly, agonizingly slowly.

水灌得很慢、惱人的慢，

It's tedious.

過程冗長 ，

Students are looking at their watches, rolling their eyes,

學生們一直看他們的錶 眼珠子不斷的轉，

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

臉上都有某程度的疑惑

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

“天啊 要多久才會灌滿啊”

(Laughter)

(笑聲)

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

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

由這個問題產生許多我認為很有趣的現象。

because, like the intro,

就像我在開頭時所說，

I teach kids -- because of my inexperience --

因為我沒有經驗，所以我教幼兒。

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

我所教都是哪些最需要矯正的幼兒，

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

他們有的不願意加入討論，

because someone else has the formula;

是因為別人有解題的公式

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

別人比他更了解解題的公式，

so I won't talk about it.

所以不願意談論。

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

但在這裡 每一個人的直覺立足點是公平的

Everyone's filled something up with water before,

因為每個人都有灌過水槽的經驗，

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

所以當我要幼兒回答，這要多久時間時，

I've got kids who are mathematically and conversationally intimidated

就讓這些有數學或是交談壓迫感覺的小孩，

joining the conversation.

共同加入了討論。

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

我把學童姓名寫在黑板，指定他們來猜，

and kids have bought in here.

所以學生就被帶入情境。

And then we follow the process I've described.

然後依據我前面所敘述的程序去做。

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

這樣做最妙的地方，或是比較優的地方就是，

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

我們不由老師版教科書後面的

in the back of the teacher's edition.

解題提要來找答案，

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

我們只要將影片一直看完。

(Laughter)

(笑聲)

And that's terrifying,

這種現象很可怕，對吧！

because the theoretical models that always work out

因為在老師版教科書後面的解題提要，

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

理論的公式總是管用的

that's great, but

那很棒呀，但是

it's scary to talk about sources of error

但是當理論和實際情況不吻合，

when the theoretical does not match up with the practical.

產生了錯誤時就會引起驚慌，

But those conversations have been so valuable,

但這正是這些對話中

among the most valuable.

最有價值的精華

So I'm here to report some really fun games

我來這裡報告一些有趣的教學心得，

with students who come pre-installed

我的學生第一天到教室時，

with these viruses day one of the class.

許多都是被錯誤數學教育所汙染的帶菌者

These are the kids who now, one semester in,

但一學期之後，他們變成這樣。

I can put something on the board,

我可以將一些教材寫在黑板上，

totally new, totally foreign,

針對新的、完全陌生的事務，

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

用比年初多約三或四分鐘。

than they would have at the start of the year,

作共同討論

which is just so fun.

這真的很有趣，

We're no longer averse to word problems,

學生們不再對數學語義問題產生反感，

because we've redefined what a word problem is.

因為我們已經對數學語義重新定義；

We're no longer intimidated by math,

學生不再對數學產生恐懼，

because we're slowly redefining what math is.

因為我們會慢慢的重新定義該情境的數學涵義是什麼

This has been a lot of fun.

這樣就變的有趣許多了！

I encourage math teachers I talk to to use multimedia,

我鼓勵數學老師們多多利用多媒體

because it brings the real world into your classroom

將真實的世界情境

in high resolution and full color;

以高解析度 全彩的方式帶入教室

to encourage student intuition for that level playing field;

用公平方式鼓勵學生運用直覺

to ask the shortest question you possibly can

盡可能 問最簡短的問題

and let those more specific questions come out in conversation;

讓特定的問題在對話中出現

to let students build the problem,

讓學生自己提出問題

because Einstein said so;

就像愛因思坦所說哪樣

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

最後一點 不要幫太多忙

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

因為教科書用錯誤的方式在幫倒忙

It's buying you out of your obligation,

它買斷你培養學生耐心解決問題

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

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

為什麼現在正是數學教師的大好時光

is because we have the tools to create

因為現在有工具可以創造

this high-quality curriculum in our front pocket.

高品質且可攜帶於口袋的教材

It's ubiquitous and fairly cheap,

它無所不在也相當便宜

and the tools to distribute it

而傳布教材的工具

freely under open licenses

因為大量供應及開放授權

has also never been cheaper or more ubiquitous.

而價格史上最低 普及性最高

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

不久之前我在我的部落格中放了一系列的影片

and it got 6,000 views in two weeks.

在兩個星期內就有六千個人來觀看

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

到現在我仍然接到一些我從未訪問過國家的老師來信

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

提到”哇!是的，我們確實用這個與學生有很好的對話

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

啊!順便告訴你，我有改進你教材的妙方”

which, wow.

效果真驚人

I put this problem on my blog recently:

我最近在部落格放了這個問題，

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

在雜貨店裡，你要排哪一個結賬櫃檯

the one that has one cart and 19 items

是要排在只有一輛購物車 但是上面有19項物品的後面

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

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

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

這是一個線性規劃的問題 是我課堂上的好教材

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

因為它，我幾星期前上了早安美國的節目

which is just bizarre, right?

答案超乎一般想像 對吧

And from all of this, I can only conclude

從前述影片受歡迎的程度 我得到一個結論就是

that people, not just students,

不只是學生

are really hungry for this.

一般人也很渴望這些內容

Math makes sense of the world.

數學彰顯真實世界的合理性

Math is the vocabulary

數學也是描述人類

for your own intuition.

直覺的詞彙

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

所以我鼓勵大家 不管你在教育上的角色是什麼

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

不管你是學生 家長老師或是政策制定者或其他

insist on better math curriculum.

都請堅持要有一個更好的數學教材

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

我們需要更多有耐心解決問題的人，非常感謝