when you really loved something --

請各位回想一下過去是否曾經

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

你非常喜歡一種東西

and you recommended it wholeheartedly

例如一部電影、一張唱片、一首歌或是一本書

to someone you also really liked,

在你全心全意的向

and you anticipated that reaction, you waited for it,

你所鍾愛的人推薦

and it came back, and the person hated it?

你預期對方會有和你一樣的反應，你等著等著

So, by way of introduction,

得到的答案卻是，他恨死那東西了

that is the exact same state

在自我介紹階段，

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

我要告訴各位，這正是

I teach high school math.

我過去六年每個工作天所面臨情形的寫照

I sell a product to a market

我在高中教數學

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

我生產一種

I mean, it's just a losing proposition.

市場不想要，但法律規定必須買的產品。

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

我想這真的是賠錢貨喔。

a useful stereotype about you all.

拿我學生對數學的刻板印象

I could give you guys

用來預測你們的行為結果也管用

an algebra-two final exam,

如果我給各位

and I would expect no higher

作一個數學(II)的期末測驗

than a 25 percent pass rate.

我不敢期望有超過

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

百分之二十五的人會通過。

than they do about what we call math education

這兩個事實並不是你或是我的學生特有的問題，

in the U.S. today.

這反應出美國今天所面臨的

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

數學教育的通病。

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

首先，我將數學分成兩類。

For example, factoring quadratics with

第一類是計算，這些內容你大概都忘光了。

leading coefficients greater than one.

例如二次方係數大於一

This stuff is also really easy to relearn,

的因式分解

provided you have a really strong grounding

這類數學是很容易學回來的。

in reasoning. Math reasoning --

假如你有堅實的推理，

we'll call it the application

數學推理的基礎，

of math processes to the world around us --

我們把它稱作數學應用。

this is hard to teach.

在生活週遭的實際應用

This is what we would love students to retain,

這很難教。

even if they don't go into mathematical fields.

我們希望學生學習這些後

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

即使將來不進入數學的領域工作也要留下的能力

all but ensures they won't retain it.

這也是我們美國數學教育之目標。

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

但幾乎可以保證最後這種能力不會被留下來

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

我會告訴各位為什麼會這樣

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

為什麼這種社會不幸會發生，我們能做些什麼

to be a math teacher.

最後，為什麼現在是數學老師

So first, five symptoms

改變現狀的大好機會

that you're doing math reasoning wrong

首先 有五種症狀

in your classroom.

顯示我們課堂上所做數學推理

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

的方式是錯誤的

You finish your lecture block

第一 缺乏主動 學生們不會自動自發的學習

and immediately you have five hands going up

當你課程進行到一個段落

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

課椅上馬上有五隻手舉起來

Students lack perseverance.

要求你整個的重新解釋

They lack retention; you find yourself

第二 學生缺乏持久力

re-explaining concepts three months later, wholesale.

學過的東西不會記得，你會發現

There's an aversion to word problems,

教過的三個月之後，你得全部都要重來一遍

which describes 99 percent of my students.

我的學生中有99%對於

And then the other one percent

解讀數學題意非常厭煩

is eagerly looking for the formula

另外1%的學生則

to apply in that situation.

熱衷於找應用在

This is really destructive.

該題目的公式

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

這真的是非常沒有建設性

has a really good description for this.

David Milch，是“米蟲”影集及其他一些電視節目的製作人

He swore off creating

對上述情況有一個很棒的形容

contemporary drama,

他誓言要停止製作

shows set in the present day,

當代戲劇

because he saw that when people fill their mind

及現今的的單元連續劇

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

因為他認為當一個人，

it shapes the neural pathways, he said,

每天花四小時，沉溺在像”二個半男人”節目的劇情中時

in such a way that they expect simple problems.

人們神經的路徑會被定型為

He called it, "an impatience with irresolution."

期待簡單的問題

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

也就是他所謂的 ”對無法做決定的焦慮 “之中

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

人們對於無法立刻解決的問題沒有耐心

three commercial breaks and a laugh track.

會期望各種問題都像情境劇一樣 可以在22分鐘

And I'll put it to all of you,

三個廣告橋段及同一個罐頭音樂中完成

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

而在真實情況中

I am very concerned about this

如大家所知，沒有一個值得解決的問題會是那麼簡單

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

對這個現象我非常擔心

I'm doing bad things

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

to my own future and well-being

如果我用現在的方式

when I teach this way.

教育他們， 那我就是在跟我自己的

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

未來及福利過不去

mass-adopted textbooks -- teach math reasoning

我在這裡要告訴大家，現今的教科書，

and patient problem solving,

尤其是那些已被大量採用的教科書，

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

它們教數學推理及耐心解決問題的方式，

(Laughter)

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

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

(笑)

It applies equally to math.

這問題要嚴肅以待，這裡是一個由物理教科書取出的例子

Notice, first of all here,

但在數學上也適用

that you have exactly three pieces of information there,

注意看，首先它

each of which will figure into a formula

提供三個片段資訊

somewhere, eventually,

而這些資訊剛好都最後套到公式

which the student will then compute.

的某一個地方

I believe in real life.

學生就將它的結果計算出來

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

我相信

that was worth solving

也請大家問問自己，在真實的生活中，有哪些你解決過的

where you knew all of the given information in advance;

有價值的問題

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

是你在事先就知道所有的資訊

or you didn't have sufficient information

是不需要你由過多的資訊中，過濾出有價值的部分

and had to go find some.

或是資訊不足，

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

你必須再找其他資訊補足的

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

我想我們都同意，沒有任何值得解決的問題是這樣處理的

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

而我認為我們的教科書很懂得讓我們的學生變成跛腳馬

When it comes time to do the actual problem set,

看看這個就知道，這是一個練習題目組

we have problems like this right here

當實際做這問題組時

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

問題像右邊這個

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

我們只要將數字切換進去， 做一些小處理答案就出來

it helpfully explains to you

如果學生仍然對於解這類型的題目的模式不懂

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

它會以解釋例題的方式

You could literally, I mean this,

讓你回頭找到解題所需的公式

pass this particular unit without knowing any physics,

所以你可以逐一

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

將各數字帶入公式， 而不必懂物理

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

只是知道解碼 就得到答案 這實在太不應該了

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

對於數學，我可以更精準的診斷它的問題點

It's about defining steepness and slope

這是我很喜歡舉出來的好問題

using a ski lift.

它是用滑雪纜車來定義

But what you have here is actually four separate layers,

斜率及坡度

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

這圖上有四不同的個階層

and, particularly, how when they're compressed together

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 presented to the student all at once,

特別是當四張圖被壓縮在一起

how that creates this impatient problem solving.

同時給學生看的時候

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

這顯示出對解題無耐心是怎麼發生的

You also have the mathematical structure,

我將對其下一個定義 這裡有圖形

talking about grids, measurements, labels,

有數學結構

points, axes, that sort of thing.

例如格子 尺寸 標示

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

點 軸 這類的元素

which section is the steepest.

但解題前需要有次步驟，

So I hope you can see.

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

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

我希望你可以看出

is taking a compelling question, a compelling answer,

我們在此進行的

but we're paving a smooth, straight path

問題及答案都是強制性

from one to the other

但我們應該由問題的這端開始

and congratulating our students for how well

舖一條筆直的道路，到解決問題那端

they can step over the small cracks in the way.

現在卻只因為學生知道

That's all we're doing here.

如何跨過路中的一個小裂縫而沾沾自喜

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

這就是我們現在的數學教育

and build them up with students,

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

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

把學生耐心解決問題

So right here I start with the visual,

的整套的能力都建立起來

and I immediately ask the question:

因此在這裡，我先展現一個圖像

Which section is the steepest?

並且立即問

And this starts conversation

哪一部份坡度最大

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

這就開始了對話

So you get people arguing against each other,

因為圖像是被設計成正反兩方都能找到論據的樣子

friend versus friend,

所以兩方開始爭辯

in pairs, journaling, whatever.

朋友和朋友

And then eventually we realize

情侶對情侶 或是讀者和期刊 等等

it's getting annoying to talk about

因此最終我們知道

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

在討論銀幕

or the skier just above the mid line.

左下方的滑雪者

And we realize how great would it be

或是線以上的滑雪者有點惱人

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

另外我們也會體會 如果

to talk about them more easily.

我們就只討論ABCD四個標示點

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

事情會簡化一些

we realize it would be nice to have some measurements

當我們給坡度下定義時

to really narrow it down, specifically what that means.

我們會了解如果能作一些量度，事情會比較好處理

And then and only then,

可以把問題做歸納，了解問題的涵義。

we throw down that mathematical structure.

只有在這時候，這個階段，

The math serves the conversation,

我們才將數學的結構鋪陳出來

the conversation doesn't serve the math.

數學提供的是對話

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

而不是用對話來服侍數學

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

我告訴各位 十分之九的課 先用這種方式

But if you need to,

再進行斜率、坡度教學效果會很好

your students can then develop those substeps together.

但假如你需要

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

你的學生們可以共同發展解題的次步驟

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

你們可以比較這邊這個 和那邊那個

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

哪一個可以產生耐心解決問題及數學推理的效果呢

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

以我過去的教學經驗 答案很明顯的

who, I believe, has paid his dues.

我將我在這裡所說的，用來呼應愛因斯坦的話

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

我相信愛因斯坦此言是經驗累積的結晶

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

他認為規畫問題的能力是無與倫比的重要

we just give problems to students;

但是在我實際的經驗，在美國

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

我們只給學生問題

So 90 percent of what I do

我們沒有讓學生參與問題的形成及規畫

with my five hours of prep time per week

所以我每星期五小時的

is to take fairly compelling elements

課前準備工作中，有百分之九十的時間，

of problems like this from my textbook

是將這一類強制提供的解題要素

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

由我的教材中去除。

And here's how it works.

然後將其重建成需要數學推理及耐心解題的形式，

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

這是它運作的方式，

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

我喜歡這和水槽有關的題目

First things first, we eliminate all the substeps.

題目是：將水槽灌滿要多少時間？

Students have to develop those,

重要的事情先做，我將教材中所有的次步驟刪除

they have to formulate those.

學生必須自己去發展出來，

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

他們必須自己規劃解題的次步驟，

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

因此他們就注意到，所有寫在那裏的都是解題所需要的，

Students need to decide, "All right, well,

沒有一個是我們可以忽略分心的。

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

學生必須自己決定 ，

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

水槽高度有關嗎？水槽尺寸有關嗎？

Such an underrepresented question in math curriculum.

閥的顏色有關嗎？哪些才是真正的重要因素？

So now we have a water tank.

以現今的數學教材而言 這題目敘述不完整

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

我們有一個水槽，

And because this is the 21st century

要多久你才會將它填滿水，題目就這樣

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

因為這是21世紀，

not in terms of line art or clip art

所以我們喜歡用真實世界的元素來敘述題目，

that you so often see in textbooks,

而不是用你常在教科書

we go out and we take a picture of it.

看到的線圖或插圖，

So now we have the real deal.

我們出去外面對實物照一張像。

How long will it take it to fill it up?

這樣我們就有真實世界，

And then even better is we take a video,

要多久才會填滿水呢？

a video of someone filling it up.

甚至可以作的更好一點，我們用影片

And it's filling up slowly, agonizingly slowly.

紀錄填加水的過程，

It's tedious.

水灌得很慢、惱人的慢，

Students are looking at their watches, rolling their eyes,

過程冗長 ，

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

學生們一直看他們的錶 眼珠子不斷的轉，

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

臉上都有某程度的疑惑

(Laughter)

“天啊 要多久才會灌滿啊”

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

(笑聲)

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

各位就知道學生們是怎麼被我騙上鉤的

because, like the intro,

由這個問題產生許多我認為很有趣的現象。

I teach kids -- because of my inexperience --

就像我在開頭時所說，

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

因為我沒有經驗，所以我教幼兒。

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

我所教都是哪些最需要矯正的幼兒，

because someone else has the formula;

他們有的不願意加入討論，

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

是因為別人有解題的公式

so I won't talk about it.

別人比他更了解解題的公式，

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

所以不願意談論。

Everyone's filled something up with water before,

但在這裡 每一個人的直覺立足點是公平的

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

因為每個人都有灌過水槽的經驗，

I've got kids who are mathematically and conversationally intimidated

所以當我要幼兒回答，這要多久時間時，

joining the conversation.

就讓這些有數學或是交談壓迫感覺的小孩，

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

共同加入了討論。

and kids have bought in here.

我把學童姓名寫在黑板，指定他們來猜，

And then we follow the process I've described.

所以學生就被帶入情境。

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

然後依據我前面所敘述的程序去做。

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

這樣做最妙的地方，或是比較優的地方就是，

in the back of the teacher's edition.

我們不由老師版教科書後面的

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

解題提要來找答案，

(Laughter)

我們只要將影片一直看完。

And that's terrifying,

(笑聲)

because the theoretical models that always work out

這種現象很可怕，對吧！

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

因為在老師版教科書後面的解題提要，

that's great, but

理論的公式總是管用的

it's scary to talk about sources of error

那很棒呀，但是

when the theoretical does not match up with the practical.

但是當理論和實際情況不吻合，

But those conversations have been so valuable,

產生了錯誤時就會引起驚慌，

among the most valuable.

但這正是這些對話中

So I'm here to report some really fun games

最有價值的精華

with students who come pre-installed