Hello.

我想要告訴你的是， 你對於自己所擁有的潛能的認知

So I'm here to tell you that what you have believed about your own potential

已經改變了你所學會的東西， 並且會持續地改變，

has changed what you have learned, and continues to do that,

持續地改變你的學習方式和你的經歷。

continues to change your learning, and your experiences.

這裡有多少人── 大家舉手示意一下，

So, how many people here -- let's get a show of hands --

有多少人曾被認為不擅長數學，

have ever been given the idea that they're not a math person,

或被認為無法在數學領域裡 進入到下一個階段，

or that they can't go onto the next level of math,

他們的大腦就是不適合學習數學？

they haven't got the brains for it?

讓我看看有多少人舉手。

Let's see a show of hands.

嗯，還蠻多人的。

So, quite a few of us.

我在這裡要告訴你的是， 這種想法完全錯誤。

And I'm here to tell you that idea is completely wrong,

它已被腦科學所推翻。

it is disproven by the brain science.

導致這種思想的迷思 在我們的社會裡出現，

But it is fueled by a single myth that's out there in our society

且非常強大和危險。

that's very strong and very dangerous.

這迷思說的是人們相信 有人天生就有數學頭腦，

And the myth is that there's such a thing as a math brain,

要麼是與生俱來的， 要麼你天生就沒有這天份。

that you're born with one, or you're not.

我們對於其他科目並不認同這種說法。

We don't believe this about other subjects.

我們從不認為自己天生就有 歷史頭腦或物理頭腦。

We don't think we're born with a history brain, or a physics brain.

我們認為這些東西是需要去學習的。

We think you have to learn those.

但對於數學，大家和學生都認為 這需要與生俱來的天賦，

But with math, people, students believe it,

老師這麼認為，家長也這麼認為。

teachers believe it, parents believe it.

直到我們改變這種迷思之前，

And until we change that single myth

進度落後的普及現象 將會一直在這個國家延續下去。

we will continue to have widespread underachievement in this country.

卡羅爾·德維克在思維上的研究顯示，

Carol Dweck's research on mindset has shown us

如果你相信自己的潛能無限，

that if you believe in your unlimited potential

你將會在數學和生活上 獲得更高的成就。

you will achieve at higher levels in maths, and in life.

一份關於錯誤的驚人研究 很好地詮釋了這種說法。

And an incredible study on mistakes show this very strongly.

傑森·莫澤和他的同事們 在磁力共振掃描中發現，

So Jason Moser and his colleagues actually found from MRI scans

當你在數學題上出錯的時候， 你的大腦會成長。

that your brain grows when you make a mistake in maths.

太棒了。

Fantastic.

當你出錯的時候，大腦會激發突觸 （神經原的神經性連接）。

When you make a mistake, synapses fire in the brain.

事實上，在他們的磁力共振掃描中

And in fact in their MRI scans

他們發現當人們出錯時， 大腦就會激發突觸。

they found that when people made a mistake synapses fired.

當人們回答正確的時候， 大腦會較少激發突觸。

When they got work correct less synapses fired.

所以犯錯是件好事。

So making mistakes is really good.

我們希望學生們知道這一點。

And we want students to know this.

但他們發現了 另一些令人難以置信的東西。

But they found something else that was pretty incredible.

這幅圖顯示著人類的腦電圖。

This image shows you the voltage maps of people's brains.

在圖中你會發現相信自己 有無限潛能的成長型思維的人，

And what you can see here is that people with a growth mindset,

他們可以學習任何東西，

who believe that they had unlimited potential,

當他們犯了一個錯誤的時候，

they could learn anything,

他們的大腦會比那些不相信自己 可以學習的人增長得多。

when they made a mistake, their brains grew more

因此，這向我們表明了 一些腦科學家早已知道的事情：

than the people who didn't believe that they could learn anything.

我們的認知和我們所學到的東西，

So this shows us something that brain scientists have known for a long time:

與我們的信念和感受有關。

That our cognition, and what we learn

不僅是對那坐在數學教室裡的孩子們， 這對我們所有人來說都很重要。

is linked to our beliefs, and to our feelings.

如果你遇到困難， 或者是面對一個充滿挑戰的情況，

And this is important for all of us not just kids in math classrooms.

你對自己說： 「我可以做到。我會去執行它。」

If you go into a difficult situation, or a challenging situation,

然後你搞砸了或者失敗了，

and you think to yourself: "I can do this. I'm going do it."

你的大腦會成長得更多， 且會有不同的反應

and you mess up or fail,

比起你在困境中想： 「我不認為我可以做到這一點。」

your brain will grow more, and react differently

所以改變孩子在教室裡 接收的信息是很重要的。

than if you go into that situation thinking:

我們知道任何人的大腦都能成長，

"I don't think I can do this."

且大腦是很有可塑性的， 它能夠學習任何程度的數學。

So it's really important that we change the messages kids get in classrooms.

我們必須讓孩子們知道這件事情。

We know that anybody can grow their brain,

他們需要知道犯錯是件好事。

and brains are so plastic, to learn any level of maths.

但數學教室需要多方面的改變。

We have to get this out to kids.

這不僅是改變 我們對孩子們傳遞的訊息。

They have to know that mistakes are really good.

我們需要從根本 去改變教室裡所發生的事情。

But maths classrooms have to change in a lot of ways.

我們想讓孩子們擁有成長型思維，

It's not just about changing messages for kids.

去相信他們可以成長， 可以學習任何東西。

We have to fundamentally change what happens in classrooms.

但要在數學上擁有成長型思維 是非常困難的。

And we want kids to have a growth mindset,

如果你經常被問及一些只需要 回答是與非的簡短、封閉式問題，

to believe that they can grow, and learn anything.

那些問題本身 就傳遞著數學的固定概念，

But it's very difficult to have a growth mindset in maths.

要麼做得到，要麼做不了。

If you're constantly given short, closed questions that you get right or wrong,

因此，我們需要開放式的數學問題，

those questions themselves

讓學生有空間去思考和學習。

transmit fixed messages about math, that you can do it or you can't.

我想舉個例子。

So we have to open up maths questions

我想請你和我一起思考一些數學題。

so that there's space inside them for learning.

這是一道在學校裡相當典型的數學題。

I want to give you an example.

我希望你從不同角度去思考這個問題。

We're actually going to ask you to think about some maths with me.

這裡有三個由正方形組成的圖形。

So this is a fairly typical problem, it's given out in schools.

圖形 2 的正方形數量比圖形 1 多，

I want you to think about it differently. So we have three cases of squares.

而圖形 3 的正方形的數量則更多。

In case 2 there's more squares than in case 1,

通常我們會這麼問：

and in case 3 there's even more.

「到圖形 100 或者圖形 n 時， 會有多少個正方形呢？」

Often this is given out with the question:

我希望你思考出另一種問法。

"How many squares would there be in case 100, or case n?"

我希望你往與數字 或代數無關的方向去思考。

I want you to think of a different question.

我想要你完全從視覺上去思考，

I want you to think without any numbers at all, or without any algebra.

我想要你思考 你從額外的正方形看到了什麼？

I want you to think entirely visually,

如果圖形 2 的正方形數量 比圖形 1 多，它們在哪裡？

and I want you to think about where do you see the extra squares?

如果我們此刻在教室裡， 我會給你更長的思考時間。

If there are more squares in case 2 than case 1, where are they?

但因為時間的關係，我會跟你們 分享人們對此的一些不同看法，

So if we were in a classroom, I'd give you a long time to think about this.

我曾經用這道題目問過不同的人，

In the interest of time, I'm going to show you some different ways

我想這是我在史丹佛大學的 大學生告訴我的答案──

people think about this, and I've given this problem to many different people,

其中一個同學告訴我：

and I think it was my undergrads at Stanford who said to me --

「噢，我覺得它就像雨點一樣， 落在正方形的上方。

or one of them said to me:

所以它們就像一層外層， 每次都添上新的一層。」

"Oh, I see it like raindrops. Where raindrops come down on the top.

我的另一個大學生這麼說：

So it's like an outer layer, that grows new each time."

「不是的，我覺得它像一個保齡球場。

It was also my undergrads who said:

你得到額外的一排，

"Oh no, I see it more like a bowling alley.

就像一排球瓶從最底層添上來。」

You get an extra row,

這是從一個非常不同的角度 去看正方形的增長。

like a row of skittles that comes in at the bottom."

我記得是一位老師告訴我 它就像一座火山：

A very different way of seeing the growth.

「從中心位置上升， 然後岩漿就流了出來。」

It was a teacher, I remember, who said to me it was like a volcano:

（笑聲）

"The center goes up, and then the lava comes out."

另一位老師說： 「不是的，這就像紅海的分離。

[Laughter]

那些正方形分開了， 而新的中心會重複地加進來。

Another teacher said: "Oh no, it's like the parting of the Red Sea.

我記得這是── 對不起，這個也一樣。

The shape separates, and there's a duplication with an extra center."

有些人將其視為三角形。

I remember this was -- Sorry, this one as well.

他們認為這是一個往外成長的三角形。

Some people see it as triangles.

然後一位新墨西哥州的老師對我說：

They see the outside growing as an outside triangle.

「哦，這就像電影《反鬥智多星》中， 『天堂的階梯，禁止進入』。

And then there was a teacher in New Mexico who said to me:

（笑聲）

"Oh it's like Wyane's World, Stairway to Heaven, access denied."

然後我們也可以用這種方式看它，

[Laughter]

如果你移動了方塊， 你隨時都可以這麼做，

And then we have this way of seeing it.

然後你稍微重新排列這個形狀，

If you move the squares, which you always can,

你會發現它其實以正方形的形式增長。

and you rearrange the shape a bit,

因此，我想用這個問題來說明這一點：

you'll see that it actually grows as squares.

「當它在數學課上被提出的時候， 這並不是最糟糕的問題，

So, this is what I want to illustrate with this question:

問題會以這種方式提出： 「一共有多少？」

"When it's given out in maths classrooms, and this isn't the worst of questions,

然後孩子們就開始計數。 所以他們會說：

it's given out with a question of: "How many?" and kids count.

「在第一個圖形裡有 4 個， 在第二個圖形裡有 9 個。」

So they'll say:

他們可能盯著那個數字欄很久， 然後說：

"In the first case there's 4. In the second there's 9."

「如果你每次都將對應的圖形數目 加 1，然後進行平方計算，

They might stare at that column of numbers for a long time and say:

你就會得到正方形的總數。」

"If you add one to the case number each time and square it,

但當我們把它交給學生和高中教師時，

then you get the total number of squares."

他們做完之後我會問他們：

But when we give it to students, and high school teachers,

「那為什麼計算平方數呢？ 你是怎麼看出來要用平方公式的呢？」

I'll say to them when they've done this:

他們會答：「不知道。」

"So why is that squared? Why do you see that squared function?"

所以，這就是它用平方的原因， 這個函數以正方形的方式增長。

They'll say: "No idea."

你從代數中可以看出來是平方。

So this is why it's squared. The function grows as a square.

所以當我們向學生提出這些問題時， 我們會向他們提出這些直觀問題。

You see that squaring in the algebraic representation.

我們會問：「他們怎麼看出來的？」

So when we give these problems to students we give them the visual question.

他們會進行非常深入的討論，

We ask them: "How they see it?"

也對這數學來說非常重要的部分 有了更深入的理解。

They have these rich discussions, and they also reach deeper understandings

其實我們在數學教室裡需要一次變革。

about a really important part of mathematics.

我們需要改變很多事情。

So we actually need a revolution in maths classrooms.

我們需要這大改革的部分原因是

We need to change a lot of things.

數學教學和學習的研究 並沒有真的進入學校和教室。

And part of the reason we need to change so much

我將和你們分享一個很好的例子。

is because research on maths teaching and learning

這真的非常有趣。

is not getting into schools and classrooms.

當我們在計算的時候── 即使是成年人在計算的時候，

And I'm going to give you a stunning example now.

「看到」手指的腦區會被點亮，

So this is really interesting.

我們沒使用手指，

When we calculate -- Even when adults calculate,

但看到手指的腦區卻亮起了。

where a brain area that sees fingers is lighting up,

所以我們使用手指和我們看到手指時 會在不同的腦區產生反應。

we're not using fingers,

這證明了看到手指這個活動 對大腦來說是非常重要的。

but that brain area that sees fingers lights up.

事實上，手指感知是──

So there's a brain area when we use fingers,

科學家通過要求他們將手 放在桌子下方來測試手指感知──

and there's a brain area when we see fingers.

他們看不到他們正觸摸著手指，

And it turns out that seeing fingers is really important for the brain.

然後看看你是否知道 哪根手指被觸摸過。

And in fact finger perception is --

那些具有良好手指感知的大學生

Scientists test for finger perception

能夠預測他們的計算分數。

by asking them to put their hands under a table --

一年級學生所擁有的手指感知數量

they can't see them touching a finger,

比考試分數能夠更好地預測 他們上了二年級的數學成績。

and then seeing if you know which finger has been touched.

就是這麼重要。

The number of university students who have good finger perception

但在學校和教室裡發生了什麼事？

predicts their calculation scores.

學生被告知他們不准使用手指來計算。

The number of finger perception grade 1 students have

他們被告知這是幼稚的。 他們被引導對此感到反感。

is a better prediction of maths achievement in grade 2

當我們阻止小孩運用手指學習數字時，

than test scores.

這等同於中止了 他們在數學認知上的發展。

It is that important.

而科學家們已經知道這個事實 很長一段時間了。

But what happens in schools and classrooms?

而且神經科學家總結出

Students are told they're not allowed to use their fingers.

手指應該被學生用來學習數字和算術。

They're told it's babyish. They're made to feel bad about it.

如果我們還沒有發表──

When we stop children learning numbers through fingers,

上週，我們在《大西洋》雜誌的 一篇論文中發表了這篇文章。

it's akin to halting their numerical development.

我認識的教育工作者中 沒有任何一個知道這個事實。

And scientists have known this for a long time.

這在教育界引起了巨大的連鎖反應。

And the neuroscientists conclude

還有許多其他研究是教師 和學校都不知道的。

that fingers should be used for students learning number and arithmetic.

我們也知道，當你開始進行計算時，

If we haven't published --

大腦中的各個區域之間 會產生複雜而動態的交流，

We published this in a paper in the Atlantic last week.

包括視覺皮層。

I don't know any educator who knew this.

然而，數學課並不屬於視覺系的， 它們是數字化且抽象的。

This is causing a huge ripple through the education community.

我想跟你們分享一下

There's lots of other research that's not known by teachers and schools.

我們在去年夏天帶了 81 名學生 到大學的時候發生的事情，

We know when you perform a calculation

我們以不同的教學方式教導他們。

the brain is involved in a complex and dynamic communication

我們教導了他們關於大腦的成長。

between different areas of the brain, including the visual cortex.

我們教導了他們關於心態和錯誤。

Yet, maths classrooms are not visual, they're numerical and abstract.

但我們也教導了他們 創意、視覺性，且動人的數學。

I want to show you now what happened

他們跟著我們上了 18 堂課。

when we brought 81 students onto campus last summer,

在他們來參與我們的課程之前， 他們已經參加了地區性的標準化考試。

and we taught them differently.

我們在他們上了 18 堂課之後 也給了他們相同的考試，

So we taught them about the brain growing.

他們的成績平均提高了50％。

We taught about mindset and mistakes.

有著不同數學水平的 81 名學生，

But we as also taught them creative, visual, beautiful maths.

在第一天都告訴我們： 「我不擅長數學。」

They came in for 18 lessons with us.

他們都可以說出他們班上 那個擅長數學的同學的名字。

Before they came to us they had taken a district standardized test.

我們改變了他們的觀念。

We gave them the same test at the end of our 18 lessons,

這片段擷取自我們以這些孩子為素材 所製作、較長一些的音樂視頻。

and they improved by an average of 50%.

（視頻播放）

Eighty one students, from a range of achievement levels,

但我們一直在討論，

told us on the first day: "I'm not a math person."

停不下來，我們不會停止解決問題，

They could name the one person in their class who was a math person.

好像有什麼東西在成長，

We changed their beliefs.

每當我們再嘗試， 就會在我們腦海中浮現。

And this is a clip from a longer music video that we made of the kids.

憎恨者還是一樣只會繼續討厭，

But we keep talking

我們都會犯錯，

Can't stop, won't stop solving

我們只要甩一甩，

It's like something is growing

把它甩掉！把它甩掉！

In our minds every time we try again.

我們的方法會突破，

'Cause the haters gonna hate, hate, hate, hate, hate.

這並不是小菜一碟，

We will make mistakes, stakes, stakes, stakes, stakes.

我們只要甩一甩，

We're just gonna shake, shake, shake, shake, shake.

把它甩掉！把它甩掉！

Shake it off! Shake it off!

我們把東西都視覺化，

Our method's gonna break, break, break, break, break.

在課堂上清晰地呈現出來，

It's not a piece of cake, cake, cake, cake, cake.

這樣他們就可以看到，

We're just gonna shake, shake, shake, shake, shake.

這樣他們就可以看到。

Shake it off! Shake it off!

我們知道，我們的大腦能夠成長，

We represent things visually,

我們才不在乎我們走得有多快，

Present them to our class clearly

真正了解我們所展示的內容，

So that they can see mmm

真正了解我們所展示的內容。

So that they can see mmm

所以我們不斷嘗試，

We know our brains can grow

突觸一直被激發，

Who cases how fast we go?

這些問題太令人興奮了，

Understanding's what we show mmm

這很酷，我想向全世界 展示我們的進步！

Understanding's what we show mmm

（視頻結束）

So we keep trying

所以，

Synapses are firing

（掌聲）

This problem's so exciting

我們需要讓老師們知道研究成果。 我們需要數學教學的革命。

It's so cool that I want to go and show the world!

如果你不相信我， 請聽聽這個孩子怎麼說。

So --

他是一名中學生， 我們曾與他的老師合作，

(Applause)

將數學練習簿轉向開放式思維 和心態相關的數學題。

We need to get research out to teachers. We need a revolution in maths teaching.

這是他對那次轉變後的反思。

If you don't believe me, come listen to this kid.

（視頻）去年的數學課只有 各種筆記和講義，

He's a middle schooler, and we had worked with his teachers

和你自己的小框框── 你只能夠躲在這小框框裡。

to shift from worksheet math to open math with mindset messages.

你感到孤獨，每個人只顧著自己。

This is him reflecting on that shift.

但現在，今年變得開放了，

Math class last year was notes, and just handouts,

我們都在一個很大的── 這就像一座城市──

and your own little box -- you were just boxed in.

我們都在一起努力創造 這個新的美麗世界。

You were by yourself, it was every man for themselves.

我覺得，擺在我面前的挑戰和未來，

But now this year is just open. We're a whole big --

如果我繼續前進，

It's like a city --

如果我堅持下去，總有一天我會成功。

we're all working together to create this new beautiful world.