A friend of mine told me recently that her six-year-old son had come from school and said he hated math.
我的一位朋友最近告诉我，她的六岁儿子从学校回来，说他讨厌数学。
And this is hard for me to hear because I actually love math.
因为我很爱数学，听到这种事我会很难过。
The beauty and power of mathematical thinking have changed my life.
数学思考的力与美改变了我的人生。
But I know that many people lived a very different story.
但我知道许多人的状况是非常不同的。
Math can be the best of times or the worst of times,
数学可能是最美好的时光，也可能是最糟糕的，
an exhilarating journey of discovery or descent into tedium, frustration, and despair.
可能是让人兴奋的发掘之旅，也可能让人厌烦、挫折、绝望。
Mathematical miseducation is so common we can hardly see it.
数学的错误教育非常常见，以致于我们几乎都不会发现。
We practically expect math class to be repetition and memorization of disjointed technical facts.
我们几乎是预期数学课就是些支离破碎的技术事实，要不断重复和记忆。
And we're not surprised when students aren't motivated, when they leave school disliking math,
不意外，学生不会有动机去学，他们离开学校时也不喜欢数学，
even committed to avoiding it for the rest of their lives.
甚至发誓说此生一定要避开数学。
Without mathematical literacy, their career opportunities shrink.
没有数学能力，他们的职业生涯的机会就会减少。
And they become easy prey for credit card companies, payday lenders, the lottery,
他们会很容易成为猎物，被信用卡公司、发薪日借钱者、彩券当成目标，
and anyone, really, who wants to dazzle them with a statistic.
其实，任何人都可以用统计数字让他们头昏上勾。
Did you know that if you insert a single statistic into an assertion,
你知道吗，只要在一项主张当中加入单一个统计数字，
people are 92 percent more likely to accept it without question?
大家不问问题就接受它的机会会提高92%？
Yeah, I totally made that up. And 92 percent is -- it has weight even though it's completely fabricated.
是啊，那是我编造的。92%是--虽然它是完全虚构的，但它还是有影响。
And that's how it works. When we're not comfortable with math, we don't question the authority of numbers.
它就是这样运作的。当我们对数学感到不舒服，我们就不会去质疑数字的权威性。
But what's happening with mathematical alienation is only half the story.
但疏远数学只不过是故事的一半而已。
Right now, we're squandering our chance to touch life after life with the beauty and power of mathematical thinking.
现在，我们在挥霍我们用数学思考的力与美来接触一个又一个生命的机会。
I led a workshop on this topic recently, and at the end,
最近，我主持了一个关于这主题的研讨会，
a woman raised her hand and said that the experience made her feel -- and this is a quote -- "like a God."
最后，有位女子举手，她说，这段体验让她觉得--引述她的话--“跟神一样。”
That's maybe the best description I've ever heard for what mathematical thinking can feel like,
对于数学思考的感受，那可能是我听过最棒的描述了，
so we should examine what it looks like.
所以我们应该来探究它是什么样子的。
A good place to start is with the words of the philosopher and mathematician René Descartes,
哲学家和数学家勒内·笛卡尔的话，是一个很好的起始点，
who famously proclaimed, "I think, therefore I am."
他有句名言：“我思，故我在。”
But Descartes looked deeper into the nature of thinking.
但笛卡尔更深入洞察了思考的本质。
Once he established himself as a thing that thinks, he continued, "What is a thinking thing?"
一旦他确立自己会思考之后，接着就问：“思考是什么东西？”
It is the thing that doubts, understands, conceives,
这样东西会质疑、了解、设想、
that affirms and denies, wills and refuses, that imagines also, and perceives.
确认、否认、愿意去做、拒绝，也会想象和感知。
This is the kind of thinking we need in every math class every day.
我们在每天的数学课中就需要这种思考。
So, if you are a teacher or a parent or anyone with a stake in education,
如果你是老师或父母，或和教育相关的任何人士，
I offer these five principles to invite thinking into the math we do at home and at school.
我要提供五项原则，将思考带入我们在家以及在学校中的数学里。
Principle one: start with a question.
原则一：从一个问题开始。
The ordinary math class begins with answers and never arrives at a real question.
传统的数学课，是从答案开始，从来没有谈到真的问题。
"Here are the steps to multiply. You repeat. Here are the steps to divide. You repeat.
“这些是乘法的步骤，你重复一次。这些是除法的步骤，你重复一次。
We've covered the material. We're moving on."
讲完了。继续下一课。”
What matters in the model is memorizing the steps.
在这种模式当中，重点是记住步骤。
There's no room to doubt or imagine or refuse, so there's no real thinking here.
没有怀疑、想象或拒绝的空间，所以这里没有真正的思考。
What would it look like if we started with a question?
如果我们从一个问题开始会变怎样？
For example, here are the numbers from 1 to 20.
比如，这里是1到20的数字。
Now, there's a question lurking in this picture, hiding in plain sight. What's going on with the colors?
在这张图中隐藏了一个问题，不是能明显看见的。这些颜色是怎么回事？
Now, intuitively it feels like there's some connection between the numbers and the colors.
从直觉来看，数字和颜色之间有某种连结。
I mean, maybe it's even possible to extend the coloring to more numbers.
或许还有可能，可以将配色方式延伸到更多的数字上。
At the same time, the meaning of the colors is not clear. It's a real mystery.
同时，颜色的意义并不清楚。这是个真的谜题。
And so, the question feels authentic and compelling.
所以，这个问题感觉很真实且引人好奇。
And like so many authentic mathematical questions,
和许多真实的数学问题一样，
this one has an answer that is both beautiful and profoundly satisfying.
这个问题也有一个又漂亮又让人能深深满足的答案。
And of course, I'm not going to tell you what it is.
当然，我不会告诉你们答案。
I don't think of myself as a mean person, but I am willing to deny you what you want.
我不觉得我是个小气的人，但我不想把你们想要的给你们。
Because I know if I rush to an answer, I would've robbed you of the opportunity to learn.
因为我知道，如果我马上给答案，我就剥夺了你们的学习机会。
Thinking happens only when we have time to struggle. And that is principle two.
只有在我们有时间挣扎的时候，思考才会发生。那就是原则二。
It's not uncommon for students to graduate from high school believing that every math problem can be solved in 30 seconds or less,
经常可以见到高中毕业的学生相信每一个数学问题都可以在三十秒以内解决，
and if they don't know the answer, they're just not a math person. This is a failure of education.
如果他们不知道答案，那他们就不是学数学的料。这是教育的失败之处。
We need to teach kids to be tenacious and courageous, to persevere in the face of difficulty.
我们得要教导孩子坚持、勇敢，在面对困难时要不屈不挠。
The only way to teach perseverance is to give students time to think and grapple with real problems.
教导不屈不挠的唯一方式，就是给学生时间，让他们去思考、处理真实的问题。
I brought this image into a classroom recently, and we took the time to struggle.
最近，我把这张图带到教室里，我们花了时间来挣扎、努力。
And the longer we spent, the more the class came alive with thinking.
我们花越多时间，班上的思考就越是活跃起来。
The students made observations. They had questions.
学生会观察。他们会问问题。
Like, "Why do the numbers in that last column always have orange and blue in them?"
比如：“为什么最后一栏的数字都一定有橘色和蓝色？”
and "Does it mean anything that the green spots are always going diagonally?"
以及“是不是有绿色的圆点都一定走对角线？”
and "What's going on with those little white numbers in the red segments?
以及“在红色区块中的那些小型白色数字是怎么回事？
Is it important that those are always odd numbers?"
那些数字都是奇数，这点重要吗？”
Struggling with a genuine question, students deepen their curiosity and their powers of observation.
为了真实的问题挣扎、努力，学生的好奇心与观察力都会加深。
They also develop the ability to take a risk.
他们也会发展出冒险的能力。
Some students noticed that every even number has orange in it, and they were willing to stake a claim.
有些学生注意到每个偶数数字上面都有橘色，他们就愿意赌一把，提出主张。
"Orange must mean even." And then they asked, "Is that right?"
“橘色一定代表偶数。”接着，他们会问：“对吗？”
This can be a scary place as a teacher.
身为老师，这是蛮吓人的情况。
A student comes to you with an original thought. What if you don't know the answer?
学生带着原创的想法来找你。如果你不知道答案怎么办？
Well, that is principle three: you are not the answer key.
那就是原则三：你不是答案。
Teachers, students may ask you questions you don't know how to answer.
老师们，学生可能会问出你们也不知道答案的问题。
And this can feel like a threat. But you are not the answer key.
这可能感觉像是威胁。但你不是答案之钥。
Students who are inquisitive is a wonderful thing to have in your classroom.
在你的教室里有好问的学生是很美好的事。
And if you can respond by saying, "I don't know. Let's find out," math becomes an adventure.
如果你能响应说：“我不知道。我们来找出答案。”数学就会变成一场冒险。

And parents, this goes for you too.
父母们，你们也是一样。
When you sit down to do math with your children, you don't have to know all the answers.
当你们坐下来陪孩子做数学时，你们不需要知道所有的答案。
You can ask your child to explain the math to you or try to figure it out together.
你们可以请孩子向你们解释数学，或是和他们一起找答案。
Teach them that not knowing is not failure. It's the first step to understanding.
要教导他们，“不知道”并不是失败。“不知道”是了解的第一步。
So, when this group of students asked me if orange means even, I don't have to tell them the answer.
所以，当这群学生问我橘色是否表示偶数时，我不需要告诉他们答案。
I don't even need to know the answer.
我甚至不需要知道答案。
I can ask one of them to explain to me why she thinks it's true.
我可以请他们其中一位来解释为什么她认为这个主张是对的。
Or we can throw the idea out to the class.
或者，我们可以把这个想法丢给全班。
Because they know the answers won't come from me,
因为他们知道答案不会从我这里说出来，
they need to convince themselves and argue with each other to determine what's true.
他们的说词就得要能说服自己，也要能和彼此争辩，用这种方式来判断对错。
And so, one student says, "Look, 2, 4, 6, 8, 10, 12. I checked all of the even numbers.
所以，有位学生说：“看，2、4、6、8、10、12。我检查过了，全部是偶数。
They all have orange in them. What more do you want?"
它们都有橘色。你还想要怎样？”
And another student says, "Well, wait a minute, I see what you're saying,
另一位学生说：“等等，我懂你的意思，
but some of those numbers have one orange piece, some have two or three.
但那些数字，有些只有一块橘色，有些有两块或三块。
Like, look at 48. It's got four orange pieces.
比如，看看48。它有四块橘色。
Are you telling me that 48 is four times as even as 46? There must be more to the story."
你的意思是，48比46还要“更偶数”四倍吗？一定不只如此。”
By refusing to be the answer key, you create space for this kind of mathematical conversation and debate.
当你拒绝提供答案，你就创造出了一个空间给这类数学对谈和辩论。
And this draws everyone in because we love to see people disagree.
这会吸引所有人，因为人都爱看别人意见不合。
After all, where else can you see real thinking out loud? Students doubt, affirm, deny, understand.
毕竟，还有什么其他地方能看到把想法大声说出来的状况？学生会怀疑、确认、否认、了解。
And all you have to do as the teacher is not be the answer key and say "yes" to their ideas.
身为老师，你要做的就只有不要提供答案，并认可他们的想法（说yes）。
And that is principle four. Now, this one is difficult.
那就是原则四。这一项很难。
What if a student comes to you and says 2 plus 2 equals 12? You've got to correct them, right?
如果学生来找你，说2+2=12怎么办？你必须要纠正他们，对吧？
And it's true, we want students to understand certain basic facts and how to use them.
的确，我们希望学生能了解某些基础事实以及如何使用它们。
But saying "yes" is not the same thing as saying "You're right."
但“认可你”不表示“你是对的”。
You can accept ideas, even wrong ideas,
在辩论中，你可以接受想法，甚至是错的想法，
into the debate and say "yes" to your students' right to participate in the act of thinking mathematically.
并认可你的学生参与数学思考这个举动的权利。
To have your idea dismissed out of hand is disempowering.
想法马上被排除会很让人气馁。
To have it accepted, studied, and disproven is a mark of respect.
接受想法、探究它、再反驳它，这是一种尊重的表现。
It's also far more convincing to be shown you're wrong by your peers than told you're wrong by the teacher.
而且，由同侪来说明你是错的，会比老师说你是错的更有说服力。
But allow me to take this a step further. How do you actually know that 2 plus 2 doesn't equal 12?
但容我再进一步谈这一点。你怎么知道2+2不会等于12？
What would happen if we said "yes" to that idea? I don't know. Let's find out.
如果你认可这个想法，会发生什么事？我不知道。咱们来找答案。
So, if 2 plus 2 equaled 12, then 2 plus 1 would be one less, so that would be 11.
所以，如果2+2=12，那么2+1就会少1，也就是11。
And that would mean that 2 plus 0, which is just 2, would be 10.
那就表示2+0，其实就是2，会等于10。
But if 2 is 10, then 1 would be 9, and 0 would be 8.
但如果2就是10，那1就是9，那0就是8。
And I have to admit this looks bad. It looks like we broke mathematics.
我得要承认，这看起来很不妙。看起来我们破坏了数学。
But I actually understand why this can't be true now.
但我现在确实了解了为什么这是错的。
Just from thinking about it, if we were on a number line, and if I'm at 0, 8 is eight steps that way,
想想看就能了解，如果我们在一条数轴上，如果我站在0的位置，8离我有8步远，
and there's no way I could take eight steps and wind up back where I started.
我不可能走了8步还回到我一开始的地方。
Unless ... well, what if it wasn't a number line? What if it was a number circle?
除非...嗯，如果不是数轴呢？如果是数圈呢？
Then I could take eight steps and wind back where I started. 8 would be 0.
那我确实有可能走了8步之后回到原点。8就是0。
In fact, all of the infinite numbers on the real line would be stacked up in those eight spots.
事实上，实数轴上的所有无限数字都可以堆垒在这八个点上。
And we're in a new world. And we're just playing here, right? But this is how new math gets invented.
我们进入了一个新世界。我们只是在玩玩，对吧？但新数学就是这样发明出来的。
Mathematicians have actually been studying number circles for a long time.
从很久以前，数学家就开始在研究数圈。
They've got a fancy name and everything: modular arithmetic.
还有很炫的名字等等：模运算。
And not only does the math work out,
这种数学不但行得通，
it turns out to be ridiculously useful in fields like cryptography and computer science.
竟还很有用，有用到不可思议，用在像是密码学和信息科学的领域。
It's actually no exaggeration to say that your credit card number is safe online
这样说并不夸张：你能在网络上安全地使用你的信用卡卡号，
because someone was willing to ask, "What if it was a number circle instead of a number line?"
是因为以前有人愿意问：“如果是数圈而不是数轴呢？”
So, yes, we need to teach students that 2 plus 2 equals 4.
是的，我们得要教学生2+2=4。
But also we need to say "yes" to their ideas and their questions and model the courage we want them to have.
但我们也得要认可他们的想法和问题，我们希望他们能拥有的勇气，我们也该以身作则展现出来。
It takes courage to say, "What if 2 plus 2 equals 12?" and actually explore the consequences.
要有勇气才能说出：“如果2+2=12呢？”并真正去探究后果。
It takes courage to say, "What if the angles in a triangle didn't add up to 180 degrees?"
要有勇气才能说出：“如果三角形的三个角度加起来不是180度呢？”
or "What if there were a square root of negative 1?" or "What if there were different sizes of infinity?"
或“如果-1可以开平方根呢？”或“如果无限大也有不同的大小呢？”
But that courage and those questions led to some of the greatest breakthroughs in history.
但正是那种勇气和那些问题在历史上引导出了一些最伟大的突破。
All it takes is willingness to play. And that is principle five.
所需要的，只是愿意参与去玩的意愿。那就是原则五。
Mathematics is not about following rules.
数学的重点并不是要遵守规则。
It's about playing and exploring and fighting and looking for clues and sometimes breaking things.
重点是在玩，还有探究、奋斗、寻求线索，有时还要打破东西。
Einstein called play the highest form of research.
爱因斯坦把玩称为研究的最高形式。
And a math teacher who lets their students play with math gives them the gift of ownership.
如果数学老师让他的学生玩数学，就是给了他们所有权这项礼物。
Playing with math can feel like running through the woods when you were a kid.
玩数学，感觉起来就像是小时候在树林里面奔跑。
And even if you were on a path, it felt like it all belonged to you.
即使跑在一条小路上，感觉也像是整条小路都属于你。
Parents, if you want to know how to nurture the mathematical instincts of your children, play is the answer.
父母们，如果你们想要知道如何培养你们孩子的数学直觉，答案就是玩。
What books are to reading, play is to mathematics.
玩之于数学，就像书之于阅读。
And a home filled with blocks and puzzles and games and play is a home where mathematical thinking can flourish.
如果家中满满都是积木、拼图、游戏、玩乐，在这样的家中，数学思考就会非常活跃。
I believe we have the power to help mathematical thinking flourish everywhere.
我相信我们有能力可以协助数学思考在各处活跃。
We can't afford to misuse math to create passive rule-followers.
我们不能误用数学来创造出只会被动遵照规则的人。
Math has the potential to be our greatest asset in teaching the next generation
数学有潜能可以成为我们最棒的资产，教育下一代，
to meet the future with courage, curiosity, and creativity.
带着勇气、好奇心和创意去面对未来。
And if all students get a chance to experience the beauty and power of authentic mathematical thinking,
如果所有的学生都有机会体验到真实数学思考的力与美，
maybe it won't sound so strange when they say, "Math? I actually love math." Thank you.
也许听到他们这么说就一点也不奇怪了：“数学？我很爱数学。”谢谢。