0:00:10.374,0:00:14.374
A friend of mine told me recently[br]that her six-year-old son

0:00:14.374,0:00:17.604
had come from school[br]and said he hated math.

0:00:18.314,0:00:21.884
And this is hard for me to hear[br]because I actually love math.

0:00:22.374,0:00:26.675
The beauty and power of mathematical[br]thinking have changed my life.

0:00:26.675,0:00:29.845
But I know that many people[br]lived a very different story.

0:00:30.005,0:00:33.276
Math can be the best of times[br]or the worst of times,

0:00:33.506,0:00:36.606
an exhilarating journey of discovery

0:00:36.606,0:00:41.796
or descent into tedium,[br]frustration, and despair.

0:00:43.518,0:00:47.798
Mathematical miseducation[br]is so common we can hardly see it.

0:00:48.088,0:00:50.137
We practically expect math class

0:00:50.139,0:00:55.291
to be repetition and memorization[br]of disjointed technical facts.

0:00:55.601,0:00:58.198
And we're not surprised[br]when students aren't motivated,

0:00:58.198,0:01:00.331
when they leave school disliking math,

0:01:00.334,0:01:03.332
even committed to avoiding it[br]for the rest of their lives.

0:01:04.179,0:01:08.709
Without mathematical literacy,[br]their career opportunities shrink.

0:01:09.028,0:01:12.738
And they become easy prey[br]for credit card companies,

0:01:12.738,0:01:15.182
payday lenders, the lottery,

0:01:16.104,0:01:17.374
(Laughter)

0:01:17.667,0:01:21.407
and anyone, really, who wants[br]to dazzle them with a statistic.

0:01:21.696,0:01:25.496
Did you know that if you insert[br]a single statistic into an assertion,

0:01:25.496,0:01:29.946
people are 92 percent more likely[br]to accept it without question?

0:01:29.946,0:01:32.656
(Laughter)

0:01:33.736,0:01:36.211
Yeah, I totally made that up.

0:01:36.211,0:01:37.341
(Laughter)

0:01:37.356,0:01:43.036
And 92 percent is - it has weight[br]even though it's completely fabricated.

0:01:43.036,0:01:44.500
And that's how it works.

0:01:44.500,0:01:46.350
When we're not comfortable with math,

0:01:46.350,0:01:49.100
we don't question[br]the authority of numbers.

0:01:51.534,0:01:56.014
But what's happening[br]with mathematical alienation

0:01:56.014,0:01:58.104
is only half the story.

0:01:58.104,0:02:02.599
Right now, we're squandering[br]our chance to touch life after life

0:02:02.599,0:02:06.337
with the beauty and power[br]of mathematical thinking.

0:02:06.597,0:02:10.861
I led a workshop on this topic recently,[br]and at the end, a woman raised her hand

0:02:10.861,0:02:13.991
and said that the experience[br]made her feel - and this is a quote -

0:02:14.291,0:02:16.001
"like a God."

0:02:16.271,0:02:19.191
(Laughter)

0:02:19.191,0:02:21.771
That's maybe the best[br]description I've ever heard

0:02:21.771,0:02:24.761
for what mathematical[br]thinking can feel like,

0:02:25.878,0:02:28.528
so we should examine what it looks like.

0:02:28.528,0:02:29.548
A good place to start

0:02:29.548,0:02:33.088
is with the words of the philosopher[br]and mathematician René Descartes,

0:02:33.088,0:02:37.107
who famously proclaimed,[br]"I think, therefore I am."

0:02:37.594,0:02:41.056
But Descartes looked deeper[br]into the nature of thinking.

0:02:41.056,0:02:44.026
Once he established himself[br]as a thing that thinks,

0:02:44.026,0:02:47.162
he continued, "What is a thinking thing?"

0:02:47.892,0:02:52.122
It is the thing that doubts,[br]understands, conceives,

0:02:52.122,0:02:55.832
that affirms and denies,[br]wills and refuses,

0:02:55.832,0:02:57.756
that imagines also,

0:02:57.756,0:02:59.296
and perceives.

0:03:00.457,0:03:05.697
This is the kind of thinking we need[br]in every math class every day.

0:03:06.372,0:03:11.322
So, if you are a teacher or a parent[br]or anyone with a stake in education,

0:03:11.322,0:03:13.381
I offer these five principles

0:03:13.384,0:03:18.580
to invite thinking into the math[br]we do at home and at school.

0:03:21.384,0:03:24.214
Principle one: start with a question.

0:03:24.864,0:03:28.280
The ordinary math class[br]begins with answers

0:03:28.280,0:03:30.370
and never arrives at a real question.

0:03:30.370,0:03:32.422
"Here are the steps[br]to multiply. You repeat.

0:03:32.422,0:03:34.372
Here are the steps to divide. You repeat.

0:03:34.372,0:03:36.472
We've covered the material.[br]We're moving on."

0:03:36.472,0:03:39.313
What matters in the model[br]is memorizing the steps.

0:03:39.313,0:03:44.372
There's no room to doubt[br]or imagine or refuse,

0:03:44.892,0:03:46.872
so there's no real thinking here.

0:03:48.230,0:03:50.845
What would it look like[br]if we started with a question?

0:03:51.495,0:03:55.015
For example, here[br]are the numbers from 1 to 20.

0:03:55.015,0:03:57.605
Now, there's a question[br]lurking in this picture,

0:03:58.285,0:04:00.175
hiding in plain sight.

0:04:00.855,0:04:03.072
What's going on with the colors?

0:04:04.592,0:04:07.142
Now, intuitively it feels like[br]there's some connection

0:04:07.142,0:04:09.622
between the numbers and the colors.

0:04:09.622,0:04:13.830
I mean, maybe it's even possible to extend[br]the coloring to more numbers.

0:04:15.140,0:04:18.890
At the same time, the meaning[br]of the colors is not clear.

0:04:19.540,0:04:20.890
It's a real mystery.

0:04:21.470,0:04:25.700
And so, the question[br]feels authentic and compelling.

0:04:26.900,0:04:31.410
And like so many authentic[br]mathematical questions,

0:04:31.418,0:04:36.838
this one has an answer that is[br]both beautiful and profoundly satisfying.

0:04:38.658,0:04:41.448
And of course, I'm not going[br]to tell you what it is.

0:04:41.448,0:04:43.698
(Laughter)

0:04:44.690,0:04:47.164
I don't think of myself as a mean person,

0:04:47.164,0:04:50.544
but I am willing to deny you[br]what you want.

0:04:50.544,0:04:52.074
(Laughter)

0:04:52.074,0:04:55.654
Because I know if I rush to an answer,

0:04:55.654,0:04:58.855
I would've robbed you[br]of the opportunity to learn.

0:04:59.705,0:05:03.475
Thinking happens only[br]when we have time to struggle.

0:05:04.735,0:05:06.675
And that is principle two.

0:05:07.738,0:05:10.868
It's not uncommon for students[br]to graduate from high school

0:05:10.868,0:05:16.158
believing that every math problem[br]can be solved in 30 seconds or less,

0:05:16.162,0:05:19.262
and if they don't know the answer,[br]they're just not a math person.

0:05:19.512,0:05:21.792
This is a failure of education.

0:05:21.792,0:05:25.662
We need to teach kids[br]to be tenacious and courageous,

0:05:25.662,0:05:28.312
to persevere in the face of difficulty.

0:05:28.652,0:05:30.911
The only way to teach perseverance

0:05:30.911,0:05:36.521
is to give students time[br]to think and grapple with real problems.

0:05:37.200,0:05:40.680
I brought this image[br]into a classroom recently,

0:05:40.680,0:05:42.900
and we took the time to struggle.

0:05:42.900,0:05:47.400
And the longer we spent, the more[br]the class came alive with thinking.

0:05:47.720,0:05:49.380
The students made observations.

0:05:49.380,0:05:51.000
They had questions.

0:05:51.000,0:05:52.140
Like,

0:05:52.140,0:05:55.890
"Why do the numbers in that last column[br]always have orange and blue in them?"

0:05:56.304,0:06:00.795
and "Does it mean anything that the green[br]spots are always going diagonally?"

0:06:01.205,0:06:03.655
and "What's going on[br]with those little white numbers

0:06:03.655,0:06:05.185
in the red segments?

0:06:05.185,0:06:07.915
Is it important that those[br]are always odd numbers?"

0:06:09.147,0:06:11.647
Struggling with a genuine question,

0:06:11.647,0:06:16.477
students deepen their curiosity[br]and their powers of observation.

0:06:17.139,0:06:23.429
They also develop[br]the ability to take a risk.

0:06:25.257,0:06:28.297
Some students noticed[br]that every even number has orange in it,

0:06:28.297,0:06:30.157
and they were willing to stake a claim.

0:06:30.157,0:06:32.467
"Orange must mean even."

0:06:33.007,0:06:35.829
And then they asked, "Is that right?"

0:06:35.829,0:06:37.239
(Laughter)

0:06:37.641,0:06:41.161
This can be a scary place as a teacher.

0:06:41.161,0:06:44.411
A student comes to you[br]with an original thought.

0:06:45.392,0:06:47.392
What if you don't know the answer?

0:06:49.335,0:06:53.805
Well, that is principle three:[br]you are not the answer key.

0:06:55.402,0:06:59.782
Teachers, students may ask you questions[br]you don't know how to answer.

0:06:59.782,0:07:02.150
And this can feel like a threat.

0:07:02.150,0:07:04.380
But you are not the answer key.

0:07:05.542,0:07:07.942
Students who are inquisitive

0:07:07.942,0:07:10.212
is a wonderful thing[br]to have in your classroom.

0:07:10.212,0:07:12.292
And if you can respond by saying,

0:07:12.902,0:07:15.782
"I don't know. Let's find out,"

0:07:16.262,0:07:18.462
math becomes an adventure.

0:07:19.882,0:07:22.652
And parents, this goes for you too.

0:07:22.652,0:07:25.642
When you sit down to do math[br]with your children,

0:07:25.642,0:07:28.242
you don't have to know all the answers.

0:07:28.652,0:07:31.522
You can ask your child[br]to explain the math to you

0:07:31.522,0:07:33.742
or try to figure it out together.

0:07:35.551,0:07:39.781
Teach them that not knowing[br]is not failure.

0:07:40.411,0:07:42.980
It's the first step to understanding.

0:07:44.242,0:07:49.892
So, when this group of students[br]asked me if orange means even,

0:07:49.892,0:07:51.972
I don't have to tell them the answer.

0:07:52.242,0:07:54.562
I don't even need to know the answer.

0:07:54.912,0:07:58.722
I can ask one of them to explain to me[br]why she thinks it's true.

0:07:59.342,0:08:02.032
Or we can throw the idea out to the class.

0:08:02.952,0:08:05.732
Because they know the answers[br]won't come from me,

0:08:05.732,0:08:09.192
they need to convince themselves[br]and argue with each other

0:08:09.192,0:08:10.912
to determine what's true.

0:08:10.912,0:08:14.302
And so, one student says,[br]"Look, 2, 4, 6, 8, 10, 12.

0:08:14.302,0:08:15.922
I checked all of the even numbers.

0:08:15.922,0:08:17.312
They all have orange in them.

0:08:17.312,0:08:18.782
What more do you want?"

0:08:18.782,0:08:21.132
And another student says,[br]"Well, wait a minute,

0:08:21.132,0:08:22.372
I see what you're saying,

0:08:22.372,0:08:24.932
but some of those numbers[br]have one orange piece,

0:08:24.932,0:08:27.092
some have two or three.

0:08:27.092,0:08:28.912
Like, look at 48.

0:08:29.492,0:08:31.632
It's got four orange pieces.

0:08:31.632,0:08:35.762
Are you telling me that 48[br]is four times as even as 46?

0:08:36.232,0:08:38.382
There must be more to the story."

0:08:39.482,0:08:41.517
By refusing to be the answer key,

0:08:41.517,0:08:45.977
you create space for this kind[br]of mathematical conversation and debate.

0:08:45.977,0:08:51.497
And this draws everyone in[br]because we love to see people disagree.

0:08:52.267,0:08:57.087
After all, where else can you see[br]real thinking out loud?

0:08:57.087,0:09:00.970
Students doubt, affirm, deny, understand.

0:09:02.180,0:09:06.240
And all you have to do as the teacher[br]is not be the answer key

0:09:06.240,0:09:08.910
and say "yes" to their ideas.

0:09:11.050,0:09:12.920
And that is principle four.

0:09:13.950,0:09:15.990
Now, this one is difficult.

0:09:16.140,0:09:19.360
What if a student comes to you[br]and says 2 plus 2 equals 12?

0:09:20.180,0:09:21.850
You've got to correct them, right?

0:09:21.850,0:09:25.200
And it's true, we want students[br]to understand certain basic facts

0:09:25.200,0:09:26.730
and how to use them.

0:09:27.140,0:09:31.420
But saying "yes" is not the same thing[br]as saying "You're right."

0:09:31.910,0:09:35.700
You can accept ideas,[br]even wrong ideas, into the debate

0:09:35.700,0:09:37.960
and say "yes" to your students' right

0:09:37.960,0:09:41.740
to participate in the act[br]of thinking mathematically.

0:09:42.840,0:09:48.590
To have your idea dismissed[br]out of hand is disempowering.

0:09:48.593,0:09:53.243
To have it accepted, studied,[br]and disproven is a mark of respect.

0:09:53.853,0:09:57.673
It's also far more convincing to be shown[br]you're wrong by your peers

0:09:57.673,0:10:00.023
than told you're wrong by the teacher.

0:10:01.293,0:10:03.933
But allow me to take this a step further.

0:10:04.873,0:10:08.353
How do you actually know[br]that 2 plus 2 doesn't equal 12?

0:10:09.263,0:10:11.573
What would happen[br]if we said "yes" to that idea?

0:10:12.703,0:10:13.943
I don't know.

0:10:14.213,0:10:15.643
Let's find out.

0:10:17.643,0:10:20.263
So, if 2 plus 2 equaled 12,

0:10:20.583,0:10:25.143
then 2 plus 1 would be one less,[br]so that would be 11.

0:10:25.592,0:10:29.382
And that would mean that 2 plus 0,[br]which is just 2, would be 10.

0:10:29.995,0:10:32.885
But if 2 is 10, then 1 would be 9,

0:10:32.885,0:10:34.595
and 0 would be 8.

0:10:35.175,0:10:37.475
And I have to admit this looks bad.

0:10:38.515,0:10:40.815
It looks like we broke mathematics.

0:10:41.575,0:10:45.035
But I actually understand[br]why this can't be true now.

0:10:45.555,0:10:47.335
Just from thinking about it,

0:10:47.335,0:10:50.845
if we were on a number line,

0:10:51.495,0:10:54.045
and if I'm at 0,[br]8 is eight steps that way,

0:10:54.045,0:10:56.485
and there's no way[br]I could take eight steps

0:10:56.485,0:10:58.565
and wind up back where I started.

0:11:01.198,0:11:02.738
Unless ...

0:11:03.188,0:11:04.418
(Laughter)

0:11:04.918,0:11:07.388
well, what if it wasn't a number line?

0:11:08.408,0:11:10.748
What if it was a number circle?

0:11:11.876,0:11:14.702
Then I could take eight steps[br]and wind back where I started.

0:11:14.702,0:11:15.732
8 would be 0.

0:11:15.732,0:11:19.792
In fact, all of the infinite numbers[br]on the real line would be stacked up

0:11:19.792,0:11:22.152
in those eight spots.

0:11:22.972,0:11:24.932
And we're in a new world.

0:11:27.348,0:11:29.518
And we're just playing here, right?

0:11:31.744,0:11:34.444
But this is how new math gets invented.

0:11:36.341,0:11:40.471
Mathematicians have actually been studying[br]number circles for a long time.

0:11:40.471,0:11:42.741
They've got a fancy name and everything:

0:11:42.741,0:11:44.711
modular arithmetic.

0:11:45.151,0:11:47.341
And not only does the math work out,

0:11:47.341,0:11:49.491
it turns out to be ridiculously useful

0:11:49.491,0:11:52.531
in fields like cryptography[br]and computer science.

0:11:52.531,0:11:54.601
It's actually no exaggeration to say

0:11:54.601,0:11:58.381
that your credit card number[br]is safe online

0:11:58.381,0:12:00.231
because someone was willing to ask,

0:12:00.231,0:12:03.781
"What if it was a number circle[br]instead of a number line?"

0:12:05.085,0:12:09.475
So, yes, we need to teach students[br]that 2 plus 2 equals 4.

0:12:10.310,0:12:14.250
But also we need to say "yes"[br]to their ideas and their questions

0:12:14.550,0:12:17.700
and model the courage[br]we want them to have.

0:12:18.240,0:12:21.330
It takes courage to say,[br]"What if 2 plus 2 equals 12?"

0:12:21.330,0:12:24.020
and actually explore the consequences.

0:12:25.057,0:12:27.127
It takes courage to say,

0:12:27.127,0:12:30.897
"What if the angles in a triangle[br]didn't add up to 180 degrees?"

0:12:31.470,0:12:34.450
or "What if there were[br]a square root of negative 1?"

0:12:34.710,0:12:38.250
or "What if there were[br]different sizes of infinity?"

0:12:39.309,0:12:41.999
But that courage and those questions

0:12:42.669,0:12:46.199
led to some of the greatest[br]breakthroughs in history.

0:12:46.999,0:12:49.779
All it takes is willingness to play.

0:12:51.409,0:12:53.649
And that is principle five.

0:12:55.269,0:12:58.039
Mathematics is not about following rules.

0:12:58.492,0:12:59.892
It's about playing

0:13:00.332,0:13:03.112
and exploring and fighting[br]and looking for clues

0:13:03.112,0:13:05.068
and sometimes breaking things.

0:13:05.638,0:13:09.188
Einstein called play[br]the highest form of research.

0:13:09.578,0:13:14.408
And a math teacher who lets[br]their students play with math

0:13:14.408,0:13:17.608
gives them the gift of ownership.

0:13:18.888,0:13:20.329
Playing with math can feel

0:13:20.332,0:13:22.968
like running through the woods[br]when you were a kid.

0:13:22.968,0:13:26.838
And even if you were on a path,[br]it felt like it all belonged to you.

0:13:27.909,0:13:30.559
Parents, if you want to know

0:13:30.559,0:13:33.519
how to nurture the mathematical[br]instincts of your children,

0:13:33.519,0:13:35.229
play is the answer.

0:13:36.001,0:13:39.551
What books are to reading,[br]play is to mathematics.

0:13:39.551,0:13:43.351
And a home filled with blocks[br]and puzzles and games and play

0:13:43.841,0:13:47.161
is a home where mathematical[br]thinking can flourish.

0:13:48.816,0:13:54.616
I believe we have the power to help[br]mathematical thinking flourish everywhere.

0:13:55.754,0:14:00.744
We can't afford to misuse math[br]to create passive rule-followers.

0:14:01.163,0:14:04.193
Math has the potential[br]to be our greatest asset

0:14:04.193,0:14:07.593
in teaching the next generation[br]to meet the future

0:14:07.593,0:14:11.633
with courage, curiosity, and creativity.

0:14:12.516,0:14:15.476
And if all students get a chance

0:14:15.478,0:14:19.636
to experience the beauty and power[br]of authentic mathematical thinking,

0:14:20.856,0:14:24.876
maybe it won't sound[br]so strange when they say,

0:14:25.871,0:14:27.261
"Math?

0:14:28.241,0:14:30.711
I actually love math."

0:14:31.783,0:14:32.973
Thank you.

0:14:32.973,0:14:35.633
(Applause)