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A friend of mine told me recently
that her six-year-old son

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had come from school
and said he hated math.

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And this is hard for me to hear
because I actually love math.

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The beauty and power of mathematical
thinking have changed my life.

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But I know that many people
lived a very different story.

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Math can be the best of times
or the worst of times,

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an exhilarating journey of discovery

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or descent into tedium,
frustration, and despair.

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Mathematical miseducation
is so common we can hardly see it.

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We practically expect math class

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to be repetition and memorization
of disjointed technical facts.

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And we're not surprised
when students aren't motivated,

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when they leave school disliking math,

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even committed to avoiding it
for the rest of their lives.

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Without mathematical literacy,
their career opportunities shrink.

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And they become easy prey
for credit card companies,

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payday lenders, the lottery,

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(Laughter)

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and anyone, really, who wants
to dazzle them with a statistic.

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Did you know that if you insert
a single statistic into an assertion,

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people are 92 percent more likely
to accept it without question?

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(Laughter)

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Yeah, I totally made that up.

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(Laughter)

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And 92 percent is - it has weight
even though it's completely fabricated.

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And that's how it works.

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When we're not comfortable with math,

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we don't question
the authority of numbers.

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But what's happening
with mathematical alienation

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is only half the story.

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Right now, we're squandering
our chance to touch life after life

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with the beauty and power
of mathematical thinking.

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I led a workshop on this topic recently,
and at the end, a woman raised her hand

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and said that the experience
made her feel - and this is a quote -

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"like a God."

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(Laughter)

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That's maybe the best
description I've ever heard

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for what mathematical
thinking can feel like,

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so we should examine what it looks like.

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A good place to start

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is with the words of the philosopher
and mathematician René Descartes,

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who famously proclaimed,
"I think, therefore I am."

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But Descartes looked deeper
into the nature of thinking.

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Once he established himself
as a thing that thinks,

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he continued, "What is a thinking thing?"

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It is the thing that doubts,
understands, conceives,

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that affirms and denies,
wills and refuses,

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that imagines also,

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and perceives.

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This is the kind of thinking we need
in every math class every day.

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So, if you are a teacher or a parent
or anyone with a stake in education,

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I offer these five principles

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to invite thinking into the math
we do at home and at school.

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Principle one: start with a question.

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The ordinary math class
begins with answers

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and never arrives at a real question.

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"Here are the steps
to multiply. You repeat.

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Here are the steps to divide. You repeat.

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We've covered the material.
We're moving on."

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What matters in the model
is memorizing the steps.

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There's no room to doubt
or imagine or refuse,

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so there's no real thinking here.

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What would it look like
if we started with a question?

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For example, here
are the numbers from 1 to 20.

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Now, there's a question
lurking in this picture,

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hiding in plain sight.

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What's going on with the colors?

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Now, intuitively it feels like
there's some connection

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between the numbers and the colors.

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I mean, maybe it's even possible to extend
the coloring to more numbers.

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At the same time, the meaning
of the colors is not clear.

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It's a real mystery.

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And so, the question
feels authentic and compelling.

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And like so many authentic
mathematical questions,

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this one has an answer that is
both beautiful and profoundly satisfying.

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And of course, I'm not going
to tell you what it is.

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(Laughter)

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I don't think of myself as a mean person,

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but I am willing to deny you
what you want.

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(Laughter)

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Because I know if I rush to an answer,

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I would've robbed you
of the opportunity to learn.

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Thinking happens only
when we have time to struggle.

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And that is principle two.

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It's not uncommon for students
to graduate from high school

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believing that every math problem
can be solved in 30 seconds or less,

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and if they don't know the answer,
they're just not a math person.

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This is a failure of education.

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We need to teach kids
to be tenacious and courageous,

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to persevere in the face of difficulty.

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The only way to teach perseverance

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is to give students time
to think and grapple with real problems.

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I brought this image
into a classroom recently,

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and we took the time to struggle.

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And the longer we spent, the more
the class came alive with thinking.

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The students made observations.

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They had questions.

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Like,

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"Why do the numbers in that last column
always have orange and blue in them?"

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and "Does it mean anything that the green
spots are always going diagonally?"

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and "What's going on
with those little white numbers

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in the red segments?

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Is it important that those
are always odd numbers?"

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Struggling with a genuine question,

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students deepen their curiosity
and their powers of observation.

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They also develop
the ability to take a risk.

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Some students noticed
that every even number has orange in it,

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and they were willing to stake a claim.

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"Orange must mean even."

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And then they asked, "Is that right?"

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(Laughter)

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This can be a scary place as a teacher.

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A student comes to you
with an original thought.

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What if you don't know the answer?

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Well, that is principle three:
you are not the answer key.

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Teachers, students may ask you questions
you don't know how to answer.

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And this can feel like a threat.

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But you are not the answer key.

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Students who are inquisitive

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is a wonderful thing
to have in your classroom.

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And if you can respond by saying,

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"I don't know. Let's find out,"

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math becomes an adventure.

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And parents, this goes for you too.

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When you sit down to do math
with your children,

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you don't have to know all the answers.

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You can ask your child
to explain the math to you

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or try to figure it out together.

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Teach them that not knowing
is not failure.

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It's the first step to understanding.

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So, when this group of students
asked me if orange means even,

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I don't have to tell them the answer.

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I don't even need to know the answer.

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I can ask one of them to explain to me
why she thinks it's true.

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Or we can throw the idea out to the class.

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Because they know the answers
won't come from me,

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they need to convince themselves
and argue with each other

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to determine what's true.

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And so, one student says,
"Look, 2, 4, 6, 8, 10, 12.

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I checked all of the even numbers.

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They all have orange in them.

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What more do you want?"

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And another student says,
"Well, wait a minute,

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I see what you're saying,

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but some of those numbers
have one orange piece,

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some have two or three.

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Like, look at 48.

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It's got four orange pieces.

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Are you telling me that 48
is four times as even as 46?

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There must be more to the story."

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By refusing to be the answer key,

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you create space for this kind
of mathematical conversation and debate.

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And this draws everyone in
because we love to see people disagree.

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After all, where else can you see
real thinking out loud?

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Students doubt, affirm, deny, understand.

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And all you have to do as the teacher
is not be the answer key

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and say "yes" to their ideas.

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And that is principle four.

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Now, this one is difficult.

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What if a student comes to you
and says 2 plus 2 equals 12?

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You've got to correct them, right?

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And it's true, we want students
to understand certain basic facts

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and how to use them.

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But saying "yes" is not the same thing
as saying "You're right."

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You can accept ideas,
even wrong ideas, into the debate

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and say "yes" to your students' right

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to participate in the act
of thinking mathematically.

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To have your idea dismissed
out of hand is disempowering.

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To have it accepted, studied,
and disproven is a mark of respect.

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It's also far more convincing to be shown
you're wrong by your peers

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than told you're wrong by the teacher.

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But allow me to take this a step further.

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How do you actually know
that 2 plus 2 doesn't equal 12?

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What would happen
if we said "yes" to that idea?

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I don't know.

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Let's find out.

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So, if 2 plus 2 equaled 12,

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then 2 plus 1 would be one less,
so that would be 11.

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And that would mean that 2 plus 0,
which is just 2, would be 10.

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But if 2 is 10, then 1 would be 9,

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and 0 would be 8.

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And I have to admit this looks bad.

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It looks like we broke mathematics.

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But I actually understand
why this can't be true now.

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Just from thinking about it,

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if we were on a number line,

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and if I'm at 0,
8 is eight steps that way,

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and there's no way
I could take eight steps

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and wind up back where I started.

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Unless ...

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(Laughter)

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well, what if it wasn't a number line?

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What if it was a number circle?

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Then I could take eight steps
and wind back where I started.

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8 would be 0.

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In fact, all of the infinite numbers
on the real line would be stacked up

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in those eight spots.

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And we're in a new world.

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And we're just playing here, right?

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But this is how new math gets invented.

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Mathematicians have actually been studying
number circles for a long time.

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They've got a fancy name and everything:

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modular arithmetic.

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And not only does the math work out,

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it turns out to be ridiculously useful

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in fields like cryptography
and computer science.

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It's actually no exaggeration to say

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that your credit card number
is safe online

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because someone was willing to ask,

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"What if it was a number circle
instead of a number line?"

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So, yes, we need to teach students
that 2 plus 2 equals 4.

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But also we need to say "yes"
to their ideas and their questions

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and model the courage
we want them to have.

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It takes courage to say,
"What if 2 plus 2 equals 12?"

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and actually explore the consequences.

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It takes courage to say,

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"What if the angles in a triangle
didn't add up to 180 degrees?"

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or "What if there were
a square root of negative 1?"

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or "What if there were
different sizes of infinity?"

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But that courage and those questions

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led to some of the greatest
breakthroughs in history.

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All it takes is willingness to play.

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And that is principle five.

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Mathematics is not about following rules.

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It's about playing

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and exploring and fighting
and looking for clues

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and sometimes breaking things.

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Einstein called play
the highest form of research.

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And a math teacher who lets
their students play with math

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gives them the gift of ownership.

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Playing with math can feel

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like running through the woods
when you were a kid.

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And even if you were on a path,
it felt like it all belonged to you.

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Parents, if you want to know

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how to nurture the mathematical
instincts of your children,

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play is the answer.

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What books are to reading,
play is to mathematics.

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And a home filled with blocks
and puzzles and games and play

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is a home where mathematical
thinking can flourish.

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I believe we have the power to help
mathematical thinking flourish everywhere.

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We can't afford to misuse math
to create passive rule-followers.

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Math has the potential
to be our greatest asset

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in teaching the next generation
to meet the future

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with courage, curiosity, and creativity.

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And if all students get a chance

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to experience the beauty and power
of authentic mathematical thinking,

247
00:14:20,856 --> 00:14:24,876
maybe it won't sound
so strange when they say,

248
00:14:25,871 --> 00:14:27,261
"Math?

249
00:14:28,241 --> 00:14:30,711
I actually love math."

250
00:14:31,783 --> 00:14:32,973
Thank you.

251
00:14:32,973 --> 00:14:35,633
(Applause)