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THE ASPEN INSTITUTE

ASPEN IDEAS FESTIVAL 2015

LEARNING REVOLUTION: HOW NEUROSCIENCE CAN CHANGE A

NATION'S MATHEMATICAL FUTURE

Koch Building, Booz Allen Hamilton Room

Aspen, Colorado

Monday, June 29, 2015

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LIST OF PARTICIPANTS

JO BOALER

Professor, Mathematics Education and

Teacher Education, Stanford University

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LEARNING REVOLUTION: HOW NEUROSCIENCE CAN CHANGE A

NATION'S MATHEMATICAL FUTURE

SPEAKER: So welcome to our first official

session on mathematics I believe it is. And every year at

the festival we try to focus on one area of science, and

we've never addressed mathematics. So the beauty of

mathematics is a session that will happen with -- a track

that will happen, we've session every single day and

tomorrow we're doing a deep dive on the subject of "Really

is Math Important?" which one of our professors suggested

we do.

Jo Boaler is at Stanford University, she is

acclaimed as one of the finest math educators and thinkers

about math education in the United States. So it's my

honor to turn this right over to Jo.

(Applause)

MS. BOALER: Thank you.

(Applause)

MS. BOALER: Thank you for that very nice

introduction. Is my mike okay, you can hear in the back?

Lovely, thank you. So I thought I'd start off just by

finding out who's here today. So could I just ask you to

put your hand up if you are a university mathematician?

(Laughter)

MS. BOALER: Few of those? Okay, how about

applied, who uses math in the workplace kind of math

person? How about an educator? Educators. Do we have

any math teachers, keep your hand up if you're a math

teacher? Perfect. Policymakers? All right, talk to you

later.

(Laughter)

MS. BOALER: All right. So what I want to talk

about today is the new science of the brain and learning,

which really is -- can be called a revolution in what we

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know about learning, and it has huge implications for math

learning. So I'm going to be sharing with you sort of six

findings that I think are extremely important for

mathematics, and the first one actually comes from the

area of mathematics -- the area of the brain science,

which you'll know as brain plasticity.

It used to be believed some people could do

math, some people can't. We now have huge amounts of

evidence that the brain is so plastic, so expandable. And

I'll show you some of it in a minute, but math educators

everywhere deal with a few myths in mathematics that

really hold students back in their learning.

So I have a book, which is here at this

conference actually, for teachers, parents, people who are

interested in math issues called, What's Math Got to Do

with It? which Penguin published. In England, we have

this sort of the same book with a different name and it's

called The Elephant in the Classroom, has different

spellings and some other differences.

(Laughter)

MS. BOALER: We called it The Elephant in the

Classroom, because I argue in both books really that it

was a big elephant standing in most math classrooms, and

that is the idea that only some kids would do well in

math. You know the expression, "the elephant in the

room," something we're all thinking about but nobody

actually says turns out that a lot of people believe that

only some kids can do well in math. Parents believe it,

students believe it, and unfortunately some teachers

believe it too.

So where does this idea come from that only some

kids can do math? Well, unfortunately we are streaming

this idea constantly through culture and media. I have

the dubious pleasure of seeing Tweeny TV shows these days,

because I have two daughters. And it is really amazing

and shocking to me how often math comes in these shows.

Probably every single day in teenage TV shows you will see

math come up always in the same light, always something

really awful, tedious, extremely inaccessible, kids pored

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over their math books unable to work. And sometimes the

TV shows even give the fixed message that only some kids

can do math.

Let's have a look at a couple of those. If

you're a parent -- how many people are parents, I meant to

ask that one. Great, if you're a parent you might

recognize this one, it's a big Disney TV show.

(Video being played)

MS. BOALER: Very strongly, hard doesn't matter,

"I'm just lousy at math." It comes through -- there's a

few more videos coming actually, it comes through many

things. Here we'll see -- you may recognize Steve Martin.

It's a bit dark, so he's sitting in a broom cupboard

talking about math.

(Video being played)

MS. BOALER: As I heard that message come out of

my living room, I jumped to get this clip. It actually

goes on in the rest of the show. We see this supposedly

sort of socially engaging girl get introduced to the math

club, and what happens is she gets all these kids who you

saw her representatives, and she's so socially awkward

that they can't even speak. Anyway she gets on to be more

social so they all leave the math club.

(Laughter)

MS. BOALER: And that is the show. So this

actually prompted me to get in touch with senior people at

Disney. It's going to be hard I think. So if any of you

know any great ways, we are transmitting, giving these

very, very powerful messages to children about who can do

math daily through children's television shows. So it's

something really important to change, I'd love help with

it.

So yeah, brain plasticity is this most amazing

new evidence that we have. So the brain is the most

complex organ in our bodies. It's made up of over 100

billion neurons and several thousand connections between

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them. And we know that when learning happens, the synapse

fire in the brain, and the synapse is like an electric

current that moves between parts of the brain. We know

that synapse is a little bit like footprints in the sand,

the brain will follow the footprints and make them deep

permanent pathways. Or if you never think about the idea

again, they'll just kind of wash away.

So some of the research that has shown

scientists this brain plasticity actually comes from my

hometown of London. And I'm sure many of you here have

traveled in black cabs in London, you probably that they

are the queen bee of taxis in London. But what you may

not have known is just how highly qualified the drivers

are. So it turns out to become a black cab driver you

study for between two and four years, and at the end of

that time you take a test, which is just beautifully

called, "The Knowledge," and in this test you have to have

memorized 25,000 streets and 20,000 landmarks in Central

London to the point that you have to be able to say where

temporary flower-stands are set up in the test. It's

extremely difficult. The average number of times it takes

to pass "The Knowledge" is 14.

So this is what they have found that after

taking "The Knowledge," the hippocampus in the brains of

the taxi drivers has significantly increased. And

scientists never knew that the brain was that flexible.

At the end of becoming a taxi driver when they retire, the

hippocampus shrinks back down again, and the hippocampus

is the area of the brain that's specialized in acquiring

and using spatial information.

More evidence. So around the time the taxicabs

studies were coming out, a six-year-old girl, you may have

seen this in the news, had half of her brain removed, a

hemisphere taken out because she was having fits that

doctors couldn't control. This was new, new surgery. And

they expected her to be paralyzed possibly for life

because that side of the brain controls your physical

movement. And she absolutely stunned doctors and

scientists who could only conclude that she had in effect

regrown the missing side of the brain, the connections

came back so quickly that she was able to recover.

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And one that is so important for educators,

especially math teachers over time, a three-week training

program where people working for 10 minutes a day changed

the permanent structure of their brain. So what all of

this tells us that every child can do well in math in

school, you know, up to calculus with the right teaching

and the right messages. If we don't give the right

teaching and the right messages, that won't happen.

Schools are set up on this notion of figuring

out who has the ability, even from when kids are about

five. And this has to change. We do not know and cannot

know what will happen to children' brains if they're given

the right opportunities.

So another myth that I think is important to

discuss and this myth I became aware of really from my

Stanford students who started to say to me there's this

thing called a "wall" in math, and I said, "wall" what's

that? And they said, well, you can take so many math

classes until you hit your wall, and then you can't go any

further because that's your limit. So big news, there is

no "wall."

(Laughter)

MS. BOALER: I didn't know who this was. One of

my teammates made this slide and I've learned now that

it's is Koolaid man.

(Laughter)

MS. BOALER: I've even learned what Koolaid is,

but -- anyway there is no "wall." So we urgently need to

shift teachers', parents' and students' ideas about who

can do well in math. And part of the evidence we have,

and I'm sure many of you know about some of this, is the

second sort of the brain area I want to talk about is the

power of mindsets.

So Carol Dweck, one of my Stanford colleagues

who I worked with wrote this book in 2006, it became a

huge bestseller. It probably had more impact on schools

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than any other research volume ever written, and it

communicates the large amount of research that shows

everyone has a mindset, you have one, I have one. People

with a growth mindset really believe that the harder they

work the smarter they'll get, people with a fixed mindset

believe that you're kind of smart or you're not.

Turns out about half people have a fixed

mindset, and half in the U.S. have a growth mindset, you

can change your mindset, and in math more people have a

fixed mindset about math than any other subject. Carol

(phonetic) has -- we found that kids with a growth mindset

-- growth mindsets are very important because those kids

achieve higher levels and they do that because a growth

mindset goes along with certain behaviors -- kids are more

persistent, they're more willing to learn from mistakes,

they keep going.

We know that the way we praise kids is very

important in which mindset they get, and this is a big one

for parents. I've had to change the way I talk to my own

kids. So kids get a fixed mindset from fixed praise.

Something we do a lot in the U.S. is praise children for

being "smart." The "smart" word. So when we praise kids

for being smart, what we now know is what they hear is, oh

good, I'm smart. And then later when they mess up and

they will, they think oh, I'm not so smart. And that

fixed idea is damaging to kids at whatever place they are.

So it's very important we don't praise kids by

telling them they're smart, we can praise them for things

they do, like it's fantastic you've learned that,

fantastic you've done that, but not issue that you are

"smart" word.

We know the fixed mindset thinking affects kids

from across the achievement spectrum and the biggest group

of fixed mindset students, most damaged group by it are

high-achieving girls. And many of us women here probably

can sort of resonate with this. Once you develop the idea

that you're smart, people become very vulnerable to giving

up that label. And for girls it means they don't choose

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it's a big part of why girls choose out of STEM subjects.

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But in this country we love the discourse of

smartness and giftedness. Same in the U.K. And we have

to really let go of these words. We know that people are

born with brain differences, but actually the brain

differences they're born with are eclipsed by experiences

they have in early life. It's very few people, minute

amount of people who are born with brain differences to

make that difference.

So we know the ideas of giftedness and smartness

harm both low and high-achieving kids. You noticed the

number of times you heard the word "smart" at this

conference. Or see it on water-bottles, not that one, and

other places? This study was so important and it came out

just very recently in Science, you may have found it --

seen it.

The more of field values the idea of giftedness,

the fewer female PhDs are on the field. And this is

amazing, field-specific beliefs about giftedness were

correlated with how many females were in the field across

all 30 disciplines they looked at. Have a look at these

graphs. The top one is STEM and the bottom one is

humanities, and what you see there is the more of field

beliefs in giftedness, and if I wasn't attached to this

chair I'd point you -- can you show where math is on there

Asif (phonetic).

The more the field beliefs in giftedness -- so

actually the two fields that interestingly do this -- oh,

I'm going to get a clicker -- math in the STEM subjects

and philosophy. Turns out philosophers all believe that

you have a natural gift or not. Thank you.

And it is another area where there are very few

women and this is why it is women and students of color

whose stereotype is lacking innate talent. So if people

think that it's innate talent they're much more likely to

think it something that white men have. Claude Steele

found that when people take math tests just marking off

your gender before you take the test causes girls to

underachieve in the math test. So that's because these

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stereotypes are so strong about women and math that as he

said they're in the air all the time.

So let's have a look at these ideas about

mindset and whether you believe you can grow your brain.

This graph just shows you -- they just found kids mindsets

and then follow them in math every two years. This top

one is the kids with the growth mindset who go onwards and

upwards all the time. The kids with the fixed mindset are

the ones whose achievement is in that lower bar.

We know the students of color and girls show the

sharpest increase in achievement with mindset

interventions. So I'm working closely with the Pisa team

at the moment at the moment. You probably know Pisa, they

are international tests that are given every three years

in math and science, they have a lot of data that isn't

just about math achievement. It actually records kids who

-- their approach math, their beliefs about math, and they

have a massive data set for us to work with, 30 million

15-year-olds.

And this is one of the big findings that we're

just publishing -- the lowest achieving kids in the world

are the memorizers. So those kids who approach math by

thinking I have to memorize all of these methods and

pathways are the lowest-achieving kids, and the highest

achieving kids are the ones who see math as a subject of

big ideas. So ideas are really important.

This graph shows you the -- first, the massive

difference with mindset -- this isn't showing up too well

-- but the two highest bars on the right are kids with the

growth mindset and the two lower ones are the kids with

the fixed mindset. That difference is two years of math

achievement across the world. And the kids in the red are

the kids with the more conceptual ideas about math, the

blue bars are the memorizers each time. And we're just

releasing that. Actually Scientific American's working

with this to get this released.

So next, findings from brain size, which is so

important for kids everywhere. We now know that mistakes

in math grow your brain. And in fact in MRI scans of

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people working with through math tests, they find every

time people make a mistake a synapse fires in the brain.

And there are actually two possible synapses, the first

one comes from people making a mistake and then the second

one comes from people who are aware they've made a

mistake.

When I told teachers that synapses fire up when

people make a mistake in a test, they say oh but children

have to work through it and get it later for the synapse

to fire. And I said you don't, you don't even have to

know you've made a mistake for the synapse to fire. And

then they'll say, well, how can a synapse fire if you

don't even know you've made a mistake, which is a

perfectly reasonable question. And the best evidence we

have on this is that the synapse fires when people make a

mistake even when they're not aware of it because it is

the time when your brain is struggling and when you're

challenged. And we know that those are the very best

times for brain growth.

So equally interesting I think -- this message

is about mistakes, they so important for kids. When we

have given kids the message that mistakes are good for

you, mistakes are when your brain's growing, that more

successful people make more mistakes, it changes

everything for them. I have had teachers sending me

videos of kids who've failed a whole life, to change their

pathways with this single message.

So but -- I've mentioned this yesterday for

those of who were in the opening session. This is so

interesting. What they found was in the study that the

brain response when you make a mistake was enhanced if you

had a growth mindset and you believed that mistakes were

good. So what this tells us I think at a deep level is

the relationship between actually how our brain responds

to things, and the amount of growth that the brain can

bring about and how that is related to what we believe

about ourselves.

So those people who are believing I can do this,

mistakes are important, I make a mistake I'm just going to

go further have more of a response and more growth from

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their brain than people who think mistakes are bad for

them. So I think that's really fascinating.

So we must encourage teachers to give kids this

message. But it also means that we need kids in math

classes -- classrooms, working on more open and

challenging work and make mistakes. Unfortunately most

math classrooms are set up in the U.S. so the kids get

most of the work right. Teachers really care about their

students, they want them to go through getting everything

right. That is not good for them. They need to be

struggling.

In fact Carol in her talks to parents says that

if your child comes home and says I got all my work right

today, you should say oh, I'm sorry.

(Laughter)

MS. BOALER: Then you didn't get an opportunity

to learn something. So we are getting these messages out

and I want to show you a couple of places how, and the

first one I'll show you is an online class that we have

for any student of math of any age, you know, elementary

up to adults.

And I designed and made this on my class, it's

six short sessions, to get these messages out to as many

people as we can. So there's six very short interactive

lessons that show math as this beautiful subject. Some

data from this course, we have over a 100,000 people have

currently taken it. I will show you a place where all

these courses are, in a minute, so don't worry.

And after taking the course we see significant

differences in these areas. Students look forward to math

class more. They believe that math is a subject of ideas

and not procedures. They stop fearing math, they go in to

-- amazing how many people are afraid of math. And they

develop a growth mindset. And I'm going to show you a

little clip from the course so you can see.

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I was very fortunate in having my undergraduate

class at Stanford. We have to present and do some of the

course with me.

(Video being played)

MS. BOALER: People who would benefit from that

course I recommend it. So the next thing I wanted to tell

you about neuroscience that again is so interesting is

this that the neuroscientists are telling us that math

should never be associated with speed. And in fact we

know the timed tests, those awful 50 questions to do in 2

minute things, giving out mad minutes, cause the early

onset of math anxiety for a large number of students

especially girls, and this is how it works.

We know that math facts are held in the working-

memory section of the brain, and when you're put under any

sort of stress, the working memory is blocked, and you

can't remember things at that time. And I don't know if

you've experienced this, I know I've experienced this, if

ever you felt under pressure, maybe under pressure doing

math, maybe you're in a restaurant and friends say you

work out the tip you're a teacher, and everyone's looking

at you.

(Laughter)

MS. BOALER: And you just have this experience

of oh my gosh, my mind's gone blank, I can't think. That

is the impact of stress hitting the working memory, and

actually stops you from being able to function. So what

do we do to kids in schools? This comes to my local

school district, 50 questions within three minutes, from

first grade upwards.

The irony of this is mathematicians are not

necessarily fast with math, and in fact some of our

greatest mathematicians are speaking out about how they're

not fast with numbers, they actually work slowly and

deeply with math.

And one that I like to share wrote an

autobiography about how he actually -- he was one of the

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slowest math thinkers in his school, and it made him feel

stupid when he was in school. He won the Fields --

another -- he won the Fields Medal (phonetic) I think --

he did? In math. So this is an excerpt from his

autobiography, from Laurent Schwartz. "I was always

deeply uncertain about my own intellectual capacity. I

thought I was unintelligent and it's true that I was and

still am rather slow. I need time to seize things because

I always need to understand them fully. Towards the end

of the eleventh grade I secretly thought to myself as

stupid, and I worried about this a long time. I'm still

just as slow. At the end of eleventh grade I took the

measure of the situation and came to the conclusion that

rapidity does not have a precise relation to intelligence.

What is important is to deeply understand things and their

relations to each other. This is where intelligence lies.

The fact of being quick or slow isn't really relevant."

And that's a great sentence -- "What is important is to

deeply understand things and their relations to each

other."

"We urgently need to dissociate math from speed

in classrooms and stop dissuading those children who think

slowly and deeply that they can't do math." And as I said

many of those children are girls. So we've been on a bit

of a campaign about this, Stanford released one of my

papers about timed tests which the U.S. News & World

picked up if you want to see more of that.

So another very interesting brain finding, I'm

giving you a lot here, is this one that we're now finding

that crossing the brain, when different pathways are lit

up in the brain, we learn most powerfully. So this also

has big implications. And delving more into the timed-

test area, this was an MRI study that was done looking at

different strategies the kids used to memorize and know

math facts.

Math facts are important, what we don't need is

kids being put under pressure to blindly memorize them.

They found there were two different ways of knowing a math

fact. So you could memorize them or you can use

strategies to work them out. And strategies, for example

you might take seven-times-eight, uh, yeah, I don't want

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to do seven times, I'm going to do seven-times-ten, and

take off to eight. Perfectly good strategies. They

taught kids the strategy way and the memorizing way and

they found that the strategy group had superior

performance in all ways. They were faster, more accurate,

they had better transfer. Their conclusion, that

automatist is a good thing, but we reach it to

understanding relationship between numbers.

And then on this brain pathway thing, these

researchers found that the more successful students on

math tests sort of math factors were those who have --

were using connections between the brain hemispheres and

that when they taught students activities that crossed the

brain their learning was powerful.

So what does this mean, to cross the brain?

Actually when we -- I collected this evidence on timed

tests -- you know as a researcher at Stanford we were

encouraged to publish papers in academic journals. And

they come out in press maybe two years later, a few

hundred people read them. I decided to write a paper and

put it up on our website, which I'll show you in a bit,

and it's a paper that combines the research evidence, but

also gives teachers things to do differently, and it's

called Fluency without Fear.

In fact this was the paper that got so much

attention the U.S. News & World and others picked it up.

But in that paper we explain good ways to teach math facts

that don't involve fear and involve this kind of brain

crossing. So the perfect brain crossing is when kids

interact with visual symbols, symbols math is always sort

of like numbers, but they also think visually.

So when they think visually about for example,

what four-times-eight looks like, a different pathway

lights up from when they look at the numbers, four-times-

eight, whatever the number is. So we gave teachers in

this paper a nice activity, it's called, "How Close to a

Hundred for Teaching Math Facts in Elementary School."

You have a 10-by-10 grid and two kids sort of play games

with each other, you role two dice and then you represent

that array as a rectangle with like 3-times-1.

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The goal is to fill the sheet, which this person

working that sheet didn't do -- it wasn't working too well

on that. But you want to strategically place your

rectangles to fill the shape. And so when kids do this

and then they write the math fact underneath as well as

the visual array, kids -- everybody loved this. It

started getting tweeted around the world. We had teachers

in all over France, everywhere using it.

So this is a perfect activity of a brain

crossing activity, where kids are working visually at the

same time as working with numbers.

So we have a lot of mindset evidence now, and

there are mindset interventions we can give kids to change

them from a fixed to a growth mindset. And my online

class does that.

But mindset interventions really won't mean

anything if we keep teaching math as a fixed subject.

Short, closed questions convey to students fixed views of

math. These are the questions you can do the more you

can't. So really if we want to give kids a growth mindset

we also have to teach them about the growth of math. And

I'll show you what I mean.

Unfortunately most math classrooms right now

offer math as a performance subject than a learning

subject, and what I mean by that is if you ask most kids

what their role is in math class, they'll tell you it's to

get questions right. They don't say that about other

subjects. They won't tell you about the beauty of the

subject or anything about it. They'll say that they had

to get questions right.

And this really came home to me recently, when a

colleague, Rachel Lambert said her six-year-old son had

come home saying he didn't like math, and she said why?

And he said, "Math is too much answer time and not enough

learning time." And you know we might think what about

six-year-olds on the board. But actually kids know it.

They know from kindergarten that math is treated

differently. So we have to give kids in math class tasks

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that have space inside them to learn, so that they see

math as a growth learning subject.

And I want to show you an example now. In fact

we're going to do some math together. The example I'm

going to show you came from a summer school class I

taught. And we're actually teaching it again right now.

I left the class on Friday, they just made boats, as you

can see some of them, and Kathy and I were teaching it,

and we're whizzing back on a plane on Tuesday night to get

back to the kids who didn't want us to go. But you can

see we got them racing their boats on the lovely Stanford

pool. So the math example I'm going to give you, and math

camp isn't all racing boats, but is -- comes from the

algebra class, we were teaching kids then as we are now,

who are pretty underserved in communities where they don't

get good math teaching. And we were teaching an algebra

class but we wanted to teach algebra as a problem-solving

tool, not as a endpoint solve X. So we gave them a lot of

algebra growth problems and I am going to get you to look

at one now.

So I'm going to give you this math problem. And

usually with the math problem, this math problem it's sort

of growing cases, usually the question that goes with that

is how many in the nth case, or how many are in the 100th

case? I want you to interact with this question

differently and I want you to think about it entirely

visually with no numbers and no algebra, just visually.

So you're going to get -- do you want to --

we'll give these out, I'm going to give this out to you

and I want you first to think about this entirely on your

own. So no talking to the people around you yet, but the

question is, this is showing the first three cases of a

growing function. And in the first one you can see these

are multi-link cubes, which is why they're like a cube

with a nodule on top. In the first one you can see there

is a certain number and in the second there's more and in

this third there's even more. So where do you see the

extra ones? The question is where are they? Where are

those extra cubes?

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So I'm going to ask you to think about that on

your own just for a couple of minutes. Okay, there should

be silence in the room right now. Thinking alone about

this math problem. Where do you see the extra cubes each

time? Where are they? I am seeing there aren't enough

papers to hand out.

It's this. You can draw or write on there or

you don't have to, it's just so you have it in front of

you. So I'm going to use a strategy we use in class

rooms, which is when you think you've had enough time and

you see where there is some extra cube, show me with a

quiet thumb like this.

We do this in the classrooms so kids don't put

their hand up, put everyone else under pressure. Okay, so

the question if you'd like it again is how is this

growing, there are extra cubes and there are one, two,

three cases here, I can't move, do you have a piece of

paper -- could I borrow that, a sec? This first case here

has these cubes, the second one has these, the third has

these, where are the extra ones each time? Where do you

see the extra cubes?

You don't need to answer just yet. Does anybody

want to share where they see extra cubes? Yeah, there is

a microphone I think somewhere that we -- oh, here, do you

want to, okay, okay. Yeah, where do you see the extra

cubes? All right, we're going to listen over here.

SPEAKER: I see them coming underneath. So with

each new pattern there, the existing pattern in the simple

structure is actually existing on the top each time, so

it's been raised up by the same amount of cubes in the

width and then being flanked by two additional ones to

increases that row horizontally.

MS. BOALER: Thank you. I don't know if you

heard all of that, did anybody see it differently to that?

Give the mic to somebody else.

19

SPEAKER: I saw it that way the second time, the

first time I also saw if you added one on the side and

then you just kept adding one at the top of every new

column, that also produced the next step.

MS. BOALER: Okay, I'm not going to take anymore

(inaudible), we're on a bit of a deadline. But I've given

this problem to a lot of different people, and I want to

show you something, which is the different ways people see

the growth of this function. So my undergrads and other

people teachers have started giving them names, some

people see the extra cubes sort of coming down on the top

and my undergrads call this the raindrop method, who saw

it in that way, the raindrop way?

Some people see it in this way, they see the

extra cubes adding on the bottom. And my undergrads call

this the bowling alley method, like a roll of pins coming

in. Who saw it in that way, the bowling alley way?

Okay. Some people, this was a teacher, said,

it's like a volcano, the top piece comes out and the lava

follows out on the side. Who saw it in that way, the

volcano way?

All right, we also have the parting of the Red

Sea. So some people see it's kind of reproducing on both

sides. Who saw it in that way, the Red Sea way? Lots of

people. Some people, this isn't (inaudible), see it as

increasing as triangles that go up, did anybody see it

triangularly? Some people see it as slices that go

across, look across the shape, did anybody see it in that

way?

And this was a teacher actually in New Mexico

who said it's like Wayne's world, stairway to heaven

access denied. I think you have to watch Wayne's World to

get that joke. And then there's this method, if you look

at the red square on the far left and rearrange the shape

each time, you can see that it actually grows as a square,

did anybody see it in that way? No squares today, okay.

20

Okay. So this is where I am going with this,

usually when this math task is given out, it's given out

with the question how many squares or how many cubes are

in the 100 case or how many in the nth case? And this is

what happens, kids count. So they say in the first there

is 4, the second is 9, and the third is 16, then they look

at it for a long time and say it's always one more than

the case number squared.

So they don't think visually, they don't have

the opportunity to think about the functional growth, and

I gave this out to a room of teachers recently, and I

actually gave them the same instruction, think only

visually about it but the high school teachers at the

table did not listen and they drew over table like that.

So I went over and I said, interesting, n+1 squared, why

do you think it's squared, why is the function growing as

a square, and they went, no idea. And this is why you see

the square it actually grows as a square.

And what happens when we give kids the visual

question, how do you see this growing, if we have time

I'll show you video, they engage in this very rich

conversation sharing each other's different methods,

they're excited to see the different ways of growing, they

also understand, they know what N is, they can tell me

what N is they can tell me why it's squared, whereas kids

who just move to symbolic notation often don't even know

what N is, they come up with something, so that is a tiny

shift in the task from find how many in the nth case to

first think visually about the growth. And we are

teaching teachers. These math tasks can all be opened up,

any math task you can have can be changed from what I

would call a fixed math task to a growth math task.

We can ask students, how do you see it? We can

ask them to represent ideas in different ways, we can have

them discuss methods, think about why they work, they can

think always about why answers make sense. I mentioned

yesterday this example, we can in a fixed math version we

can ask what's one divided by two thirds, kids will cross

multiply, get confused, I have a wonderful video of a

21

teacher who says you may know a rule for working this out

but the rule doesn't matter today, I want you to make

sense for your answer and show me visually the answer to

one divided by two thirds, and what you see is a rich

great conversation where kids are representing it

visually, we can do it with math facts, we can have kids

memorize, or we can ask, we can open up the tasks. So

this is part of the work we're doing at YouCubed which I

want to show you in a sec.

The last thing, the last brain idea I want to

share with you is this one, the butterfly effect, you've

probably heard of this butterfly effect, that was written

by the chaos theory and other things, the flapping of the

wings of a butterfly in one state can cause a hurricane in

another, turns out the things we believe about ourselves

are that powerful too.

And I want to share a couple of things about the

messages that kids get. First, we know that when mothers

tell their daughters I wasn't good at math in school,

their daughter's achievement immediately goes down the

same term. That is a crushing message for kids, that is

okay not to do well in math, like your mom, unfortunately

teachers also, some elementary teachers give messages that

are similar.

And then this study, I look at research studies

for living but this one really shocked me, I think it was

hard, I could hardly believe it, it was an experimental

study they had hundreds of students in two experimental

groups doing high school English, all of them were given

critical diagnostic feedback at the end of their English

essay. Half the students, an extra sentence was put on by

researchers, teachers didn't know who got it. And what

they found was that kids who got the extra sentence

achieved at significantly higher levels a year later,

particularly students of color just because they got one

extra sentence and nothing else changed. The sentence

they got at the end of their feedback was this, I am

giving you this feedback because I believe in you. And

22

the kids who got that did significantly better a year

later.

Teachers didn't know who got it. These tell us

the power of these ideas that kids have, that they can do

math, that their teachers believes in them, super

important. We know that kids are sitting in math

classrooms all the time thinking does my teacher believe

me? Why did she give that question to that kid and

different one to me? They're thinking like this from the

age of four, and those believes they have about whether

their teacher believes in them are probably more important

in some ways than the math teaching they get.

So what do all these findings from Brain Science

mean and how do we get them out to teachers and others? I

am going to show you a clip from the film now, we're very

fortunate in getting the messages out to be working with a

filmmaker, Vicky Abeles who is sitting over there, and

she's making a film because she also knows how important

it is, math, to change math. And Vicky was the producer,

director of Race to Nowhere, which is a film you've

probably heard of, which documented the huge pressure kids

are under in education, and I watched that film and I just

saw math all the way through it, math is the subject that

causes kids most to drop out of college.

And so Vicky fantastically is making a short

film about math change. And this is a two-minute clip I'm

going to show you and you're going to see in it the work I

was doing with Kathy to change math teaching in a school

district which had huge failure rates and we got teachers

to shift from kind of worksheet math to enquiry math and

you're going to hear from some teachers and kids in this

two-minute clip.

(Video being played)

MS. BOALER: -- more there. This is going to be

out soon and available on websites I think, so look out

for that.

23

Other resources, if you know math teachers or

you're a parent, I have an online class for parents and

teachers too. The people are very happy with that causes

the teachers to believe in their students more, and this

was the first thing I did, I did this online class two

summers ago. I didn't know if anybody would take it but I

thought we've got to get this brain science out to

teachers, 45,000 people took it. And that caused me to

get 45,000 e-mails from people just saying, please can we

have more, can you keep these ideas up.

So we started our site, YouCubed, at Stanford,

it is a website that gives teachers tasks, videos, shows

them how to open up math, both of the online classes can

be found there if you go there, we have had over a million

hits just in the last six months on this site, and there

is huge energy, parent -- teachers love the ideas. They -

- what we teach teachers is, you can do this. We can't

wait for publishers to start producing good materials.

You can take math and you can give kids the right

messages, you can open up the questions, you can -- and we

have teachers who are now doing this and who are telling

us and showing us amazing things with failure rate cut in

half. It's fantastic.

Other resources are two new books. The Penguin

book is out now and here in the book store. This is my

new book that's coming out in the fall. Carol Dweck has

written the forward called Mathematical Mindset which I am

excited about and it's just come on Amazon for pre-order.

And then something else we thought, how do we get the --

how to we get teachers to change? It's so hard to get

teachers to change, you probably know. And so we decided

that what we are going to do is, we going to gave teachers

everywhere and it is worldwide, our website is -- you

know, our (inaudible) is read in 95 countries.

We're giving the world of math teachers the

first week of the school year in the fall. It's called

the week of inspirational math, five lessons, super easy

to download, we were kind of inspired by the hour of code

that go so many people coding. Super easy, just get on

the site, you can download all the lessons. They have

24

little videos about mindset that show the kids wonderful

tasks. And what we are hoping is that hundreds of

thousands of teachers will use this in the fall and they

will just try and see what happens to their kids. We are

betting on the hope that teachers in the first week of

school would just try something.

It's already gone around to hundreds of

thousands of teachers. The U.K. is using it, China is

using it. So if you would like to share that with

teachers and I want to finish just before we have question

time by saying this that inquiry math, when we open up

math and kids role is to ask questions to investigate, you

know, they are still learning those standards methods but

they are learning them through using them is so powerful

and when we do that with kids, what amazes me is, I keep

hearing the same words from kids and I heard these words

again in my summer camp last week. They said the math is

open and you heard a girl in a video say it makes me feel

alive. You heard her use that word. And free, kids are

started to say this to me. On our website there is a clip

of kids who are taught number talks in third grade and I

asked him in the interviews, so how you -- what you think

about these number talks and the first in the boys says to

me, we are free. We can use numbers in any way we want.

And this -- you're going to hear a message of hope that I

am going to end on. What inquiry math does for kids, when

you combine it with mindset messages is give them an

intellectual empowerment that they take throughout their

lives. It's not just about doing when a math class, and

we want this for all students.

So please help us in our work. I am going to

end with a clip which also Vicky has made from the movie.

One of the boys in the school we already saw where we were

to the teachers to change and his words, I just think is

so important for all of us to listen to.

So I am going to end on this and then we've got

about ten minutes for questions, I think.

SPEAKER: Math class last year was notes and

just hand out and your own little box, you're just boxed

in. You were like by yourself, it was every man for

25

themselves. But now this year is just open, we are whole

big like, it's like a city, we are all working together to

create this new beautiful world. I think the challenges

and like the future that lies ahead of for me like all if

I keep on pushing, if I keep on doing this some day, I

will make it.

MS. BOALER: And that's why we've to do this for

kids. So --

(Applause)

MS. BOALER: So I love that. It shows us

exactly why we have to do this for kids and we got a bit

of time for questions. So I see two, I don't if someone

can keep track of this. How are you? Well, thought you

want to start here -- he just needs a mike and then I go

to you. And then to you.

SPEAKER: Thank you. I thought your talk was

fantastic and --

MS. BOALER: Thank you.

SPEAKER: -- I speak as a father of two young

kinds including a daughter and also board member of a

Charter school that teaches thousands of under-privilege

youth.

MS. BOALER: Uh-huh.

SPEAKER: And that really is kind of a little bit

of my question which is, I think we've done a great job

instilling a growth mindset but against a standardized

testing procedure.

MS. BOALER: Yeah, right.

SPEAKER: So, we talk people to believe and we

score very highly.

MS. BOALER: Uh-huh.

SPEAKER: But they don't really learn math, to

26

learn how to take a test.

MS. BOALER: Right.

SPEAKER: And so, really two questions. How

labor intensive is the new method? And can it really

succeed all the way through society without a kind of

modification of or kind of testing issues in the U.S.?

MS. BOALER: Yeah, great question. So this is

the problem. There are a lot of schools that are building

up growth mindsets in kids. But then they go into math

class and sit and do procedural questions and take tests,

which give the opposite message.

The -- I mean, the testing regime has a lot to

answer for in our nation of kids who don't believe they

can do math and I would love to get rid of it all

together. But the -- what we do have is a new system --

the tests that going with the coming core, the smarter

balanced and park do require. They are more open ended,

more kids are going to have interact with visuals, explain

their thinking. They won't be up to do that if they're

being brought up on this drill practice math and we'll see

probably a lot of failure for kids when they go into

those.

So, yeah, I agree, the testing regime has to

change and so does the math teaching in classroom. So

hopefully, we are in a good place for that and we are

going to move into a better place with our new assessment.

Yeah.

SPEAKER: (Off mic).

MS. BOALER: Great.

SPEAKER: (Off mic) having a barrier in math and

I don't want to wait for teachers. Is there any way that

--

MS. BOALER: Yeah.

SPEAKER: -- as students -- is there any

27

possible way we can convince ourselves to get over it?

MS. BOALER: Sure. Yeah, great question.

SPEAKER: Because this is a part -- always

needed but --

MS. BOALER: Yeah, great question.

SPEAKER: -- really takes --

MS. BOALER: So it turns out, people's mindset

can change and actually I teach a class of undergraduates

at Stanford, many of whom are math traumatized, even

though they've done well enough to get to Stanford. And

we learn about the brain science, the growth of the brain

and they do shift during few weeks. And what I would say

about that is it's so important even for students like

Stanford students. Many of them come to Stanford, having

never got anything less than an A in their lives and they

are crushed when they get a B. And many of them fall

apart, and these knowing that challenge and failure is

good. So to your question of how do you get help without

the teacher, the online classes super help will come to

YouCubed. We would like actually to build more of a

student revolution. We say on YouCubed that, well, this

is a math revolution we are trying to bring about and we

are have a lot of teachers behind us, we could do with

some students too. But read about, read Carol's book,

read my book, you can totally do on your own.

Yeah. And then I come to you.

SPEAKER: Hi.

MS. BOALER: Hi.

SPEAKER: My name is (inaudible) and I'm a

business scholar. And during your talk, you said that

when students make mistakes, it helps develop their brain

and I do agree with you. But then I wanted to ask that

can't that message be misinterpreted as that of the

butterfly on that. It's okay not to do well because if I

fail, after all my brain is going to develop. So I might

28

just want to put an efforts.

MS. BOALER: Uh-huh.

SPEAKER: So you know to get the answers right.

MS. BOALER: Yeah.

SPEAKER: So how would you send that message and

not to get misinterpret it?

MS. BOALER: Yeah, good question, good question.

So actually you hear me give the message in the online

class is what I found in the study was that the brain grew

more when people made a mistake and when they go work

correct. And I actually feel fine about giving that

message. The mistakes are powerful because students are

so batted down with the opposite message. I don't know

any students who is going to, oh, great, I am going to go,

I am failing now. Because they actually believe to their

core that they have to get work right and that when they

make a mistake, it means they're not a math person. So I

think we've to almost over give, you know not over but we

can safely give the message that mistake and failure are

really good and they are important, they are an important

part of learning and we are finding that it's really

inspirational for kids. And we are not finding that they

are just say, okay, I will just go to failing. Because

every child, no matter what they put out in terms of their

behavior and defense mechanisms, every child wants to

learn and it's a powerful message.

I don't where we are, I know you are next and

then I guess we should go back, I guess you, the guy in

the green. Yeah.

SPEAKER: Thank you so much. I am a mom of a

three-year-old and a four-year-old a boy and a girl and I

am very interested in pre-school education in my

community.

MS. BOALER: Uh-huh.

SPEAKER: What can you tell us about how we

29

reach those children at the very earliest stages and how

can we do our best for them?

MS. BOALER: Yeah, it's a great question. This

may scare you, this piece of evidence, but we know that

the mindset messages given to babies between the ages of 0

and 3, like whether we telling kids they are smart or not,

predicts their mindset five years later. So babies and

little ones upto the age of 3 are soaking up the messages

we give them about math and about themselves. So it's

really important to start those growth messages right from

the beginning. You know, everyone can learn and that

don't use the smart word because the smart word is the one

that sends kids down a negative path. And then math, you

know, math in the early should be all the through playful

and about shapes and ideas. And we know that countries

like Finland don't teach any form of mathematics until

kids are 7 and they taught a world of math achievement.

So our system of trying to teach 4-year-olds times math

test is no really part of the problem we have. Yeah, the

back.

SPEAKER: (Off mic)

MS. BOALER: Thank you.

SPEAKER: My question is about -- you are kind

of talking about teaching approach and certainly if you

look at piece of results, other countries blow us away in

terms of the actual results.

MS. BOALER: Yeah.

SPEAKER: Finland being one of them but also if

you look at countries like South Korea --

MS. BOALER: Uh-huh.

SPEAKER: -- very, very different sort of style

but they also have exceptional results. And I guess I am

just curious from your study of this.

MS. BOALER: Yeah.

30

SPEAKER: Have you looked at the kind of

pedagogy in other countries --

MS. BOALER: Uh-huh.

SPEAKER: -- and the relative kind of gender

performance in other countries and if you could just get

comment on --

MS. BOALER: Yeah. Sure.

SPEAKER: -- how that informs the way you think

about maths.

MS. BOALER: I mean it does get very complicated

when you look at performance in other countries just

because of there are big cultural differences. We know

that eastern countries performed very well. There are

also countries that have no fixed mindset messages. They

see learning as a process that takes time, that everyone

can learn. I have quotes from Japanese educators in my

book saying we want balance, we are not there to -- in

Japan, they don't think about pushing somebody ahead as we

do in America, there about everybody doing well together.

We visited China. I've been to China a number

of times, everybody thinks China, Singapore topped the

last pizza chart by a significant margin and China is a

very interested country where they teach math very

conceptually against what many people think that kids

giving drill and practice, they do get drill in the after

school arena. We went to a number of high school math

classes and I am -- we are about to publish one on our

site.

In our log lessons, the kids did no more in any of

the classes and we actually run away and went to the

classes we wanted to see, not the ones they took us to.

But in no class did kids work among three questions in an

hour. And in those classes, the majority of the time,

what you heard was students talking to each other and the

teacher in inquiry that going deeper, deeper, deeper.

They were doing the mathematical topics in high school.

They don't even though as you might think of the most

31

engaging but they were going into such depth around them

that if it we'd been in American high school, you would

have seen kids were come 40 questions on that topic.

So it's complicated with looking at other

countries but definitely we see China, Japan they have a

much better process of teacher learning and they really

believe that learning is a process and it's pretty

consistent that we see that the higher performing

countries and what we can see in the pizza data is who are

the higher performing kids in each country and that's

where we see that is always the more conceptual thinkers.

In this, in America, we know that memorizes are

the lowers achievers in the world and we know that America

has more memorizes than most countries in the world and

that's part of the problem.

I don't know where to go to another, there are

so many hands. How are you? It's probably the last

question I think because it's 11:19 I see.

SPEAKER: Especially given what you said about

time and speed.

MS. BOALER: Uh-huh.

SPEAKER: How does this track with blended

learning and things like the Khan Academy and that sort of

thing.

MS. BOALER: Good question again. Blended

learning can be good I think if what we're doing is having

kinds engage actively in class and we are having them

review things. The Khan Academy materials, I am not a big

fan of. It's a great system that teachers like because it

gives us nice feedback but it's basically very traditional

math instruction -- listen to somebody talk to you, do

lots of questions.

So I am a friend of Sal Khan and we've talked

about this lots of times and there is a place, I mean I

think it's great, the technology can help get math out to

many more countries and people. But if you have that

32

quality to develop things, that technical quality, I would

love a system that engage the kids better with math where

it's more active and different ways of approaching math.

So one of the things we know that not many people do know

is there were two ways of teaching math. You can teach

kids methods and then they could apply them if they lucky,

in some context or you can give kids an applied math

context where they don't know the methods but they need

them and then later you teach the methods. Those kids do

very better in every test. And we never do that in the

U.S., it's a flip of what happens in the U.S. and the

reason for that is, if you give kids complex problems and

then later you teach them the method they need, the point

you teach them the method that brains are trying to learn

it. They need it, they are curious. So this upfront give

you methods, sit and watch is -- it's not what we know,

it's not all we know works.

SPEAKER: But with blended work.

MS. BOALER: So blended learning, I would say

can work, if you give kids the active experience in the

classroom and then maybe they can review and see methods

out of the classroom you want to. My personal preference

and what we're doing in summer school is to engage kids

enrich tasks where they need to learn the four mathematics

to solve the task inside it.

So we spend last week introducing algebra to our

kids when they were working on growth visual and they

ended up needing to use N and they were saying, wow, this

is algebra, you know, this is, I can do this, I know what

it is. We don't do that in math classrooms, we have them

go through lots of questions. So yeah, definitely, this

blended learning has some interesting opportunities.

And I know we have to stop because of the

timing, but thanks.

(Applause)

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