Jo Boaler: How to Learn Math
嘉宾：Jo Boaler
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本篇目录11 章节
§01Transcript · 00:00–08:24§02Transcript · 08:27–17:25§03Transcript · 17:29–25:55§04Transcript · 25:59–34:49§05Transcript · 34:51–44:04§06Transcript · 44:06–52:49§07Transcript · 52:50–1:02:05§08Transcript · 1:02:07–1:11:10§09Transcript · 1:11:14–1:20:44§10Transcript · 1:20:47–1:29:57§11Transcript · 1:30:01–1:30:36§01Transcript · 00:00–08:24
The following is a conversation with Jo Bowler,
a mathematics educator at Stanford
and co-founder of ucubed.org
that seeks to inspire young minds
with the beauty of mathematics.
To support this podcast,
please check out our sponsors in the description.
This is the Lex Friedman Podcast,
and here is my conversation with Jo Bowler.
What to you is beautiful about mathematics?
- I love a mathematics that some people
don't even think of as mathematics,
which is beautiful, creative mathematics,
where we look at maths in different ways,
we visualize it,
we think about different solutions to problems.
A lot of people think of maths as
you have one method and one answer.
And what I love about maths
is the multiple different ways you can see things,
different methods, different ways of seeing, different...
In some cases, different solutions.
So that is what is beautiful to me about mathematics,
that you can see and solve it in many different ways.
And also the sad part that many people think
that maths is just one answer and one method.
- So to you, the beauty emerges
when you have a problem with a solution
and you start adding other solutions,
simpler solutions, weirder solutions, more interesting,
some that are visual, some of their algebraic,
geometry, all that kind of stuff.
- Yeah, I mean, I always say that you can take any maths area
and make it visual.
And we say to teachers,
give us your most dry, boring maths
and we'll make it a visual, interesting, creative problem.
And it turns out you can do that with any area of maths.
And I think we've given...
It's been a great disservice to kids and others
that it's always been numbers,
lots and lots of numbers.
Numbers can be great,
but you can think about maths in other ways besides numbers.
- Do you find that most people are better visual learners
or is this just something that's complimentary?
What's the kind of the full spectrum of students
and the way they like to explore math, would you say?
- There's definitely people who come into the classes I do
who are more interested in visual thinking
and like visual approaches.
But it turns out what the neuroscience is telling us
is that when we think about maths,
there are two visual pathways in the brain
and we should all be thinking about it visually.
Some approaches have been to say,
well, you're a visual learner, so we'll give you visuals
and you're not a visual learner.
But actually, if you think you're not a visual learner,
it's probably more important that you have a visual approach
so you can develop that part of your brain.
- So you were saying that there's some kind
of interconnected aspect to it,
so the visual connects with the non-visual.
- Yeah, so this is what the neuroscience has shown us,
that when you work on a maths problem,
there are five different brain pathways
and that the most high-achieving people in the world
are people who have more connections
between these pathways.
So if you see a maths problem with numbers,
but you also see it visually,
that will cause a connection to happen in your brain
between these pathways.
And if you maybe write about it with words,
that would cause another connection,
or maybe you build it with something physical,
that would cause a different connection.
And what we want for kids is,
we call it a multidimensional experience of maths,
seeing it in different ways,
experiencing it in different ways,
that will cause that great connected brain.
- You know, there's these stories
of physicists doing the same.
I find physicists are often better
at building that part of their brain
of using visualization for intuition building,
'cause you ultimately want to understand
the deepest secret underneath this problem.
And for that, you have to intuit your way there.
And you mentioned offline that one of the ways
you might approach a problem
is to try to tell a story about it.
And some of it is like legend,
but I'm sure it's not always.
Is you have Einstein thinking about a train
and the speed of light,
and that kind of intuition is useful.
You start to imagine a physical world,
like how does this idea manifest itself
in the physical world,
and then start playing in your mind
with that physical world,
and think, is this going to be true?
Is this going to be true?
- Right, right.
Einstein is well known for thinking visually.
And people talk about how he really
didn't want to go anywhere with problems
without thinking about them visually.
But the other thing you mentioned
that sparked something for me
is thinking with intuition,
like having intuition about math problems.
That's another thing that's often absent in math class,
the idea that you might think about a problem
and use your intuition,
but so important.
And when mathematicians are interviewed,
they will very frequently talk about
the role of intuition in solving problems,
but not commonly acknowledged
or brought into education.
- Yeah, I mean, that's what it is.
If you task yourself with building
an intuition about a problem,
that's where you start to pull in,
like what is the pattern I'm seeing?
In order to understand the pattern,
you might want to then start utilizing visualization.
But ultimately, that's all in service
of like solving the puzzle,
like cracking it open
to get the simple explanation
of why things are the way they are,
as opposed to, like you said,
having a particular algorithm
that you can then execute to solve the problem.
Yeah, but it's hard.
It's hard.
- Yeah. - Like reasoning
is really hard.
- Yeah, it's hard.
I mean, I love to value what's hard in maths
instead of being afraid of it.
We know that when you struggle,
that's actually a really good time for your brain.
You want to be struggling
when you're thinking about things.
So if it's hard to think intuitively about something,
that's probably a really good time for your brain.
I used to work with somebody called Sebastian Thrun,
who is a great sort of mathematician,
you might think of him, an AI person.
And I remember in one interview I did with him,
he talked about how they'd built robots,
I think for the Smithsonian,
and how they were having this trouble
with them picking up white noise.
And he said they had to solve it.
They had to work out what was going on,
and how he intuitively worked out what the problem was.
But then it took him three weeks to show it mathematically.
I thought that was really interesting,
that how you can have this intuition
and know something works.
It's kind of different from going through
that long mathematical process of proving it.
But so important.
- Yeah, I think probably our brains are evolved
as like intuition machines.
And the math of like showing it like formally
is probably an extra thing that we're not designed for.
You see that with Feynman and his,
all of these physicists definitely you see
starting with intuition,
sometimes starting with an experiment,
and then the experiment inspires intuition.
But you can think of an experiment
as a kind of visualization.
Just like let's take whatever the heck we're looking at
and draw it, and draw like the pattern as it evolves
as the thing grows for N equals one,
for N equals two, N equals three,
and you start to play with it.
And then in the modern day, which I loved doing,
is you can write a program that then visualizes it for you.
And then you can start exploring it programmatically.
And then you can do so interactively too.
I tend to not like interactive
because it takes way too much work.
'Cause you have to click and move and stuff.
I love to interact through writing programs.
But that's my particular brain, software engineer.
So you can do all these kinds of visualizations.
And then there's the tools of visualization like color,
all of those kinds of things.
That you're absolutely right,
they're actually not taught very much.
§02Transcript · 08:27–17:25
Like the art of visualization.
- Not taught.
And we love as well color coding.
Like when you represent something mathematically,
you can show color to show the growth.
And kind of code that.
So if I have an algebraic expression for a pattern,
maybe I show the X with a certain color,
but also write in that color
so you can see the relationship.
Very cool.
And yeah, particularly in our work with elementary teachers,
many of them come to our workshops
and they're literally in tears
when they see things making sense visually.
Because they've spent their whole lives
not realizing you can really understand things
with these visuals.
It's quite powerful.
- You say that there's something valuable to learning
when the thing that you're doing is challenging,
is difficult.
So a lot of people say math is hard,
or math is too hard, or too hard for me.
Do you think math should be easy or should it be hard?
- I think it's great when things are challenging,
but there's something that's really key
to being able to deal with challenging maths,
and that is knowing that you can do it.
And I think the problem in education
is a lot of people have got this idea
that you're either born with a maths brain or you're not.
So when they start to struggle,
they think, oh, I don't have that maths brain.
And then they will literally sort of switch off
in their brain and things will go downhill
from that point.
So struggle becomes a lot easier,
and you're able to struggle if you don't have that idea.
But you know that you can do it.
You have to go through this struggle to get there,
but you're able to do that.
And so we're hampered in being able to struggle
with these ideas we've been given about what we can do.
- Can I ask a difficult question here?
- Yeah.
- So there's kind of, I don't know what the right term is,
but some people struggle with learning in different ways.
Like their brain is constructed in different ways.
And how much should, as educators,
should we make room for that?
So how do you know the difference between this is hard,
and I don't like doing hard things,
versus my brain is wired in a way
where I need to learn in very different ways.
I can't learn it this way.
How do you find that line?
How do you operate in that gray area?
- So this is why being a teacher is so hard.
And people really don't appreciate
how difficult teaching is when you're faced with,
I don't know, 30 students who think in different ways.
But this is also why I believe it's so important
to have this multidimensional approach to maths.
We've really offered it in one way,
which is here's some numbers and a method.
You follow me, do what I just did, and then reproduce it.
And so there are some kids who like doing that,
and they do well.
And a lot of kids who don't like doing it,
and don't do well.
But when you open up maths,
and you let kids experience it in different ways,
maybe visually, with numbers, with words,
what happens is there are many more kids who can access it.
So those different brain wirings you're talking about,
where some people are just more able to do something
in a particular way, that's why we want to,
that's one of the reasons we want to open it up,
so that there are different ways of accessing it.
And then that's not really a problem.
- So I grew up in the Soviet Union,
and fell in love with math early.
I was forced into math early,
and fell in love through force.
- That's good.
Well, good that you fell in love.
(laughing)
But something we talked about a little bit,
is there's such a value for excellence.
It's competitive, and it's also everybody kind of looks up,
the definition of success is being,
in a particular class, is being really good at it.
And it's not improving, it's being really good.
I mean, we are much more like that with sports,
for example, we're not,
it's like it's understood,
you're going to star on the basketball team,
if you're gonna start on the basketball team,
if you're going to be better than the other guys,
the other girls on the team.
So that coupled with the belief,
this could be partially a communist belief, I don't know,
but the belief that everybody is capable of being great.
But if you're not great, that's your fault,
and you need to work harder.
And I remember I had a sense that, probably delusional,
but I could win a Nobel Prize,
I don't even know what that entails.
But I thought, like my dad early on told me,
just offhand, and it always stuck with me,
that if you can figure out how to build a time machine,
how to travel back in time,
it will probably give you a Nobel Prize.
And I remember early in my life thinking,
I'm going to invent the time machine.
And the tools of mathematics were in service of that dream
of winning the Nobel Prize.
And it's silly, I didn't really think
in those concrete terms,
but I just thought I could be great, that feeling.
And then when you struggle,
the belief that you could be great,
struggle is good.
- Right, pushes you on, yeah.
- And so the other thing about the Soviet system
that I'd love to hear your comments about,
is just the sheer hours of math.
Like the number of courses,
you're talking about a lot of geometry, a lot more geometry.
I think in the American system,
you take maybe one year of geometry.
- In high school, yeah.
- In high school, first of all,
geometry is beautiful, it's visual.
And then you get to reason through proofs
and stuff like that.
In Russia, I remember just being nailed over and over,
it was just nonstop.
And then of course,
there's different perspectives on calculus
and just the whole, the sense was that math is like,
fundamental to the development of the human mind.
So math, but also science and literature, by the way,
was also hit very hard.
Like we read a lot of serious adult stuff.
America does that a little bit too.
They challenge young adults with good literature,
but they don't challenge adults very much with math.
So those two things, valuing excellence
and just a lot of math in the curriculum.
Do you think, do you find that interesting?
'Cause it seems to have been successful.
- Yeah, I think that's very interesting.
And there is a lot of success,
people coming through the Soviet system.
I think something that's very different to the US
and other countries in the world
is this idea that excellence is important
and you can get there if you work hard.
In the US, there's an idea that excellence is important,
but then kids are given the idea in many ways
that you can either do it
or you're one of the people who can't.
So many students in the school system
think they're one of the kids who can't.
So there's no point in trying hard
because you're never going to get there.
So if you can switch that idea, it would be huge.
And it seems from what you've said,
that in the US, in the Soviet Union,
that idea is really different.
Now, the downside of that idea,
that anybody can get there if you work hard,
is that thought that if you're not getting there,
it's your fault.
And I would add something into that.
I would say that anybody can get there,
but they need to work hard
and they also need good teaching
because there are some people who really can't get there
because they're not given access to that good teaching.
But that would be huge, that change.
As to doing lots of maths,
if maths was interesting and open and creative
and multidimensional, I would be all for it.
We actually run summer camps at Stanford
where we invite kids in
and we give them this maths that I love.
And in our camp classrooms, they were three hours long.
And when we were planning,
the teachers were like, "Three hours?
Are we going to be able to keep the kids excited
for three hours?"
Turned out, they didn't want to go to break or lunch.
They'd be so into these mathematical patterns.
We couldn't stop them.
It was amazing.
So yeah, if maths was more like that,
then I think having more of it would be a really good thing.
§03Transcript · 17:29–25:55
- So what age are you talking about?
Could you comment on what age is the most important
when people quit math or give up on themselves
or on math in general?
And perhaps that age or something earlier
is really an important moment for them to discover,
to be inspired to discover the magic of math.
- I think a lot of kids start to give up on themselves
and maths around, from about fifth grade.
And then those middle school years are really important.
And fifth grade can be pivotal for kids
just because they're allowed to explore
and think in good ways in the early grades
of elementary school.
But fifth grade teachers are often like,
"Okay, we're going to prepare you now for middle school.
And we're going to give you grades and lots of tests."
And that's when kids start to feel really badly
about themselves.
And so middle school years,
our camps are middle school students.
We think of those years as really pivotal.
Many kids in those years are deciding,
"Yes, I'm going to keep going with STEM subjects."
Or, "No, I'm not, this isn't for me."
So, I mean, all years are important.
And in all years, you can kind of switch kids
and get them on a different pathway.
But I think those middle school years are really important.
- So what's the role of the teacher in this?
So one is the explanation of the subject,
but do you think teachers should almost do like one-on-one,
you know, little Johnny, I believe in you kind of thing?
Like that energy of like--
- Turns out it's really important.
There's a study that was done,
it was actually done in high school English classrooms,
where all kids wrote an essay for their teacher.
And this was done as an experiment.
Half of the kids got feedback from their teacher,
diagnostic feedback, which is great.
But for half of the kids,
it said an extra sentence at the bottom
that the researchers had put on.
And the kids who read that extra sentence
did significantly better in English a whole year later.
The only change was this one sentence.
- What did the sentence say?
- So what did the sentence say?
The sentence said, I'm giving you this feedback
because I believe in you.
And the kids who read that did better a year later.
- Yeah.
- So when I share this with teachers, I say, you know,
I'm not suggesting you put on the bottom of all kids' work,
I'm giving this feedback because I believe in you.
One of the teachers said to me, we don't put it on a stamp?
I said, no, don't put it on a stamp.
But your words are really important.
And kids are sitting in classrooms all the time thinking,
what does my teacher think of me?
Does my teacher think I can do this?
So it turns out it is really important to be saying to kids,
I know you can do this.
And those messages are not given enough by teachers.
- And really believe it.
- And believe it, yeah.
- It's like--
- You can't just say it, you have to believe it.
- I sometimes, 'cause it's like,
it's such a funny dance,
'cause I'm such a perfectionist,
I'm extremely self-critical,
and I have, when I have students come up to me,
and it's clear to me that they're not even close to good.
And it's tempting for me to be like,
to sort of give up on them mentally.
But the reality is, like,
if you look at many great people throughout history,
they sucked at some point.
- Yeah, exactly.
- And some of the greatest took non-linear paths
to where they sucked for long into later life.
And so always kind of believing
that this person can be great.
- Exactly.
- You have to communicate that,
plus the fact that they have to work hard.
- That's it, yeah.
Yeah, and you're right.
Silicon Valley, where I live,
is filled with people who were dropouts at school,
or who had special needs, who didn't succeed.
It's very interesting that have gone on
to do amazing work in creative ways.
I mean, I do think our school system is set up
to value good memorizers,
who can reproduce what a teacher is showing them,
and push away those creative, deep thinkers,
often slower thinkers, they think slowly and deeply.
And they often get the idea early on
that they can't be good at maths or other subjects.
So yeah, I think many of those people
are the ones who go on and do amazing things.
- So there's a guy named Eric Weinstein.
I know many mathematicians like this,
but he talks a lot about having
a non-standard way of learning.
I mean, a lot of great mathematicians,
a lot of great physicists are like that.
And he felt like he became quickly,
he got his PhD at Harvard,
became quickly an outcast of the system.
Like the education, especially early education system
didn't help him.
Is there ways for an education system
to support people like that?
Is it this kind of multi-dimensional learning
that you're mentioning? - Yeah, absolutely.
I mean, I think our education system
still uses an approach that was in classrooms
hundreds of years ago.
The textbooks have a lot to answer for,
and producing is very uninspiring mathematics.
But yeah, if you open up the subject
and have people see and solve it in different ways
and value those different ways.
Somebody I appreciated a lot
is a mathematician called Mary Mizzikani.
I don't know if you've heard of her.
She won the Fields Medal.
She was from Iran.
First woman in the world
to win the Fields Medal in mathematics.
She died when she was 40.
She was at Stanford.
But her work was entirely visual.
And she talked about how her daughter
thought she was an artist
because she was always visualizing.
And I attended, she asked me to chair the PhD defense
for one of her students.
And I went to the defense in the math department.
And it was so interesting
because this young woman spent like two hours
sharing her work.
All of it was visual.
In fact, I don't think I saw any numbers at all.
- That's awesome.
- And I remember that day thinking,
wow, I could have brought her like 13 year old
into this PhD defense.
They would not recognize this as maths.
But when Mary Mizzikani won the Fields Medal,
all these other mathematicians were saying
that her work had connected
all these previously unconnected areas of maths.
And so, but when she was,
she also shared that when she was in school,
when she was about 13,
she was told that she couldn't do maths.
She was told that by her teacher.
- This is Iran?
- Mm-hmm, in Iran.
Yeah.
So I love that, you know,
to be told you can't be good at maths
and then go on and win the Fields Medal is cool.
- I've been told by a lot of people in my life
that I can't do something.
I'm very definitely non-standard.
But all it takes, that's why people talk about
like the one teacher that changed everything.
All it takes is one teacher.
- That's right.
- That's the power of that.
So that should be inspiring to teachers.
- I think it is.
- You as a single person,
given the education system, given the incentives,
you have the power to truly change lives.
And like 20 years from now,
a Fields Medalist will walk up to you and say thank you.
- You did that for me.
Yeah, absolutely.
And I share that with teachers
that even in this broken system
of what they have to do for districts and textbooks,
a single teacher can change kids' maths relationship
or other subjects and forever.
- What's the role of the parents in this picture?
Let's go to another difficult subject.
- Yeah, that is a difficult subject.
One study found that
the amount of maths anxiety parents had
predicted their child's achievement in school,
but only if they helped with homework.
(both laughing)
- That's so funny.
- Yeah, there are some interesting implications for this.
§04Transcript · 25:59–34:49
I mean, you can see how it works.
If you have maths anxiety
and you're helping your kids with homework,
you're probably communicating things like,
oh, I was terrible at this at school,
and that's how it gets passed on to kids.
So one implication is
if you have a really bad relationship with maths,
you hate maths, you have maths anxiety,
just don't do maths homework with your kids.
But on our website, we have a little sheet for parents
of ways to interact around maths with your kids.
- That's ucubed.org?
- That's ucubed.org, yes.
So one of the things I say to parents
when I give parent presentations is,
even if you hate maths,
you need to just fake it with your kids.
You should be always endlessly optimistic
and happy about doing maths.
- I'm always curious about this.
So I hope to have kids one day,
I don't have kids currently.
Are parents okay with sucking at math
and then trying to get their kid
to be better than them, essentially?
Is that a difficult thing for a lot of parents?
- It is difficult.
To have like, it's almost like an ego thing.
Like I never got good at this
and I probably should have.
Yeah, I mean, to me, you wanna celebrate that,
but I know a lot of people struggle with that.
Like coaches in sports,
to make an athlete become better than them,
it can be hard on the ego.
- Yeah.
- Do you experience the same with parents?
- Parents too.
I haven't experienced parents worrying
that their kids will be better than them.
I have experienced parents
just having a really bad relationship with maths
and not wanting to help,
not knowing how to help, saying things.
Like another study showed
that when mothers say to their daughters,
"I was bad at maths in school,"
their daughter's achievement goes down.
So we know that kids pick up on these messages,
which is why I say you should fake it.
But also I know that lots of people
have just had a really bad relationship with maths,
even successful people.
The undergrads I teach at Stanford
have pretty much always done well in maths,
but they come to Stanford thinking maths
is a set of methods to memorize.
And so do many parents believe that.
There's one method that you memorize
and then you reproduce it.
So until people have really had an experience
of what I think of as the other maths,
where until they've really seen
that it's a really different subject,
it's hard for them to be able to shift their kids
to see it differently.
- Is there for a teacher,
if we're to like systematize it,
is there something teachers can do
to do this more effectively?
So you mentioned the textbook.
- Yeah.
- So what are the additional things
you can add on top of this whole old school,
traditional way of teaching that can improve the process?
- So I do think there's a way of teaching maths
that changes everything for kids and teachers.
So I'm one of five writers of a new framework
for the state of California, a new maths framework.
It's coming out next year.
And we are recommending through this maths framework
that people teach in this way.
It's called teaching to big ideas.
So at the moment, people have standards
that have been written,
and then textbooks have taken these standards
and made not very good questions.
And if you look at the standards,
like I have some written down here,
just reading the standards,
it makes maths seem really boring and uninspiring.
- What are the kind of, can you give a few examples?
- So this is an interesting example.
In third grade, there are three different standards
about unit squares.
- Okay.
- So this is one of them.
A square with side length one unit, called a unit square,
is said to have one square unit of area
and can be used to measure area.
- And that's something you're expected to learn.
- That is something, so that's a standard.
The textbook authors say,
oh, I'm gonna make a question about that.
And they translate the standards into narrow questions.
- And then you measure success
by your ability to deliver on these standards.
- So the standards themselves,
I think of maths, and many people think of maths in this way,
as a subject of a few big ideas
and really important connections between them.
So you could think of it as a network map
of ideas and connections.
And what standards do is they take that beautiful map
and they chop it up like this into lots of little pieces,
and they deliver the pieces to schools.
And so teachers don't see the connections between ideas,
nor do the kids.
So anyway, this is a bit of a long way of saying
that what we've done in this new initiative
is we have set out maths as a set of big ideas
and connections between them.
So this is grade three.
So instead of there being 60 standards,
we've said, well, you can pull these different standards
to get in with each other,
and also value the ways these are connected.
- And by the way, for people who are just listening,
we're looking at a small number of big concepts
within mathematics, square tiles,
measuring fraction, shape, and time,
and then how they're interconnected.
And so the goal is for,
this is for grade three, for example.
- Yeah.
And so we've set out for the state of California,
the whole of mathematics K10
as a set of big ideas and connections.
So we know that teachers, it works really well
if they say, okay, so a big idea in my grade is measuring.
And instead of reading five procedural statements
that involve measuring, they think,
okay, measuring is a big idea.
What rich, deep activity can I use
that teaches measuring to kids?
And as kids work on these deep, rich activities,
maybe over a few days,
turns out a lot of maths comes into it.
So we're recommending that let's not teach maths
according to all these multiple, multiple statements
and lots and lots of short questions.
Instead, let's teach maths by thinking about
what are the big ideas and what are really rich,
deep activities that teach those big ideas.
- So that's the, like how you teach it
and maximize learning.
What about like from a school district perspective,
like measuring how well you're doing,
grades and tests and stuff like that?
Do you throw those out or is it possible?
- I'm not a fan of grades and tests myself.
I think grades are fine
if they're used at the end of a course.
So at the end of my maths course,
I might get a grade because a grade
is meant to be a summative measure.
It kind of describes your summative achievement.
But the problem we have in maths classrooms across the US
is people use grades all the time,
every week or every day even.
My own kids, when they went through high school,
technology has not helped with this.
When they went through high school,
they knew they were being graded
for everything they did, everything.
And not only were they being graded for everything,
but they could see it in the grade book online
and it would alter every class they went into.
So this is the ultimate,
what I think of as a performance culture.
You're there to perform, somebody's measuring you,
you see your score.
So I think that's not conducive for deep learning.
And yes, have a grade at the end of the year,
but during the year,
you can assess kids in much better ways.
Like teachers can, a great way of assessing kids
is to give them a rubric that kind of outlines
what they're learning over the course of a unit
or a few weeks.
So kids can actually see the journey they're on,
like this is what we're doing mathematically.
Sometimes they self-assess on those units.
And then teachers will show what the kids can do
with a rubric and also write notes.
Like, in the next few weeks,
you might like to learn to do this.
So instead of kids just thinking about,
I'm an A kid or a B kid,
§05Transcript · 34:51–44:04
or I have this letter attached to me,
they're actually seeing mathematically what's important.
And they're involved in the process
of knowing where they are mathematically.
At the end of the year, sure, they can have a grade,
but during the year,
they get these much more informative measures.
- I do think this might be more for college,
but maybe not.
Some of the best classes I've had
is when I got a special set aside.
The professor clearly saw that I was interested
in some aspect of a thing.
And then I have a few in mind,
and one in particular,
he said that he challenged me.
So this is outside of grades and all that kind of stuff.
That basically it's like reverse psychology.
I don't think this can be done.
And so I gave everything to do that particular thing.
So this happened to be in an artificial intelligence class.
But I think that special treatment of taking students
who are especially excellent at a particular little aspect,
that you see their eyes light up.
I often think maybe it's tempting for a teacher
to think you've already succeeded there,
but they're actually signaling to you
that you could really launch them on their way.
And I don't know, that's too much to expect from teachers,
I think, to pay attention to all of that,
'cause it's really difficult.
But I just kind of remember who are the biggest,
the most important people in the history
of my life of education.
And it's those people who really didn't just inspire me
with their awesomeness, which they did,
but also just, they pushed me a little.
They gave me a little push.
And that requires focusing on the quote unquote excellent
students in the class.
- I think what's important though,
is teachers to have the perspective that they don't know
who's gonna be excellent at something
before they give out the activity.
- Exactly.
- And in our camp classes that we ran,
sometimes students would finish ahead of other students.
And we would say to them,
"Can you write a question that's like this, but different?"
Oh, and over time, we encouraged them
to like extend things further.
I remember we were doing one activity
where kids were working out the borders of a square
and how big this border would be in different case sizes.
And one of the boys came up at the end of the class
and said, "I've been thinking about how you do this
"with the Pentagon."
And I said, "That's fantastic.
"How do you, what does it look like with Pentagon?
"Go find out, see if you can discover."
So I didn't know he was gonna come up and say that.
And I didn't have in my head,
like this is the kid who could have this extension task.
But you can still do that as a teacher.
When kids get excited about something
or they're doing well in something,
have them extend it, go further.
It's great.
- And then you also, like this is like teacher and coach,
you could say it in different ways to different students.
Like for me, the right thing to say is almost to say,
I don't think you could do this, this is too hard.
Like that's what I need to hear.
'Cause it's like, no, there's an immediate push.
But with some people, if they're a little bit more,
I mean, it all has to do with upbringing,
just how your genetics is.
They might be much more, that might break them.
- Yeah, that might break them.
- And so you have to be also sensitive to that.
I mean, teaching is really difficult.
- It is really difficult.
- For this very reason.
- It is.
- So what is the best way to teach math,
to learn math at those early few days
when you just wanna capture them?
- I do something, actually there's a video of me
doing this on our website that I love
when I first meet students.
And this is what I do.
I show them a picture, this is the picture I show them.
And it's a picture of seven dots like this.
And I show it for just a few seconds.
And I say to them, I'd like you to tell me
how many dots there are, but I don't need to count them.
I want you to group the dots.
And I show it them and then I take it away
before they've even had enough time to count them.
And then I ask them, so how did you see it?
And I go around the room and amazingly enough,
there's probably 18 different ways
of seeing these seven dots.
And so I ask people, tell me how you grouped it.
And some people see it as like an outside hole
with a center dot.
Some people see like stripes of lines.
Some people see segments.
And I collect them all and I put them on the board.
And at the end I say, look at this,
we are a class of 30 kids and we saw these seven dots
in 18 different ways.
There's actually a mathematical term for this.
It's called groupitizing.
- Groupitizing?
- Yeah.
(laughing)
- I like it.
- It's kind of cool.
So turns out though that how well you groupitize
predicts how well you do in math.
- Is it a raw talent or is it just something
that you can develop?
- I don't think you're born groupitizing, I think.
But some kids have developed that ability if you like.
And you can learn it.
You can, so this to me is part of how wrong we have math.
That we think to tell whether a kid's good at math,
we're gonna give them a speed test on multiples.
But actually, seeing how kids group dots
could be a more important assessment
of how well they're gonna do in math.
Anyway, I diverge.
What I like to do when I start off with kids is show them,
I'm gonna give you math problems.
I'm gonna value the different ways you see them.
And turns out you can do this kind of problem
asking people how they group dots with young children
or with graduate students.
And it's engaging for all of them.
- You talk about creativity a little bit
and flexibility in your book "Limitless".
What's the role of that?
So it sounds like there's a bit of that kind of thing
involved in groupitizing.
- Yeah, yeah.
- I love this term.
So what would you say is the role of creativity
and flexibility in the learning of math?
- I think what we know now is that what we need
for this 21st century world we live in is a flexible mind.
School should not really be about teaching kids
particular methods, but teaching them to approach problems
with flexibility.
Being creative, thinking creatively is really important.
So people don't think the words math and creativity
come together, but that's what I love about math
is the creative different ways you can see it.
And so helping our kids, there's a book I like a lot
by, I think by physicists.
You probably know this book called "Elastic".
You might know it.
And it's about how we want elastic minds.
Same kind of thing, flexible, creative minds.
And schools do very little on developing that kind of mind.
They do a lot of developing the kind of mind
that a computer now does for us.
- Memorization.
- Memorization, doing procedures, a lot of things
that we spend a lot of time in school on.
In the world, when kids leave school,
a computer will do that.
And better than they will.
But that creative, flexible thinking,
we're kind of at ground zero at computers
being able to engage in that thinking.
Maybe we're a little above ground zero,
but the human brain is perfectly suited
for that creative, flexible thinking.
That's what humans are so great at.
So I would like the balance to shift in schools.
Maybe you still need to do some procedural kind of thinking,
but there should be a lot more
of that creative, flexible thinking.
- And what's the role of other humans in this picture?
So collaborative learning, so brainstorming together.
So creativity as it emerges
from the collective intelligence of multiple humans.
- Yeah, super important.
And we know that also helps develop your brain,
that social side of thinking.
And I love mathematics collaboration
where people build on each other's ideas
and they come up with amazing things.
I actually taught 100 students calculus
at Stanford recently, undergrads.
And we taught them to collaborate.
So these students came in Stanford
and most of them were against collaboration in math.
- This is before COVID in person?
§06Transcript · 44:06–52:49
- Yeah, it was just before COVID hit.
It was 2019.
And this summer--
- Sorry, you said they're against?
- Yeah, so it's really interesting.
So they'd only experienced maths individually
in a kind of competitive individual way.
And if they had experienced it as group work,
it had been a bad experience.
Like maybe they were the one who did it all
and the others didn't do much.
So they were kind of against collaboration.
They didn't see any role for it in maths.
And we taught them to collaborate.
And it was hard work because as well as the fact
that they were kind of against collaboration,
they came in with a lot of like social comparison thinking.
So I'm in this room with other Stanford undergrads
and they're better than me.
So when we set them to work on a maths problem together,
the first one was kind of a disaster
because they were all like, they're better than me,
they're faster than me.
They came up with something I didn't come up with.
So we taught them to let go of that thinking
and to work well together.
And one of the things we did,
we decided we wanted to do a pre and post test
at the end of this teaching.
It was only four weeks long,
but we knew we didn't want to give them
like a time test of individual work.
So we gave them an applied problem to do at the beginning
and we gave them to do in pairs together.
And we gave each of them a different colored pen
and said, work on this activity together
and keep using that pen.
So then we had all these pieces of student work.
And what we saw was they just worked
on separate parts of the paper.
So there's a little like red pen section
and a green pen section.
And they didn't do that well on it,
even though it was a problem
that middle or high school kids could do,
but it was like a problem solving kind of problem.
And then we gave them the same one to do at the end,
gave them the same colors
and actually they had learned to collaborate.
And not only were they collaborating the second time around,
but that boosted their achievement.
And the ones who collaborated did better on the problem.
Collaboration is important, having people,
and what was so eye opening for these undergrads
and they talked about it in lovely ways
was I learned to value other people's thinking on a problem.
And I learned to value
that other people saw it in different ways.
And it was quite a big experience for them
that they came out thinking,
I can do maths with other people,
people can see it differently,
we can build on each other's ways of thinking.
- I got a chance to,
I don't know if you know who Daniel Kahneman is,
got a chance to interact with him.
And like the first, 'cause he had a few,
but one famous collaboration throughout his life
with Tversky and just like,
he hasn't met me before in person,
but just the number of questions he was asking,
just the curiosity.
So I think one of the skills,
the collaboration itself is a skill.
And I remember my experience with him was like,
okay, I get why you're so good at collaboration
because he was just extremely good at listening
and genuine curiosity about how the other person
thinks about the world, sees the world.
And then together, he pulled me in,
in that particular case,
he doesn't know in particular
like that much about autonomous vehicles,
but he kept like asking all of these questions.
And then like 10 minutes in,
we're together trying to solve the problem
of autonomous driving.
And like, and that, I mean, that's really fulfilling,
that's really enriching,
but it also in that moment made me realize
it's kind of a skill
is you have to kind of put your ego aside,
put your view of the world aside
and try to learn how the other person sees it.
- And the other thing you have to put aside
is this social comparison thinking.
Like if you are sitting there thinking,
wow, that was an amazing idea,
he's so much better than I am,
that's really gonna stop you
taking on the value of that idea.
So there's a lot of that going on
between these Stanford students when they came.
And trying to help them let go of that.
- One of the things I've discovered
just because being a little bit more in the public eye,
how rewarding it is to celebrate others.
And how much is going to actually pay off in the long term.
So this kind of silo thinking of like,
I want to prove to a small set of people around me
that I'm really smart
and do so by basically not celebrating
how smart the other people are,
that's actually maybe short term,
it seems like a good strategy, but longterm it's not.
And I think if you practice at the student level
and then at the career level, at every single stage,
I think that's ultimately--
- I agree with you,
I think that's a really good way of thinking about it.
- You mentioned textbooks.
And you didn't say it,
maybe textbooks isn't the perfect way to teach mathematics.
But I love textbooks.
They're like pretty pictures and they smell nice.
I mean, I talk about like physical,
some of my greatest experiences have been just like,
'cause they're really well done.
When we're talking about basic,
like high school, calculus, biology, chemistry,
those are incredible.
It's like Wikipedia, but with color and nice little--
- You must have seen some good textbooks
if they had pretty pictures and color.
- Yeah, I mean, I remember,
I guess it was very, very standard,
like AP calculus, AP biology, AP chemistry.
I felt those are like some of the happiest days of my life
in terms of learning was high school.
'Cause it was very easy, honestly.
It felt hard at the time,
but you're basically doing a whirlwind tour
of all the science.
Without having to pick, you do literature,
you do like Shakespeare, calculus, biology, physics,
chemistry, what else?
Anatomy, physiology, computer science,
without like nobody's telling you what to do with your life.
You're just doing all of those things.
- That's a good thing, you're right.
- But I remember the textbooks weren't,
I mean, maybe I'm romanticizing the past,
but I remember they weren't, they're pretty good.
But so you think, what role do you think they play still?
And like in this more modern digital age,
what's the best materials
with which to do these kinds of educations?
- Well, I'm intrigued that you had such a good experience
with textbooks.
I mean, I can remember loving some textbooks
I had when I was learning and I love books.
I love to pick up books and look through them.
But a lot of math textbooks
are not good experiences for kids.
We have a video on our website
of the kids who came to our camp
and one of the students says,
"In maths, you have to follow the textbook.
"The textbook's kind of like the Bible.
"You have to follow it."
And every day, it's slightly different.
Like on Monday, you do 2.3.2
and on Tuesday, you do 2.3.3 and on Wednesday.
And you never go off that.
That's like every single day.
And that's not inspiring for a lot of the kids.
So one of the things they loved about our camp
was just that there were no books.
Even though we gave them sheets of paper instead,
they still felt more free
because they weren't just like trotting through exercises.
- Like what a textbook allows you is like,
the very thing you said they might not like,
the 2.3, 2.3, it feels like you're making progress
and like it's a little celebration
'cause you do the problem and it seems really hard
and you don't know how to do it.
And then you try and try and then eventually succeed.
And then you make that little step and further progress.
And then you get to the end of a chapter
and you get to like, it's closure.
You're like, "All right, I got that figured out."
And then you go on to the next chapter.
- I can see that.
I mean, I think it could be in a textbook.
You can have a good experience with a textbook,
but what's really important is what is in that textbook?
What are you doing inside it?
And I mean, I grew up in England and in England,
we learn maths.
§07Transcript · 52:50–1:02:05
We don't have this separation of algebra and geometry.
And I don't think any other country
apart from the US has that.
But I look at kids in algebra classes
where they're doing algebra for a year.
And I think I would have been pretty bored doing that.
(laughing)
- By the way, can we analyze your upbringing real quick?
Why do British folks call mathematics maths?
Why is it the plural?
Is it because of everything you're saying
or is it a bunch of sub-disciplines?
- Yeah, I mean, mathematics is supposed to be
the different maths that you look at,
whether you think of that as topics
like geometry and probability,
or I think of it as maths is just multi-dimensional,
lots of ways, but that's why it was called mathematics.
And then it was shortened to maths.
And then for some reason, it was just math in the US,
but to me, math has that more singular feel to it.
And there's an expression here, which is do the math,
which basically means do a calculation.
That's what people mean by do the math.
So I don't like that expression
'cause math could be anything.
It doesn't have to be calculation.
So yeah, I like maths
'cause it has more of that broad feel to it.
- Yeah, I love that.
Maths kind of emphasizes the multi-dimensional,
like a variety of different sub-disciplines,
different approaches, yeah.
Okay, but outside of the textbook,
what do you see broadly being used?
You mentioned Sebastian Thrun and MOOCs, online education.
Do you think that's an effective set of-
- Can be.
I mean, online, having great teachers online
obviously extends those teachers to many more people,
and that's a wonderful thing.
I have quite a few online courses myself.
I got the bug working with Sebastian
when he had released his first MOOC.
And I thought, hmm, maybe I could do one in maths education.
And I didn't know if anybody would take it.
And I remember releasing it that first summer,
and it was a free online class,
and 30,000 maths teachers took it that first summer.
And they were all talking about it with each other
and sharing it, and it was like, took off.
In fact, it was that MOOC that got me to create Ucubed
with Kathy Williams, who's the co-founder,
because people took the MOOC,
and then they said, "Okay, what now?
"I finished, what can I have next?"
So that was where we made our website.
But so yeah, I think online education can be great.
I do think a lot of the MOOCs don't have great pedagogy.
They're just a talking head.
And you can actually engage people in more active ways,
even in online learning.
So I learned from the Udacity principle
when I was working at Udacity
never to talk more than like five minutes,
and then to ask people to do something.
So that's the sort of pedagogy of the online classes I have,
is a little bit of presenting something,
and then people do something,
and then there's a little bit more.
Because I think if you have a half hour video,
you just switch off and start doing other things.
- So the way Udacity did it is like five, 10 minute,
like bit of teaching with some visual stuff, perhaps,
and then there's like a quiz almost.
- Then you answer a question, yeah.
- Yeah, that's really effective.
You mentioned Ucubed.
So what's the mission, what's the goal?
You mentioned how it started, but what's,
yeah, where are you at now, and what's your dream with it?
Or what are the kind of things
that people should go and check out on there?
- Yeah, we started Ucubed,
I guess it was about five years ago now,
and we've had over 52 million visitors to the site,
so I'm very happy about that.
And our goal is to share good ideas for teaching
with teachers, students, parents in maths,
and to help, we have a sort of sub-goal
of a raising maths anxiety, that's important to us,
but also to share maths as this beautiful creative subject.
And it's been really great.
We have lessons on the site,
but one of the reasons I thought this was needed
is there's a lot of knowledge in the academy
about how to teach maths well.
Loads and loads of research and journals
and lots of things written up, but teachers don't read it.
They don't have access to it.
They're often behind pay walls.
They're written in really inaccessible ways,
so people wouldn't want to read them or understand them.
So this I see as a big problem.
You have this whole industry of people
finding out how to teach well,
not sharing it with the people who are teaching.
So that's why we made U-Cubed.
And instead of just putting articles up saying,
here's some things to read about how to teach well,
we translated what was coming from research
into things that teacher could use.
So lessons, there were videos to show kids,
and there were tips for parents.
There were all sorts of things on the site.
And it's been amazing.
We took inspiration from the Week of Code,
which got teachers to focus on coding for a week.
And we have this thing called
the Week of Inspirational Maths.
And we say, just try it for a week.
Just give us one week and try it and see what happens.
And so it's been downloaded millions of times.
Teachers use it every year.
They start the school year with it.
And what they tell us is, it was amazing.
The kids' lights were on.
They were excited.
They loved it.
And then the week finished, and I opened my textbooks,
and the lights went out, and they were not interested.
- Yeah, but getting that first inspiration
is still powerful.
- It is.
I wish, I mean, what I would love
is if we could actually extend that for the whole year.
We're a small team at Stanford,
and we're trying to keep up with great things
to put on the site.
We haven't the capacity to produce
these creative visual math tasks
for every year group for every day,
but I would love to do that.
- How difficult is it to do?
I mean, to come up
with visual formulations of these big, important topics
you need to think about in a way that you could teach.
- I mean, we can do it.
We actually, we went from the Week of Inspirational Maths,
and we made K-8 maths books with exactly that.
Big ideas, rich activities, visuals.
We just finished the last one.
We've been doing it for five years,
and it's been exhausting, and we just finished.
So now there's a whole K-8 set of books,
and they're organized in that way.
These are the big ideas, here are rich, deep activities.
They're not, though, what you can do every day for a year.
So some teachers use them as a kind of supplement
to their boring textbook, and some people have said,
"Okay, this is the year.
"This book tells us what the year is,
"and then we'll supplement these big activities with."
So they're being used, and teachers really like them
and are really happy about them.
I just always want more.
And I guess one of the things I would like for U-Cubed,
one of my personal goals, is that every teacher of maths
knows about U-Cubed.
At the moment, lots of teachers who come to us
are really happy they found it,
but there's a lot of other teachers
who don't know that it exists.
- I hope this helps.
- Yeah.
- From a student perspective, and not in the classroom,
but at home, studying,
is there some advice you can give
on how to best study mathematics?
So what's the role of the student outside the classroom?
- Yeah, I think one thing we know is a lot of people,
when they review material,
whether it's maths or anything else,
don't do it in the best way.
I think a problem a lot of people have
is they read through maybe a teacher's explanation
or a way of doing maths, and it makes sense.
And they think, "Oh yeah, I've got that," and they move on.
But then it's not until you come to try and work on something
and do a problem that you actually realize
you didn't really understand it,
it just seemed to make sense.
So I would say this is also something
that neuroscientists talk about,
to keep giving yourself questions
is a really good way to study,
rather than looking through lots of material.
It's almost like giving yourself lots of tests
is a good way to actually deeply understand things
and know what you do when you don't understand.
§08Transcript · 1:02:07–1:11:10
- So would the questions be in the form of the material
you're reviewing is the answer to that question,
or is it almost like beyond,
it's the polygon thing that you mentioned,
or square, is it almost like,
"I wonder what is the bigger picture?"
Always kind of asking, "How is this extended?"
And so on.
- Yeah, that would be great.
And it's a similar, I mean,
a question I get asked a lot is about homework,
what is a good thing for kids to do for homework.
And one of the recommendations I give
is to not have kids just do lots of questions for homework,
but to actually ask them to reflect on what they've learned,
like, "What was the big idea you learned today?"
Or, "What did you find difficult?
"What did you struggle with?
"What was something that was exciting?"
Then kids go home and they have to kind of reflect
in a deeper way.
A lot of times, I don't know if you have this experience
as a math student, lots of people do,
kids are going through math questions,
they're successful, they get them right,
but they don't even really know what they're about.
And a lot of kids go through many years of math like that,
doing lots of questions,
but not really knowing what even the topic is
or what it's about, what it's important for.
So having students go back and think at the end of the day,
"What was the big idea from this math lesson?
"Why is it important?
"Where would I find that in real life?"
Those are really good questions
for kids to be thinking about.
- It's probably for everybody to be thinking about.
I think most of us go through life
never asking the bigger question.
Almost like those layers of why questions
that kids ask when they're very young.
We need to keep doing that.
- We do.
- Like what, that's the, whatever the term is,
you call first principles thinking,
some people call it that,
which is like, why are we doing it this way?
- But one nice thing is to do that
'cause there's usually a good answer.
Like the reason we did it this way
is because it works for this reason.
But then if you want to do something totally novel,
is you'll say, "Well, we've been doing it this way
"because of historical reasons,
"but really this is not the best way to do it.
"There might be other ways."
And that's how invention happens.
And then you get, that's really useful
in every aspect of life.
Like choosing your career, choosing your,
I don't know, where you live,
who your romantic partner is,
like everything. - Everything, yeah.
- And I think it probably starts doing that in math class.
- That would be good if we started doing that.
- I mean, I wonder,
I probably didn't do very much of that
for most of my education, asking why,
except for later, much later,
in the subjects on like grad school
when you're doing research on them.
When your first task of doing something novel using this
or solving a problem really outside the classroom,
they have to publish on it.
It's the first time you think,
"Wait, why are these things interesting, useful?
"Which are the things that are useful?"
And yeah, I guess that would be nice
if we did that much earlier,
that the quest of invention.
- Yeah, yeah.
I mean, one of the sad pieces of research data I think about
is the questions kids ask in school goes down
like in a linear progression from in the early years,
you can't stop kids asking those questions,
but they learn not to ask the questions.
- I think you told somewhere about an early memory you had
in your own education where you asked the question,
or maybe that was an example you gave,
but it was shut down.
- Oh yeah.
You've listened to something I said, yeah.
(laughing)
- I don't remember where it was, but it caught me.
- Yeah, I remember it really vividly.
- Well, can you tell the memory?
- Yeah, it's funny, I can remember.
It must have really impacted me in that moment
because you know how there's lots of hours of school
you don't remember at all.
But anyway, I can remember where I was sitting
and everything.
I was in a high school maths class,
although they don't call it that in England,
and the teacher said, and it was like the first class
of this teacher's class, and he said,
"Ask if you have any questions."
So at one point I put my hand up and I said,
"I have a question."
And he said something like, "That's your question?"
(laughing)
And I was like, "Okay, I'm not asking any more questions
"in this class."
- And that hit hard in a way where you didn't wanna,
the lesson you learned from that is I'm not gonna ask
that question. - Yeah, that was absolutely
that's the last question I'm asking.
And that was, yeah, he was the chair of the maths department
and I remember that really well.
So maybe because of that experience,
one of the things we encourage when we teach kids
is asking questions and we value it when they ask questions
and we put them up on walls and celebrate.
- It's funny 'cause I wish there was a feedback signal
because he probably, to put a positive spin on it,
he probably didn't realize the negative impact
he's had in that moment, right?
If he only knew.
See, this is probably when you're more mature in grad school.
Had an amazing professor named Alicia Kafande
in computer science and he would get,
he encouraged questions, but then he would tell everybody
how dumb their questions are.
But it was, I guess if you show,
if you say it with love and respect behind it,
then it's more like a friendly, humorous encouragement
for more questions.
It's an art, right?
- Yeah, teaching is very hard.
- You have to time it right because that kind of humor
is probably better for when you're in grad school
versus when you're in early education.
- Right, well, and I guess kids or young people
get whether somebody's doing it to be funny or has it.
I mean, this is why teaching is so hard.
Even your tone can be impactful.
- It's so sad because like for that particular human,
the teacher, you could have just had a bad day
and one statement can have a profound negative impact.
- I know, sadly, that maths, there's a lot of maths teachers
who have that kind of approach and they,
I think they're suffering from the fact
that they think people are math people, not math people.
And that comes across in their teaching.
- But on the flip side, one positive statement
to keep them going.
- That's right, that is the flip side of that.
And I myself had like one teacher
who was really amazing for me in maths
and she kept me in the subject.
- Who was she?
- She was, her name was Mrs. Marshall.
She was my A-level maths teacher.
So I was in England. - What's that mean?
- You do lots of subjects until you're 16
and then you choose like three or four subjects.
So I had chosen maths and you go to--
- Higher levels.
- Probably equivalent more to a master's degree in the US
'cause you're more specialized.
But anyways, she was my teacher.
And for the first time in my whole career in maths,
she would give us problems
and then tell us to talk about them with each other.
And so here I was sitting there at like 17
talking with friends about how to solve a math problem.
And that was it, that was the change that she made.
But it was profound for me
because like those calculus students,
I started to hear other people's ways of thinking
and seeing it and we would talk together
and come up with solutions.
And I was like, that was it, that changed maths for me.
- And so it wasn't some kind of personal interaction
with her, it was more like she was the catalyst
for that collaborative experience.
- I mean, yeah, the many ways teachers can inspire kids.
I mean, sometimes it's a personal message
but it can be your teaching approach
that changes maths for kids.
- You know, Cal Newport, he wrote a book called "Deep Work"
and he's a mathematician,
theoretical computer scientist.
And he talks about the kind of the focus required
to do that kind of work.
Is there something you can comment on?
You know, we live in a world full of distractions.
That seems like one of the elements that makes studying
and especially the studying of subjects
that require thinking like math does difficult.
§09Transcript · 1:11:14–1:20:44
Is there something from a student perspective,
from a teacher perspective that encourages deep work
that you can comment on?
- Yeah, I think giving kids really inspiring deep problems
and we have some on our website,
is a really important experience for them.
Even if they only do it occasionally,
but it's really important.
They actually realize, I give a problem out often
when I'm working with teachers and I say to them,
all right, I'm gonna check in with you after an hour.
And they're like, an hour?
They think it's shocking.
And then they work on this problem and after an hour,
I say, okay, how are we doing?
They're like, an hour has gone by?
How is this possible?
And so everybody needs those like rich, deep problems.
Most kids go through their whole maths experience
of however many years never once working on a problem
in that kind of deep way.
So the undergrad class I teach at Stanford, we do that.
We work on these deep problems every session
and the students come away going,
okay, I never wanna go back to that maths relationship
I had where it was just all about quick answers.
I just don't wanna go back to that.
And so we can all, all teachers can incorporate
those problems in their classrooms.
Maybe they don't do them every day,
but they at least give kids some experience
of being able to work slowly and deeply
and to go to deeper places and not be told
they've got five minutes to finish 20 questions.
- But part of it is also just the exercise
of sitting there and maintaining focus
for prolonged periods of time.
That's not often, I mean, that's a skill.
It's a skill that also could be discouraging.
Like if you don't practice it,
just sitting down for 10 minutes straight
and maintaining deep focus could be exceptionally challenging
like if you're really thinking about a problem.
And I think it's really important to realize
that that's a skill that you can just like a muscle,
you can build, you can start with five minutes
and it goes to 10 minutes to 30 and to an hour.
And to be successful, I think in certain subjects
like mathematics, you wanna be able to develop that skill.
Otherwise, you're not going to get
to the really rewarding experience
of solving these problems.
- Definitely.
There was a survey done of kids in school
where they were asked,
how long will you work on a math problem
before you give up and decide it's not possible to solve it?
And the result on average across the kids was two minutes.
- Yeah, that's a bad sign,
but that was a powerful sign that they need to learn
to not give up so quickly.
We mentioned offline because we've been talking so much
about visualization, Grant Sanderson, 3Luwan Brown.
So he's inspired millions of people
with exactly the kind of way of thinking
that you've been talking about.
- Yeah, I love his work.
- Converting sort of mathematical concepts into visual,
like visually representing them, exploring them in ways
that help you illuminate like the concepts.
What do you think is the role of that?
So he uses mostly programmatic visualization.
So it's the thing I mentioned where there's like animations
created by writing computer programs.
Like, what do you think, how scalable is that approach?
But in general, what do you think about his approach?
- I think it's amazing.
I should work with him.
I can share some of our visuals
and he can make them in that amazing way.
- So part of his storytelling,
part of it is like, it's creating the visuals
and then weaving a story with those visuals
that kind of builds, like there's also,
I mean, there's also drama in it.
You start with a small example and then you kind of,
all of a sudden there's a surprise.
- Yeah, yeah, yeah.
- And it really, I mean, it makes you fall in love
with the concept.
He does talk about that.
His sense is like some of the stuff,
he doesn't feel like he's teaching,
like the core curriculum, which is something,
he sees himself as an inspirational figure,
but because I think it's too difficult
to kind of convert all of the curriculum
into those elements.
- And probably you don't need to.
I mean, if people get to experience mathematical ideas
in the way that he shares them,
that will change them and it will change the way they think
and maybe they could go on to take
some other mathematical idea and make it that beautiful.
- Well, he does that.
He created a library called Manum and he open sourced it.
And that library is the, people should check it out.
It's written in Python and uses some of those same elements.
Like it allows you to animate equations
and animate little shapes, like people that,
he has a very distinct style in his videos
and what that resulted in,
even though from a software engineer perspective,
the code you release is not like super well-documented
or perfect, but him releasing that,
now there's all of these people educating it.
And the cool, to me personally,
the coolest thing is to see like people,
they're not, don't have like a million subscribers
or something, they have just a few views in the video,
but it just seems like the process of them creating a video
where they teach is like transformative to them
from a student perspective.
It's the old Feynman thing,
the best way to learn is to teach.
And then him releasing that into the wild is,
it shows that impact.
- Yeah, absolutely.
I think just giving people that idea
that you can do that with maths and other subjects,
there's bound to be people all around
who can create more, which is cool.
- Yeah, I definitely, so I recommend people do
like JavaScript or Python.
You can build like visualizations of most concepts
in high school math,
you can do a lot of kinds of visualizations
and doing that yourself.
Plus if you do that yourself, people will really love it.
People actually, people love visualizations of math.
- Yeah.
- 'Cause they, I mean, it's something in us
that loves patterns, loves figuring out difficult things
and the patterns in there
then are unexpected in some way.
- Yeah, have you ever noticed that hotels
are always filled with patterns?
I was just noticing it at the hotel.
I mean, now all of their carpets are pattern carpets
and then they have patterns on the walls.
- Yeah. - So, yeah.
- We humans love the symmetry in patterns,
the breaking of symmetry in patterns.
- Yeah.
- And it's funny that we don't see mathematics
as somehow intricately connected to that, but it is, right?
- I mean, that's one of the perspectives
I love students to take is to be a pattern seeker.
- In everything.
- Yeah, certainly in all of maths.
I mean, you can think of all of maths
as a kind of subject of patterns
and not just visual patterns,
but when you think about multiplying by five
and the fact you can, if you're multiplying 18 times five,
you can instead think of nine times 10.
That's a pattern that always works in mathematics.
You can have a number and double the number.
And so, yeah, I just think there are patterns everywhere.
And if kids are thinking their role is to see patterns
and find patterns, it's really exciting.
- What do you think about like MIT OpenCourseWare
and the release of lectures by universities?
- I think it's good.
I think it's good.
I think that is what started the MOOC I did
was using that platform.
- So you ultimately think like the Udacity models
is a little bit more effective
than just a plain two-hour lecture?
- I think there's definitely,
you can bring in good pedagogy into online learning.
And I think the idea of putting things online
so that people all over the world can access them is great.
I don't think the initial excitement around MOOCs
sort of democratizing education and making it more equal
came about because they found that the people taking MOOCs
tended to be the more privileged people.
So that was, I think there's still something
to be found in that.
There's still more to be done to help
that online learning reach those principles.
But definitely I think it's a good invention.
And I have an online class that's for kids
that's a little free class that gives them--
- What's the topic?
- It's called How to Learn Maths.
- How to Learn Maths.
- It shows maths as this visual creative subject
and it shares mindset and some brain science.
§10Transcript · 1:20:47–1:29:57
And kids who take it do better in maths class.
We've studied it with like randomized controlled trials
and given it to middle school kids
and other middle school kids who don't take it
but are taught by the same teachers.
So their teachers are the same.
And the kids who take the online class
end up 68% more engaged in their maths class
and do better at the end of the year.
So that's a little six session, 15 minute class
and it changes kids' maths relationships.
So it is true that we can do that with some words
that aren't, it's not a huge change to the education system.
- Do you have advice for young people?
We've been talking about mathematics quite a bit
but in terms of their journey through education,
through their career choices, through life,
maybe middle school, high school, undergrad students
of how to live a life they can be proud of.
- I think if I were to give advice to people,
especially young people, my advice would be to always,
it sounds really corny, but always believe in yourself
and know that you can achieve
because although that sounds like obvious,
of course we want kids to know that they can achieve things.
I know that millions of kids who are in the school system
have been given the message they cannot do things.
And adults too, they have the idea,
oh, I did okay in this, I went into this job
because those other things I could never have done okay in.
So actually when they hear,
hey, maybe you could do those other things,
even adults think, you know, maybe I can.
And they go back and they encounter this knowledge
and they relearn things and they change careers
and amazing things happen.
So for me, I think that message is really important.
You can learn anything.
Scientists try and find a limit.
They're always trying to find a limit.
Like how much can you really learn?
What's the limit to how much you can learn?
And they always come away not being able to find it.
People can just go further and further and further.
And that is true of people born with brain,
you know, areas of their brain that aren't functioning well
that have what we call special needs.
Some of those people also go on to develop
and do amazing things.
So I think that really experiencing that,
knowing that, not just saying it, but knowing it deeply,
you can learn anything,
is something I wish all people would have.
- Actually also applies when you've achieved
some level of success too.
What I find, like in my life with people that love me,
when you achieve success,
they keep celebrating your success
and they want you to keep doing the thing
that you were successful at,
as opposed to believing in that you can do something else,
something big, whatever your heart says to do.
And one of the things that I realized the value of this,
you know, quite recently, which is sad to say,
is how important it is to seek out,
when you're younger, to seek out mentors,
to seek out the people,
like surround yourself with people that will believe in you.
- Yeah.
- It's like a little bit is on you.
It's like, you don't get that,
sometimes if you go to like grad school,
you think you kind of land on a mentor,
maybe you pick a mentor based on the topic
they're interested in.
But the reality is the people you surround yourself with,
they're going to define your life trajectory.
So select people that--
- That's really true.
And get away from people who don't believe in you.
- Sometimes parents can be that.
- Yeah.
- They can love you deeply, but they set,
it's the math thing we mentioned,
they might set certain constraints
on the beliefs that you have.
And so in that, if you're interested in mathematics
and your parents are not that interested in it,
don't listen to your parents on that one dimension.
- Right.
Exactly.
Yeah, and if people tell you you can't do things,
you have to hear from other people who believe in you.
I think you're absolutely right about that.
It's so sad the number of people
who've had those negative messages from parents.
In my Limitless Mind book,
I interviewed quite a few people
who'd been told they couldn't do math,
sometimes by parents, sometimes by teachers.
And fortunately, they had got other ideas
at some point in their life
and realized there was this whole world
of mathematical thinking that was open to them.
So it's really important that people
do connect with people who believe in them.
However hard that might be to find those people.
- What do you hope the education system,
education in general, looks like 10, 20, 50,
100 years from now?
Are you optimistic about this future?
- Yeah, I definitely have hope.
There is, change can happen in the education system.
In recent years, it's been microscopically slow.
But I do actually see change happening.
Like we were talking earlier that data science
is now a course you can take in high school
instead of algebra two.
And that's pretty amazing
because that content was set out in 1892
and hasn't changed since then.
And so now we're actually seeing a change
in the content of high school.
So I'm amazed that that's happening
and very happy it's happening.
So change is very slow in education usually,
but when you look ahead and think about all that we know
and all that we can offer kids in terms of technology,
you've got to think that 100 years from now,
education will be totally different to the way it is now.
Maybe we won't have subject boundaries anymore
because those don't really make much sense.
- And it's interesting to think how certain tools
like programming, maybe they'll be deeply integrated
in everything we do.
- You would think, yeah.
You would think that all kids are growing up
learning to program and create.
So I just think, I mean, the system of schooling
we have now, people call it a factory model.
It's not designed to inspire creativity.
And I feel like that will also change.
People might look back on these days
and think they were hilarious,
but maybe in the future,
kids will be doing their own programming
and they'll be able to learn things
and find out things and create things
even as they're learning.
And maybe the individual subject boundaries will go.
Data science itself coming into the education system
kind of illustrates that because people realize
it doesn't really fit inside any of the subjects.
So what do we do with it?
Where does it go?
And who teaches it?
So it's already raising those kind of questions
and questioning how we have
these different subject boundaries.
- So you've seen data science be integrated
into the curriculum?
- Yes, it's happening across the United States as we speak.
- I wonder how they got initiated.
Like how does change happen in the education system?
Is it just a few revolutionary leaders?
- I think so, I think so.
It's been an interesting journey
seeing data science take off, actually.
There was a course that was developed in 2014
by some people who thought data science
was a good idea for high schoolers.
And then after some kids took the course
and nothing bad happened to them,
they went to college and people started to accept it more.
And then this was a big piece of the change in California.
The UC system communicated.
They sent out an email last year
to 50,000 high schools saying,
"We now accept data science.
Kids can take it instead of Algebra II.
That's a perfectly legitimate college pathway."
So that was like a big green light for a lot of schools
who were like wondering about whether they could teach it.
So I think it happens in small spaces and expands.
So now-- - It goes viral.
- Yeah. - In this modern age.
- Then it goes viral.
California's ahead, I think, in creating courses
and having kids go through it.
But it's, certainly when I last looked,
there were 12 states that were allowing data science
as a high school course.
And I think by next year, that will have doubled or more.
So change is happening.
- Joe, as I said, I think mathematics
is truly a beautiful subject.
And you having an impact on millions of people's lives
by educating them, by inspiring teachers
to educate in the ways that you've talked about
in multi-dimensional ways, in visual ways,
§11Transcript · 1:30:01–1:30:36
I think is incredible.
So you're spreading beauty-- - Thank you, I really
appreciate that. - Into the world.
So I really, really appreciate that you spent
your valuable time with me today.
Thank you for talking. - Thank you.
It was really good to talk to you.
- Thanks for listening to this conversation with Joe Bowler.
To support this podcast, please check out our sponsors
in the description.
And now let me leave you with some words
from Albert Einstein.
Pure mathematics is the poetry of logical ideas.
Thanks for listening and hope to see you next time.
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