Kind: captions
Language: en
the following is a conversation with joe
buller a mathematics educator at
stanford and co-founder of youcubed.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 joe 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 beautiful the beauty
emerges when you have a problem with a
solution and you start adding other
solutions
simpler solutions
weirder solutions more interesting some
of their visual some of their algebraic
geometry all that kind of stuff yeah i
mean i 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
turns out you can do that with any area
of maths
and i think we've given
p 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
complementary
what's the kind of the full spectrum of
students in 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
multi-dimensional experience of math
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 because you ultimately want to
understand
the like the deepest secret underneath
this problem and for that you have to
intuit your way there yeah and you you
mentioned offline that um 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 know you
have einstein
uh thinking about a train
you know and the speed of light and you
know that kind of intuition is useful
yeah you start to like 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
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 maths problems
that's another thing that's often absent
in maths 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 like
if if you
task yourself with building an intuition
about a problem
that's what
that's where you start to pull in like
um
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 yeah you get the
simple explanation of what why
why things are the way they are
as opposed to um
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
reasoning is really hard yeah it's 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 i used to work with somebody called
sebastian thoran who
it's 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's 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 uh the 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
i mean it's just all of these physicists
definitely you see um
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
right just like let's
let's take whatever the heck we're
looking at and draw it
and and draw like uh the pattern as it
evolves as the thing grows for n equals
one for n equals two n equals three and
you start uh to play with it and then in
the modern day which i loved uh doing
is you know you can write a program that
then visualizes it for you right and
then you can start exploring it
programmatically
and that that and
then you can do so interactively too i
tend to not
uh like interactive because the way it
takes to way too much work because you
have to click and move and stuff i love
to interact through writing programs
that's my particular brain software
engineer so like
you can you can uh do all these kinds of
visualizations uh and then there's the
tools of visualization like color
uh all those kinds of things yeah that
you're absolutely right they're actually
not
taught very much right 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
yeah 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 um yeah we 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
think not realizing you can really
understand things with these visuals
it's quite powerful
you say that uh there's something about
there's something valuable to learning
when the thing that you're doing is
challenging it's difficult so a lot of
people say you know 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 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 math sprain 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 you can do it you
have to go through this struggle to get
there but you're 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 i ask a
difficult question here yes so there's
kind of um
i don't know what the right term is but
some people
are um
struggle with learning in different ways
like their brain is
constructed in different ways
and um
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 know 30
students who think in different ways and
but this is also why i believe it's so
important to have this multi-dimensional
approach to maths we've really offered
it in one way which is
here's some numbers in 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 give
you let kids experience it in different
ways maybe visually with numbers with
words
what happens is kids
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 uh fell in love with math early i
was forced into math early and fell in
love
through force that's good well you fell
in love with other people well but there
uh something we talked about a little
bit is there's such a
value for excellence
uh it's competitive
and it's also everybody kind of looks up
the the definition
of
success is being
in a particular class is you know being
really good at it
um and like it's not improving it's like
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 know
you're going to star on the basketball
team
if uh you're going to start on the
basketball team if you're going to be
better than the other guys the other
girls on the team
uh
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 um probably
delusional but i could win a nobel prize
i don't even know what that entails
um but i thought um
like uh my dad early on told me
just off hand and it always stuck with
me that if you 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 like like 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 feeling
and then then when you struggle the
belief that you could be great
is like struggle is good right pushes
you on yeah and so the other thing about
the soviet system
that might
love to hear your comments about is just
the sheer like hours of math like the
number of courses you're talking about
a lot of geometry a lot more 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 in
russia i remember just being nailed over
and over would you it was just non-stop
and then of course there's different
perspectives on calculus and just the
whole
the sense was
that math is like
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 math so those two things um
valuing
excellence
and just a lot of math in the curriculum
do you think do you think do you find
that interesting because 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 that idea that excellence is
important and you can get there if you
work hard
in the u.s
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 uh 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 i would add something into that i
would say that anybody can get there
but they 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
so
but that would be huge that change as to
doing lots of maths if um
if maths was interesting and open and
creative and multi-dimensional i would
be all for it
we we actually run summer camps at
stanford where we invite kids in and we
give them this maths that i love
and
the in our camp classrooms they were
three hours long
and
when we were planning the teachers were
like three hours are we gonna 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 so what uh
what age are you talking about is there
um could you comment on what age
is like 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
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 we our camps are
middle school students we think of those
years as really pivotal many kids in in
those years are deciding yes i'm gonna
keep going with stem subjects
or no i'm not that 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 yeah that energy of like
turns out it's really important there's
um 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 are the
sentences 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 and giving
this feedback 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 it's um
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
um
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 i
sometimes cause like
it's such a funny dance because i'm such
a perfectionist i'm extremely
self-critical and i have when 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 uh
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 and some of the greatest took
non-linear paths to where they sucked
for long into la into later life and so
always kind of believing that this
person uh 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
are dropouts at school or who had
special needs who didn't succeed
um 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
um 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
um 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
he talks a lot about not having us
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
the education especially early education
system didn't help them
is there
ways for an education system to support
people like that is it this kind of
multi-dimensional learning that you're
missing 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
in producing this very uninspiring
mathematics
um 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 appreciate it a lot is a
mathematician called marion miz akani
you 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's 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 it'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 and ms kearney 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 is this
is 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
um but all it takes is that's
that's why people talk about like the
one teacher that changed everything
exactly all it takes is one teacher
that's right that's that's the power of
that
so that that's like that should be
inspiring to teachers like i think it is
you as a single person given the
education system given the incentives
you have the power to truly change lives
in like 20 years from now that's right i
feel this medalist will walk up to you
and yeah 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 um
one study found that
um
the amount of maths anxiety parents had
predicted their child's achievement in
school
but
only if they helped with homework
so
that's so funny yeah
but there are some interesting
implications for this 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
i was terrible at this at school and
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 math anxiety just don't
don't do math say well with your kids
um but we have a on our website we have
a little
sheet for parents of
ways to interact around maths with your
kids
and
that's uh
youtube.org that's youtube.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 and um
i'm always curious about this so i you
know
i hope to have kids one day i don't have
kids currently
um are parents okay with like sucking at
math and then
trying to get their kid to be better
than them essentially like is that
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
and
yeah i mean to me this you want to
celebrate that but i know a lot of
people struggle with that like coaches
in sports
to to make an athlete
become better than them it can be hard
on the ego
yeah
so is that do you experience the same
with parents too i think i mean i
haven't experienced parents worrying
that their kids will be better than them
i have experienced
i have experienced parents
just having a really bad relationship
with maths and yeah you know 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
and um 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 like 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
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
um
it's hard for them to be able to shift
their kids to see it differently is
there for a teacher
if we were to like systematize it is
there something teachers can do to do
this more effectively so you must you
mentioned the textbook yeah so so what
what are the additional things you can
add on top of this whole
old school traditional way of teaching
that can improve the 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 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 um 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 math
seem really boring and uninspiring what
what what are the kind of can you give a
few examples what so
this is an interesting example in third
grade there are three different
standards about unit squares
okay um 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 that so that's a
standard the textbook authors say oh i'm
going to make a question about that and
they translate the standards into narrow
questions
and then you measure success by your
ability to deliver on on these standards
so the standards themselves uh i think
of maths and many people think maths in
this way is a subject of like a few big
ideas and really important connections
between them
um so like you could think of it as like
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
a
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
um
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 uh like big concepts within
mathematics
square towels 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 k
10 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 you know grades
and tests and stuff like that do you
throw those out or is there is not a fan
of grades and tests um 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
summit of achievement but the problem we
have in maths classrooms across the uh
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 had been 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
um
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
um will show where what they can what
the kids can do with a rubric and also
write notes like you know in the next
few weeks you might like to learn to do
this so
um so instead of kids just thinking
about i'm an a kid or a b kid 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 um during the year they
get these much more informative measures
i do think
this this might be more for college
but maybe not i
some of the best classes i've had is
when
i got a special
like set aside
like the the professor
clearly saw that i was interested in
some aspect of a thing and then um i
have a few in mind and one in particular
but he said that um
he kind of challenged me so this is
outside of grades and all that that kind
of stuff
that
basically it's like reverse psychology i
don't i don't think this can be done
and so i gave everything to do that
particular thing so this was happened to
be in an artificial intelligence class
but
i i think that like special treatment of
taking students
who are especially like excellent at a
particular little aspect that you see
their eyes light up
i i often think like maybe it's tempting
for a teacher to think you've already
succeeded there but they're actually
signaling to you that
like you could really launch them on
their way yeah um and
i don't know that's too much to expect
from teachers i think to to to
pay attention to all of that because
it's really difficult but i 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 that who really
didn't just like inspire me with their
awesomeness which they did but also just
they pushed me a little yeah like it
gave me a little push
and that requires focusing on the quote
unquote excellent
uh yes in the class yeah i think what's
important though is teachers to have the
perspective that they don't know who's
going to 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
um
oh and and over time we encourage them
to like extend things further
i remember we were doing one activity
where kids were working at 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'd do
this with a pentagon
and i said that's fantastic how do you
how what does it look like with pentagon
go you know find out see if you can
discover
so i didn't know he was going to 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 uh
almost to say
uh i don't think you could do this this
is too hard like that's what i need to
hear it's just like no i you know you
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
how your genetics is
they might be much more that might break
them yeah that might break them so you
have to be also sensitive to that i mean
teaching is really difficult it's really
difficult for this very reason it is
so um what is the best way
to teach math to learn math at those
early few days when you just want to
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'm going to group the
dots and i show it them and then i
i i take it away before they've even had
enough time to count them and then i
asked 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 asked people tell me how 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
i like it it's kind of cool so
turns out though that how well you
groupitize predicts
how well you do in maths
is is it uh is it a raw talent or is it
just something that you can develop i
think it's i don't think you're born
groupitizing i think but some kids have
developed that
um ability if you like and you can learn
it you can so
this to me is part of how wrong we have
maths that we think to tell whether a
kid's good at maths we're going to give
them a speed test on
fact on multiples but actually
seeing how kids group dots could be a
more important assessment of how well
they're going to do maths anyway i
diverge what i like to do though when i
start off with kids is show them i'm
going to give you math problems i'm
going to 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
is uh 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
grupotizing yeah
yeah i love this term so what's what
would you say is the role of creativity
and flexibility in 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
it's 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 maths
and creativity come together but i
that's what i love about maths is the
creative different ways you can see it
and so helping our kids
there's a book
i like a lot but i've been by physicists
you probably know this book could
elastic
you might know it um
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
ground zero at computers being able to
engage in that thinking
maybe we're a little above ground zero
but um 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 uh what's the role of other humans
in this picture so
collaborative
yeah learning so brainstorming together
so creativity as it emerges from
the collective intelligence of multiple
humans yeah
super important and
um 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 a hundred
students calculus at stanford recently
undergrads and
we taught them to collaborate so these
students came into stanford and most of
them were against collaboration
in maths this is before covered in
person yeah it was just before covert
hit it was 2019.
and um
you said they're against uh so yeah so
it's really interesting so they'd only
experience maths individually as 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 or
so when we send 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 timed 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 were on separate parts of the paper
uh so this 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
it 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 you know i
can do maths with other people people
can see it differently we can build on
each other's way
ways of thinking i got a chance to i
don't know if you know who daniel
kahneman has
got a chance to interact with him
and like the first because he had
a few but
one famous collaboration throughout his
life
uh with taversky and
just like
you know he hasn't met me before uh 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's 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 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 because you have to kind of put
your ego aside yep put your view of the
world aside and try to learn how the
other person right
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
and so there's a lot of that going on
between these stanford students when
they came
and
yeah but 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
yeah and how much is going to actually
pay off in the long term yeah 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 long
term it's not and i think if you
practice at the student level and then
at the career la at every single stage i
think that's ultimately i i agree with
you i think that's a really good way of
thinking about it
you mentioned textbooks you said and you
didn't say it
you know
um
maybe textbooks isn't the perfect way to
teach mathematics but i i love textbooks
there's they're like pretty pictures and
they smell nice and they open i mean i
talk about like physical some of my
greatest experiences have been just like
oh
like because they're really well done
when we're talking about basic like high
school
calculus
biology chemistry those are like those
are incredible
it's like wikipedia but with color and
and nice little you must have seen some
good textbooks if they had pretty
pictures in color
yeah i mean i remember i guess it was
very very standard
like ap
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 because it was it's very
easy honestly
it felt hard at the time but you're
basically doing a um
whirlwind tour of all of science yeah
yeah without having to pick you do
literature you do like shakespeare your
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 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
um but so you think what role do you
think they play still
and like in this more modern
digital age what what's the best
materials with which to do these kinds
of educations well i'm intrigued that
you have 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 maths textbooks are not good
experiences for kids they
um
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 textbooks 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 exercises
so um like what a textbook allows you is
like you're
the the very thing you said they might
not like the two point yeah point three
two point if you feels like you're
making progress and like it's little
celebrations because 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
then you make that little step and yeah
further progress and then you get to the
end of a chapter and you get to like
it's closure you're like yeah 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 it what's really important is what
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 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
um
by the way can we can we analyze your
upbringing real quick um
why
do british folks call
mathematics maths
why is it the plural is it is it because
of everything you're saying where it's a
bunch of sub-disciplines yeah i mean
mathematics is sure is supposed to be
the
uh
the map the different maths that you
look at
whether you think of that as topics like
geometry and probability or
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 because
no math could be anything it doesn't
have to be calculation and
so yeah i like maths because it has more
of that broad feel to it yeah i love
that maths kind of emphasizes the
multi-dimensional like yeah variety of
different different disciplines
different approaches yeah
okay uh what so but outside of the
textbook what do you
see like broadly being used you
mentioned sebastian through and moocs
online education do you think that's an
effective so can be
i mean online
having great teachers online obviously
extends those teachers to many more
people and that's a wonderful thing
um
i have
quite a few online courses myself i got
the bug working with sebastian when he
was
had released his first mooc
and i thought maybe i could do one in
maths education and i didn't know if
anybody would take it um 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 caught
got me to
create you cubed with kathy williams
who's the co-founder um because people
took the mooc and then they said okay
what now
i finished what what can i have next and
so that was where we made our website
but
um 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
it you can actually engage people in
more active ways even in online learning
so i learned from the udacity principle
when i was working udacity never to talk
more than like five minutes and but then
and then to ask people to do something
so
that's the sort of pedagogy of the
online classes i have there's a little
bit of
presenting something and then people do
something and there's a little bit more
because i think if you have a half hour
video you just you know
switch off and start doing other things
so the the way udaci did it is uh like
five ten minute like bit of
teaching
and then with some visual stuff perhaps
and then there's like a quiz then you
answer a question yeah yeah now that
that's that's yeah that's really
effective you mentioned youtube so
what's the mission what's the goal you
mentioned how it started but what's uh
um yeah where are you at now and what do
you 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 you cubed 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 subgoal of erasing
maths anxiety that's important to us but
also
to share maths as this beautiful
creative subject
and um
it's been really great we have lessons
on the site but one of the things 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 they don't have access to it
they're often behind pay walls
they
they're written in really inaccessible
ways so people wouldn't want to read
them or understand them so this i see is
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
um that's why we made youtube 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
um there were tips for parents there
were all sorts of things on the site and
it's been amazing
as we had we
took inspiration from the week of code
which
got teachers to focus on coding for a
week and um we have this thing called
the week of inspirational maths
and we say just try it for a week just
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 uh but getting that first
inspiration is still powerful i mean it
is i i wish i mean my
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 um
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 uh visual formulations of these uh
big important topics you need to think
about
in a way that you know
that uh that that you could teach i mean
it 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
um
they're not though what you can do every
day for a year that 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
um 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 you cubed
one of my personal goals is that every
teacher of maths knows about youtube
at the moment um a lot 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
and you know
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 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 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 you
mentioned from a square is it almost
like i wonder
what is the bigger picture i was kind of
asking
like how's this extended and so on yeah
that that that would be great and it's a
similar i mean a question i get asked a
lot is about homework well it's 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 were you learned today or
what did you
find difficult what did you struggle
with what was something that was
exciting
um
then kids go home and they have to kind
of reflect in a deeper way a lot of
times i don't if you have this
experience as a math student lots of
people do kids are going through maths
questions they're successful they get
them right but they don't even really
know what they're about they and a lot
of kids go through many years of maths
like that doing lots of questions but
that 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 maths 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
like the bigger question almost like
you know those like layers of why
questions that kids ask when they're
very young yeah we need to keep doing
that
we do like what uh that's the you know
whatever the term is you call first
principles thinking some people call it
that which is like
why
are we doing it this way
so one nice thing
is to do that because 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
um 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 right
and then you get you know that's really
useful in every aspect of life like
choosing your career
choosing your um
i don't know what where you live yeah
who your like 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 i
want i mean i wonder
i
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 you're first tasked with doing
something novel
using this
or solving a problem really outside the
classroom they have to publish on it's
the first time you think
wait why are these things
uh interesting useful which are the
things that are useful
and yeah i guess that was that would be
nice if we did that much earlier that uh
the quest of invention yeah yeah
i mean one of the sad pieces of research
data i think about is the questions kids
ask
um in school goes down like in a linear
uh you know 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
i don't remember where it was it was
something yeah it caught me yeah i
remember it really vividly
what can you tell uh the memory yeah it
was 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 um i i remember where i
was sitting and everything i was in a
high school maths class although they
don't call it in england
and um
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
um
[Laughter]
and i was like oh okay i'm not asking
any more questions
hard in a way where you didn't wanna you
the lesson you learned from that is i'm
not gonna ask yeah that was absolutely
that's nice that's the last question i'm
asking
and um
[Music]
i was
yeah he was the chair of the maths
department 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 we put them up
on walls and celebrate and
um it's funny because
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
i had an amazing professor named ali
shakafande in uh computer science and he
would get he encouraged questions but
then he would tell everybody how dumb
their questions are
but it was
it was done 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
yeah it's an art right yeah to do it
you're right and then it's very hard you
have to time it right because that's
probably that kind of humor is probably
better for when you're
uh in grad school versus when you're in
the early education right
well and i guess kids
or young people get whether somebody's
doing it to be funny or you know
has it there's i mean this is why i
teach you so hard even your tone
can be impactful
it's so sad because like for
for that particular human the teacher
you could 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
are not math people and that comes
across in their teaching
but on the flip side one positive
statement yeah 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
like
who was she she was
um
her name was mrs marshall
[Laughter]
um she was my a level maths teacher so i
was in i mean in england you do lots of
subjects so you're 16 and then you
choose like three or four subjects so i
chosen maths and
you go to
higher levels
probably equivalent more to a master's
degree in the us because 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 tell us
to talk about them with each other
and so here i was sitting there 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 i because
like those
calculus students i started to hear
other people's ways of thinking and
seeing it we would talk together and
come up with solutions and i was like
that was it that changed maths for me
and i know so it wasn't some kind of
personal interaction with her it was
more like
she uh
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 uh cal newport
he uh he wrote a book called deep work
and he's a mathematician a theoretical
computer scientist and he talks about
the kind of the focus required
yeah to do that kind of work is there
something
you can comment on
you know we live in a in a world full of
distractions
that that seems like one of the elements
that makes studying and especially the
studying of subjects that require
thinking like math does
difficult
is there something from a student
perspective from a teacher perspective
that encourages deep work
that you can come up with yeah i think
giving kids
really inspiring deep problems and we
have some on our website
is a really important experience for
them
um even if they only do it occasionally
but
it's really important they actually
realize i i do i give a problem out
often when i'm working with teachers and
i say to them all right i'm going to
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 like an hour's gone by
how is this possible yeah 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
i the 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 want to go back to that maths
relationship i had where it was just all
about quick answers i
i just don't want to 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 yeah well part of it is
also just the um
the exercise of sitting there
maintaining focus for prolonged periods
of time
that's not often
i mean um that's a skill yeah it's a
skill
that uh 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 to re 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 it goes to 10
minutes to 30 into an hour and
and to be successful i think in certain
subjects like mathematics you want to be
able to develop that skill otherwise
you're not going to get to the 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 maths problem before
you give up and decide it's not
possible to solve that question and the
result on average across the kids was
two minutes
yeah
so that's
a bad sign but that's that was the
powerful sign that uh they need to learn
to not give up so quickly
yeah
uh we mentioned offline
because we've been talking so much about
visualization
uh grant sanderson three luanne brown
so
he's inspired millions of people with
the kind of uh exactly the kind of way
of thinking that you've been talking
about yeah i love his
work converting sort of uh
mathematical concepts into visual
uh
like uh visually representing them
exploring them in ways that uh help you
illuminate like the concepts
um
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 uh computer programs
um like what what do you think how
scalable is that approach but in general
what do you think about his impression i
think it's amazing i should work with
him
i i can share some of our visuals and he
can make them in that amazing way um
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 that's right with
the concept yeah he does talk about that
his sense is like some of the stuff he
he he doesn't feel like he's teaching
like the core curriculum which is
something you know
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
you
if
people get to experience mathematical
ideas in the way that he shares them
um that will change them and it will
change the way they they think and maybe
they could go on to
take some other mathematical idea and
make it that beautiful well he does that
uh there's a
he created a library called manum
and he open-sourced it and that library
is the uh people should check it out
it's
written in python
and it uses some of those same elements
like it allows you to animate equations
and animate little shapes like people
that
you know he has a very distinct style in
his videos and what that resulted in
even though from a software engineer
perspective the code he released is not
like
super well documented or perfect
but him releasing that
now there's all of these other people
educating it and the cool to me
personally the coolest thing is to see
like
people they're not you know
don't have like a million subscribers or
something is they they they have just a
few views on the video but it just seems
like
the process of them creating a video
where they teach
is like transformative to them yeah from
a student perspective it's the old
feynman thing the best way to learn is
to teach right and then him releasing
that into the wild is yeah
it shows that that impact yeah
absolutely
i think just giving people that idea
that you can do that with maths and
other subjects they're
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 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
really love it people actually
people love visualizations of math
yeah because they i mean it's something
in us that loves patterns loves figuring
out difficult things and the patterns in
that are then are unexpected in some way
yeah have you ever noticed that hotels
are always filled with patterns
i was just noticing at the hotel i mean
now all of their carpets are patent
carpets and then they have patterns on
the walls
yeah so
yeah we humans love
the symmetry and patterns the breaking
of symmetry patterns
yeah and it's funny that we don't see
mathematics is 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 in in 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
you know when you think about
multiplying by five
and the fact you can
you know if you if you're multiplying 18
times 5
you can instead think of 9 times 10.
that's a pattern that always works in
mathematics you can have a number and
double 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 open
courseware
and the release of lectures by
universities i think it's good
i think it's good i think the
that is what started
the
mooc i did was
using that platform so you ultimately
think like the udacity model 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 it's great i don't think
the initial excitement around moocs
sort of democratizing education and
making it more equal
um
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
um definitely i think it's it's a good
invention and i have an online class
that's for kids that's little free class
that gives them a topic it's called how
to learn maths
it shows maths as this visual creative
subject and it shares mindset and some
brain science and
um
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 you know not 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
how to live a life that 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 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 earn 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
um 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 feeling not just saying it
but knowing it
deeply
you can learn anything
is um
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 right and one of
the things that i realized the value of
this
um
you know quite recently which is sad to
say
is how important it is to seek out uh
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 of is on you
it's like uh you don't get that um
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 yeah trajectory so select
people they're really true and get away
from people who don't believe you don't
yeah
sometimes parents can be that they love
you deeply but they be you know 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
exactly
yeah and if people tell you you can't do
things you have to
hear from other people who who believe
you i think you're absolutely right
about that
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
and
um
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 2
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 but
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 it's interesting to think
how certain tools like programming maybe
they'll be deeply integrated and
everything you think yeah you would
think that all kids are growing up
learning to program
and
create
so um 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
um maybe we'll 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 subjects 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 that got
initiated like how does change happen in
the education system is it just a few
revolutionary like it does
i think so i think so it's been an
interesting journey seeing data science
take off actually it um
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 two that's a
perfectly legitimate college pathway so
that was like a big green light for a
lot of schools who are 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 latter age then it goes viral
california's ahead
um i think in creating courses and
having kids go through it but it's
suddenly 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 i as i said i think
mathematics is uh
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 um
i think is incredible so you're
spreading beauty appreciate that
so i really really appreciate that you
spend your valuable time with me today
thank you for talking thank you it's
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
you