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