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What Makes People Engage With Math | Grant Sanderson | TEDxBerkeley | TEDx Talks | YouTubeToText
YouTube Transcript: What Makes People Engage With Math | Grant Sanderson | TEDxBerkeley
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I want to ask what makes people engage
with math I mean we all seem very intent
that our children should learn math
right that we should learn math that it
somehow puts us in a better position to
understand Sciences technology and when
you have someone sitting there in a
classroom it's not a given that they're
engaged now I work on a YouTube channel
right and on YouTube this question is
put to an unusual level of let's just
say an extreme stress test right because
if you're bored with what you're
watching or you're debating whether or
not to watch something there are
billions of hours of content sitting
there waiting for you some of the most
entertaining things humanity has ever
created are sitting there just one click
away so if you're trying to teach math
on YouTube and someone's not engaged
they're not sticking around so what I
want to do is answer this question
through a youtubers lens in a way that's
hopefully helpful to more traditional
teaching contexts and I was asked to
talk about some of what I do so I
figured what we would do here is take a
look at some of the content that I've
made that by the extremely coarse metric
of view count is in a sense more
engaging than others and part of the
reason I choose to do this is the four
specific videos at the top paint a very
interesting picture to answer our
question so sitting at number four was a
video about Fourier transforms now this
it's a beautiful piece of math
absolutely wonderful the whole idea is
about understanding functions in terms
of pure frequencies so when you hear a
musical note played something like the a
440 that's used to tune an orchestra
what the air pressure over time would
look like if you were to graph it if
something maybe like this yellow graph
you know it's a pure sine wave
it Wiggles at a nice steady rate pitches
that are higher or lower also wiggle
according to pure sine waves but maybe
faster or slower now when you play them
all together to get a chord what happens
is that at each point in time the
strength of each of those individual
notes is getting added together but
because there are different frequencies
you end up with this very complicated
looking graph in this case I've only
added together four different
frequencies but the one at the top
it's notably more complicated it's
definitely not a clean sine wave and the
question that Fourier transforms try to
answer is how do you do this in Reverse
how do you start with a signal that
something like your microphone would
pick up and reverse engineer what the
pure frequencies that went into it are
now it's not just relevant for sound
engineering if you ask any electrical
engineer or someone who works with
quantum mechanics are all kinds of
physics it turns out that being able to
break up functions as pure frequencies
is kind of a problem-solving superpower
but if you all bought into the idea that
this is a very neat thing to learn it's
a wonderful beautiful piece of math and
you go to look it up what you would find
is something that looks like this which
is very intimidating right I mean first
of all there's an integral there so you
at least need to know calculus that's
just a bare minimum but if you look
closer you see - this e to the PI stuff
so we're doing calculus with complex
numbers I mean the very first line of
the Wikipedia page is sort of screaming
at you you need to be at a certain elite
level before you're going to be able to
understand this but if you look past the
symbols and the formalisms it turns out
there's a very nice way to understand
what the Fourier transform is actually
doing this isn't sugarcoating it it's
showing what it's actually doing but in
a very visual way and this is what I
tried to make a video about I'll just
give you the very high level here the
the idea is to take this graph that
might be a mixture of different
frequencies and sort of wind it around a
circle and to really talk through the
details of what's going on in this
animation and how it pulls out the exact
frequencies it takes maybe 10 15 minutes
it's not too bad but it's a complicated
image but the idea is that even if you
don't have a deep technical background
you can come to a substantive
understanding of what the Fourier
transform is doing before you see the
calculus and the complex numbers and at
that point once you bring in the
formalisms it has a way of articulating
an idea that's already in your mind
explaining the usefulness of those
calculus and complex member terms so
that's Fourier transforms now if we jump
up to number two I did a video on neural
networks and I think I hardly need to
tell anyone in this audience just how
useful neural networks have proved to be
in the last couple decades but if you
really drill in on what specifically is
going on when we reference machines
learning you're giving them training
examples in what sense is it learning in
this context how to recognize
handwritten digits there is a ton of
wonderful math to be had in there and
again it's highly visualizable it's
something where you can show what's
happening before you bring in the
formalisms of matrix operations and
nonlinearities but
Queen them in gradient descent and all
of that delicious stuff you can get to
the substance before you get to the
intimidating formalism so what makes
people engage with math I personally am
a big fan of visualizations I think
animation can play a big role
but that's only once you've got them
bought into learning a topic if we think
about these two Fourier transforms and
neural networks I think a big part of
what draws people in is that they answer
a question that often goes woefully
unaddressed for most people in their
math classes when am I ever gonna use
this I mean you all know this feeling
right I hear that kind of murmur of
agreement you're in an algebra class
you're doing something like the
quadratic formula you're just working
through worksheet after worksheet and
it's also unrelated to your life or
anything that you could imagine being in
your life but if you can't answer this
question I think it elevates math to the
status of going to the gym it's still
gonna take work right we're not going to
sugarcoat things but you know what
you're getting for that work and instead
of being something that's kind of nerdy
exclusively for the realm of school it's
something that you can feel proud of
doing where after you do it you feel
good you feel powerful
and you feel smug too you kind of want
to boast to your friends so what makes
people engage with math relevance you
know connect it to the world preferably
connected to the audience's world but
that's almost too obvious I think you
all know that that's the answer I would
like to argue it's actually not the
complete answer though I think there's
an ingredient that people don't really
talk about when they set curriculums
when they decide what their class is
going to look like and I think it's a
very important ingredient if we're
thinking about this question of
engagement and what leads me to think of
it is looking at some of the content
that I've made that people have seemed
most engaged with so let's take a look
at number three and one here because
they paint a very different story and
what I want to do is just talk about the
topic itself the math that you're going
to learn in the video without any of the
context okay just what problem does each
one talk about so sitting at number
three the problem is let's say we have
two different blocks and they're sitting
on a frictionless surface okay in this
case I have one that's one kilogram and
one that 16 kilograms and we're going to
send that right block slide
towards the left one they're gonna
bounce off of each other a couple times
there's a wall to their left a bunch of
bounces happen and eventually they sail
off never to touch again and we're going
to be very idealized you know no energy
is lost to friction no energy is lost to
the collisions between them they don't
act on each other with gravity we just
want to count the collisions in this
idealized situation that's it that is
the question that is the number three
video and this might seem like a joke
and I promise it's not but here's the
number one here's the question that it's
asking if you take a sphere and you
choose four random points on the surface
of that sphere okay so a uniform
probability all points on the sphere are
equally likely and we're going to form a
tetrahedron which is sort of a
triangular pyramid shape it's what you
get to make those four points its
vertices the question is what is the
probability that this tetrahedron this
weird shape contains the center of the
sphere so you know sometimes the four
points are kind of on opposite sides and
it contains it other times they're
bunched up together and it doesn't that
is it that is the question somehow
that's more popular than known that
works right and I can hear some of you
scratching your heads in the audience
because if you look at the question when
am I ever going to use this you might
think that you won't use the answer to
this question I know you might think
that you'd be absolutely right I promise
you're never gonna need to know the
number of times that two blocks bounce
off of each other in a frictionless
situation and you are never gonna need
to know the probability that tetrahedron
formed by four random points on the
surface of a sphere contains it's a
weird question it's not a natural
question so why on earth do more people
care about block collisions than Fourier
transforms and why on earth do more
people care about the strange sphere
question the neural networks okay so I
said that if you can answer this
question that elevates math to the
status of going to the gym now let me
ask in the audience today among you how
many of you have gone to the gym in the
last 24 hours by raise of hands
okay so raise your hand if we've been to
the gym in the last 24 hours and it
looks like maybe the rooms 20 percent
most muscular arms are all rising at
once okay
so set them down in contrast how many of
you in this audience today again by
raise of hands in the last 24 hours
have consumed some piece of fiction so
maybe a book or a movie or that Netflix
series that you've been binging some
piece of fiction it's a lot more a lot
more now what's funny is fiction makes
no attempt to answer this question I
don't know about you but when I was
reading Harry Potter I didn't find
myself asking little I ever used
Wingardium Leviosa when am I going to
apply the newfound knowledge I have of
the rules of Quidditch or the newfound
knowledge I have of the intimate
personality quirks of each individual
wisly child no I didn't ask that because
we understand fiction Appeals for an
entirely different reason it's about
emotion it's about wonder it's about
establishing a mystery that you just
need to see resolved it's about
introducing a romance that you really
want to see come to fruition it's a warm
escape from a world that to a lot of us
can be cold and sometimes lonely and
before you go thinking that math plays
you know plays by different rules it
absolutely does not if you look at some
of the people who are most engaged with
the subject professional mathematicians
the way that they describe their subject
sometimes seems almost callously removed
from the idea of reality there was this
one English mathematician G H Hardy and
he wrote a book in 1940 called a
mathematicians apology and it might be
best summed up by the following
quotations we have concluded that
trivial mathematics is on the whole
useful and that the real mathematics on
the whole is not so yeah you know
Fourier transforms neural networks all
that trivial stuff it might be useful
but leave it to the engineers pure stuff
number theory topology analysis yeah
that's the good stuff but not useful so
why would he care well let's turn to an
earlier mathematician another giant of
the field on Reap on Kure
he writes the mathematician does not
study pure mathematics because it is
useful he studies it because he delights
in it and he delights in it because it's
beautiful it's funny that sounds more like
like
people talk about art than how they talk
about science what makes people engage
with math I think the thing not enough
people talk about is what I'm just going
to call story and when I use that word
I mean appeals to emotion I mean having
comedy having some notion of characters
that you care about I mean having a
mystery you need to see resolved really
anything that pulls you in for the math
for what it is now not what it promises
to give you later let's take a look at
the block collisions because context
here is crucial yes the question is
useless but let me show you what would
pull you in this is really a mystery
novel and like any good mystery novel
you open with a crime scene smoking gun
a fingerprint of someone who's kind of
familiar in a way that suggests there's
something deeper at play
if each block has the same mass it's not
too hard to see what's going to happen
they transfer their momentum entirely
with each collision you end up getting
three total clacks now if we increase
one of those masses by a factor of 100
it gets more interesting because once it
hits that block it retains a lot of its
momentum and it ends up taking a lot
more collisions to turn it around it is
a legitimately hard problem
I'll tell you that it's a hard problem
to figure out but I'm just gonna tell
you the answer because the pattern is
what's going to be interesting here all
in all when the dust settles it ends up
being 31 total collisions so we had
three and then 31 if we up it by another
factor of 100 to 10,000 most collisions
happen in a very big unrealistic burst
and it's dependent on the idealism of
the situation and what I love about it
is you get a beautiful dramatic pause
before the final because you remember
our pattern was three then 31 and then
finally it's going to be 3 1 4 and you
might not see it I wouldn't blame you
it's a very surprising result but it
turns out if you keep playing this game
and you upped by various powers of 100
what ends up happening and again I want
to emphasize this depends on the
idealism of the situation the total
number of digits in the collision are
the same as pi 3 1 4 1 5 9 2 and at this
point it does not matter if the physics
is idealized if you have a soul you have
to know why
right it's a one-dimensional situation
there's no circle I don't see a circle
and pies digits are counting something
that is a very weird thing for pi to do
that's not what it does so what follows
is a detective story tracking down the
circle and you're not shying away from
the math to get a satisfying answer to
ministry you dive right into that math
and you learn what you need to learn but
it's not because it's useful it's
because the story has drawn you in now
what about that weird sphere problem I
will admit that maybe most of the
popularity there has more to do with a
mildly clickbait t title that I gave it
you see I called it the hardest problem
on the hardest test which is actually
kind of the point you see there's this
contest given to some of the most
ambitious math students in colleges
around the United States and Canada it's
called the Putnam it is famously hard
you know the mean score on this is 2 out
of 120 it's a very hard test and it's
given in these two parts each with six
questions number one is hard because
it's the Putnam and they get
progressively more challenging so the
pint that you know you get to five and
six it ends up being real it ends up
being crushing let's be honest and this
problem that I talked about earlier the
sphere probability tetrahedron situation
showed up as number six on one of these
tests so the video is not about the
problem per se you do see how to solve
it but it's a story about how you dear
viewer whoever you are whatever your
background in math you're not actually
that different from the top students
because what we can do is walk through
step by step the problem-solving tactics
that could lead you to find the clever
insight to answer this question that is
you know maybe a stretch to call the
hardest problem on the hardest test but
it's positioned as the hardest problem
on a famously hard test and in the same
way that people watching Star Wars I
think get a little buzzed within them by
thinking what if I had the force I like
to think that people watching something
like this get that same buzz thinking
hmm what if I were to solve the hardest
problem on a Puttnam test
yeah it's a fiction it might be a
fiction but that's exactly what pulls
you in and I know I've been a little bit
focused on my own channel here but I
guarantee if you look at any of the most
successful math outlets out there they
succeed by leveraging some component of
story maybe the most popular math
channel on YouTube numberphile great
channel one of the best things that it
does is it exposes the humanity and the
character of different mathematicians if
we look at stand-up math by Matt Parker
he leverages comedy and wit in order to
talk about very technical topics but in
a very laughable way a personal favorite
of mine is a channel called
looking-glass universe and when you
watch a video you almost hear in the
narration the smile behind each word and
the whole channel is a sort of
scientific omage to Lewis Carroll and
Alice in Wonderland so you want to talk
about incorporating fiction into science
this exemplifies it but even then even
if you buy me that there's some
storytelling component to be had in math
that it can be genuinely entertaining I
know that some people are going to be
thinking yeah but when am I ever going
to use that math surely the stuff we
should teach our students isn't that
playful puzzle stuff it's the useful
stuff that's the reason we emphasize map
and put it core in the education system
who cares about puzzles but here's the
thing about math even if it's not useful
even if it's almost trying not to be
useful it has a way of coming back
around do you remember our friend Hardy
from earlier well one of the reasons he
was not only okay with but like weirdly
proud of the fact that his worth had no
applications is that he had just lived
through two world wars so at that time
utility and morality were not exactly
synonyms and shortly after the quote we
saw earlier he is what he writes he says
no one has discovered any warlike
purpose to be served in the theory of
numbers or relativity and it seems very
unlikely that I know any will do so for
many years now in hindsight we can
almost laugh at this because relativity
is critical for most physics to include
GPS GPS guided weaponry Anniston number
theory I'm sorry Hardy as pure and
platonic as your primes might have
seemed that's the backbone of modern
cryptography so even when he's trying
to make it useful it had a way of coming
back around and you know those block
collisions I would have put money on the
fact that you would never need to know
the solution to this problem you would
never actually need to apply the fact
that you get a circle out of this and
yet a couple months after I made it a
quantum computing researcher came up to
me and pointed out a discovery he made
that the math behind that is identical
to not similar to but identical to the
math behind a very famous quantum search
algorithm so bizarrely tracking down the
circle in that detective story puts you
in a better position to understand
quantum computation I wouldn't have
guessed that
that's what math does it shines a light
on unexpected connections so what makes
people engage with math well honestly I
think the most compelling answer is
neither the usefulness nor the story but
understanding the bizarre way that they
intertwine with each other you know the
easy half here is that sometimes the
best narrative is rooted in a really
good application but much more
counterintuitive and just as true is
that some of the most useful math that
you'll ever find or that you can teach
has its origins and someone who is just
looking for a good story thank you very much
much [Applause]
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