0:06 I want to ask what makes people engage
0:09 with math I mean we all seem very intent
0:10 that our children should learn math
0:12 right that we should learn math that it
0:14 somehow puts us in a better position to
0:17 understand Sciences technology and when
0:18 you have someone sitting there in a
0:20 classroom it's not a given that they're
0:22 engaged now I work on a YouTube channel
0:25 right and on YouTube this question is
0:28 put to an unusual level of let's just
0:30 say an extreme stress test right because
0:32 if you're bored with what you're
0:34 watching or you're debating whether or
0:36 not to watch something there are
0:38 billions of hours of content sitting
0:40 there waiting for you some of the most
0:41 entertaining things humanity has ever
0:43 created are sitting there just one click
0:45 away so if you're trying to teach math
0:47 on YouTube and someone's not engaged
0:49 they're not sticking around so what I
0:51 want to do is answer this question
0:53 through a youtubers lens in a way that's
0:54 hopefully helpful to more traditional
0:56 teaching contexts and I was asked to
0:57 talk about some of what I do so I
1:00 figured what we would do here is take a
1:01 look at some of the content that I've
1:03 made that by the extremely coarse metric
1:05 of view count is in a sense more
1:08 engaging than others and part of the
1:10 reason I choose to do this is the four
1:11 specific videos at the top paint a very
1:13 interesting picture to answer our
1:15 question so sitting at number four was a
1:17 video about Fourier transforms now this
1:19 it's a beautiful piece of math
1:21 absolutely wonderful the whole idea is
1:23 about understanding functions in terms
1:25 of pure frequencies so when you hear a
1:27 musical note played something like the a
1:29 440 that's used to tune an orchestra
1:32 what the air pressure over time would
1:34 look like if you were to graph it if
1:35 something maybe like this yellow graph
1:37 you know it's a pure sine wave
1:39 it Wiggles at a nice steady rate pitches
1:41 that are higher or lower also wiggle
1:43 according to pure sine waves but maybe
1:45 faster or slower now when you play them
1:48 all together to get a chord what happens
1:50 is that at each point in time the
1:51 strength of each of those individual
1:54 notes is getting added together but
1:55 because there are different frequencies
1:57 you end up with this very complicated
1:58 looking graph in this case I've only
2:00 added together four different
2:01 frequencies but the one at the top
2:03 it's notably more complicated it's
2:05 definitely not a clean sine wave and the
2:07 question that Fourier transforms try to
2:09 answer is how do you do this in Reverse
2:11 how do you start with a signal that
2:12 something like your microphone would
2:15 pick up and reverse engineer what the
2:17 pure frequencies that went into it are
2:18 now it's not just relevant for sound
2:20 engineering if you ask any electrical
2:21 engineer or someone who works with
2:23 quantum mechanics are all kinds of
2:25 physics it turns out that being able to
2:26 break up functions as pure frequencies
2:28 is kind of a problem-solving superpower
2:31 but if you all bought into the idea that
2:33 this is a very neat thing to learn it's
2:34 a wonderful beautiful piece of math and
2:36 you go to look it up what you would find
2:39 is something that looks like this which
2:42 is very intimidating right I mean first
2:44 of all there's an integral there so you
2:45 at least need to know calculus that's
2:47 just a bare minimum but if you look
2:49 closer you see - this e to the PI stuff
2:52 so we're doing calculus with complex
2:54 numbers I mean the very first line of
2:56 the Wikipedia page is sort of screaming
2:58 at you you need to be at a certain elite
2:59 level before you're going to be able to
3:02 understand this but if you look past the
3:04 symbols and the formalisms it turns out
3:06 there's a very nice way to understand
3:08 what the Fourier transform is actually
3:10 doing this isn't sugarcoating it it's
3:12 showing what it's actually doing but in
3:14 a very visual way and this is what I
3:16 tried to make a video about I'll just
3:17 give you the very high level here the
3:20 the idea is to take this graph that
3:21 might be a mixture of different
3:23 frequencies and sort of wind it around a
3:25 circle and to really talk through the
3:26 details of what's going on in this
3:29 animation and how it pulls out the exact
3:31 frequencies it takes maybe 10 15 minutes
3:33 it's not too bad but it's a complicated
3:35 image but the idea is that even if you
3:37 don't have a deep technical background
3:39 you can come to a substantive
3:40 understanding of what the Fourier
3:43 transform is doing before you see the
3:45 calculus and the complex numbers and at
3:46 that point once you bring in the
3:48 formalisms it has a way of articulating
3:50 an idea that's already in your mind
3:52 explaining the usefulness of those
3:55 calculus and complex member terms so
3:57 that's Fourier transforms now if we jump
3:59 up to number two I did a video on neural
4:01 networks and I think I hardly need to
4:02 tell anyone in this audience just how
4:04 useful neural networks have proved to be
4:07 in the last couple decades but if you
4:10 really drill in on what specifically is
4:12 going on when we reference machines
4:13 learning you're giving them training
4:15 examples in what sense is it learning in
4:17 this context how to recognize
4:19 handwritten digits there is a ton of
4:21 wonderful math to be had in there and
4:23 again it's highly visualizable it's
4:24 something where you can show what's
4:26 happening before you bring in the
4:29 formalisms of matrix operations and
4:30 nonlinearities but
4:31 Queen them in gradient descent and all
4:33 of that delicious stuff you can get to
4:35 the substance before you get to the
4:38 intimidating formalism so what makes
4:42 people engage with math I personally am
4:43 a big fan of visualizations I think
4:44 animation can play a big role
4:46 but that's only once you've got them
4:48 bought into learning a topic if we think
4:50 about these two Fourier transforms and
4:53 neural networks I think a big part of
4:55 what draws people in is that they answer
4:57 a question that often goes woefully
4:59 unaddressed for most people in their
5:01 math classes when am I ever gonna use
5:04 this I mean you all know this feeling
5:05 right I hear that kind of murmur of
5:06 agreement you're in an algebra class
5:08 you're doing something like the
5:10 quadratic formula you're just working
5:12 through worksheet after worksheet and
5:14 it's also unrelated to your life or
5:16 anything that you could imagine being in
5:19 your life but if you can't answer this
5:21 question I think it elevates math to the
5:24 status of going to the gym it's still
5:25 gonna take work right we're not going to
5:28 sugarcoat things but you know what
5:30 you're getting for that work and instead
5:31 of being something that's kind of nerdy
5:34 exclusively for the realm of school it's
5:35 something that you can feel proud of
5:38 doing where after you do it you feel
5:40 good you feel powerful
5:42 and you feel smug too you kind of want
5:45 to boast to your friends so what makes
5:47 people engage with math relevance you
5:49 know connect it to the world preferably
5:53 connected to the audience's world but
5:54 that's almost too obvious I think you
5:57 all know that that's the answer I would
5:58 like to argue it's actually not the
6:00 complete answer though I think there's
6:02 an ingredient that people don't really
6:04 talk about when they set curriculums
6:06 when they decide what their class is
6:08 going to look like and I think it's a
6:11 very important ingredient if we're
6:12 thinking about this question of
6:14 engagement and what leads me to think of
6:16 it is looking at some of the content
6:17 that I've made that people have seemed
6:19 most engaged with so let's take a look
6:21 at number three and one here because
6:22 they paint a very different story and
6:24 what I want to do is just talk about the
6:26 topic itself the math that you're going
6:27 to learn in the video without any of the
6:30 context okay just what problem does each
6:32 one talk about so sitting at number
6:34 three the problem is let's say we have
6:36 two different blocks and they're sitting
6:38 on a frictionless surface okay in this
6:40 case I have one that's one kilogram and
6:42 one that 16 kilograms and we're going to
6:43 send that right block slide
6:45 towards the left one they're gonna
6:46 bounce off of each other a couple times
6:48 there's a wall to their left a bunch of
6:50 bounces happen and eventually they sail
6:52 off never to touch again and we're going
6:54 to be very idealized you know no energy
6:56 is lost to friction no energy is lost to
6:58 the collisions between them they don't
7:00 act on each other with gravity we just
7:01 want to count the collisions in this
7:03 idealized situation that's it that is
7:05 the question that is the number three
7:06 video and this might seem like a joke
7:08 and I promise it's not but here's the
7:10 number one here's the question that it's
7:12 asking if you take a sphere and you
7:15 choose four random points on the surface
7:17 of that sphere okay so a uniform
7:18 probability all points on the sphere are
7:20 equally likely and we're going to form a
7:22 tetrahedron which is sort of a
7:24 triangular pyramid shape it's what you
7:25 get to make those four points its
7:28 vertices the question is what is the
7:30 probability that this tetrahedron this
7:33 weird shape contains the center of the
7:35 sphere so you know sometimes the four
7:37 points are kind of on opposite sides and
7:38 it contains it other times they're
7:40 bunched up together and it doesn't that
7:42 is it that is the question somehow
7:44 that's more popular than known that
7:47 works right and I can hear some of you
7:49 scratching your heads in the audience
7:51 because if you look at the question when
7:52 am I ever going to use this you might
7:54 think that you won't use the answer to
7:55 this question I know you might think
7:58 that you'd be absolutely right I promise
8:00 you're never gonna need to know the
8:02 number of times that two blocks bounce
8:04 off of each other in a frictionless
8:06 situation and you are never gonna need
8:09 to know the probability that tetrahedron
8:11 formed by four random points on the
8:13 surface of a sphere contains it's a
8:14 weird question it's not a natural
8:16 question so why on earth do more people
8:18 care about block collisions than Fourier
8:20 transforms and why on earth do more
8:22 people care about the strange sphere
8:26 question the neural networks okay so I
8:27 said that if you can answer this
8:29 question that elevates math to the
8:31 status of going to the gym now let me
8:34 ask in the audience today among you how
8:35 many of you have gone to the gym in the
8:37 last 24 hours by raise of hands
8:39 okay so raise your hand if we've been to
8:41 the gym in the last 24 hours and it
8:44 looks like maybe the rooms 20 percent
8:46 most muscular arms are all rising at
8:48 once okay
8:51 so set them down in contrast how many of
8:53 you in this audience today again by
8:55 raise of hands in the last 24 hours
8:57 have consumed some piece of fiction so
9:00 maybe a book or a movie or that Netflix
9:02 series that you've been binging some
9:03 piece of fiction it's a lot more a lot
9:06 more now what's funny is fiction makes
9:08 no attempt to answer this question I
9:10 don't know about you but when I was
9:11 reading Harry Potter I didn't find
9:15 myself asking little I ever used
9:18 Wingardium Leviosa when am I going to
9:24 apply the newfound knowledge I have of
9:27 the rules of Quidditch or the newfound
9:28 knowledge I have of the intimate
9:30 personality quirks of each individual
9:34 wisly child no I didn't ask that because
9:35 we understand fiction Appeals for an
9:38 entirely different reason it's about
9:42 emotion it's about wonder it's about
9:43 establishing a mystery that you just
9:46 need to see resolved it's about
9:47 introducing a romance that you really
9:50 want to see come to fruition it's a warm
9:52 escape from a world that to a lot of us
9:54 can be cold and sometimes lonely and
9:56 before you go thinking that math plays
9:58 you know plays by different rules it
10:01 absolutely does not if you look at some
10:03 of the people who are most engaged with
10:05 the subject professional mathematicians
10:08 the way that they describe their subject
10:10 sometimes seems almost callously removed
10:14 from the idea of reality there was this
10:16 one English mathematician G H Hardy and
10:18 he wrote a book in 1940 called a
10:20 mathematicians apology and it might be
10:21 best summed up by the following
10:23 quotations we have concluded that
10:25 trivial mathematics is on the whole
10:28 useful and that the real mathematics on
10:31 the whole is not so yeah you know
10:33 Fourier transforms neural networks all
10:35 that trivial stuff it might be useful
10:37 but leave it to the engineers pure stuff
10:40 number theory topology analysis yeah
10:44 that's the good stuff but not useful so
10:46 why would he care well let's turn to an
10:48 earlier mathematician another giant of
10:49 the field on Reap on Kure
10:51 he writes the mathematician does not
10:53 study pure mathematics because it is
10:55 useful he studies it because he delights
10:57 in it and he delights in it because it's
11:01 beautiful it's funny that sounds more like
11:01 like
11:03 people talk about art than how they talk
11:06 about science what makes people engage
11:08 with math I think the thing not enough
11:11 people talk about is what I'm just going
11:13 to call story and when I use that word
11:16 I mean appeals to emotion I mean having
11:18 comedy having some notion of characters
11:20 that you care about I mean having a
11:21 mystery you need to see resolved really
11:23 anything that pulls you in for the math
11:26 for what it is now not what it promises
11:29 to give you later let's take a look at
11:31 the block collisions because context
11:33 here is crucial yes the question is
11:34 useless but let me show you what would
11:37 pull you in this is really a mystery
11:38 novel and like any good mystery novel
11:41 you open with a crime scene smoking gun
11:43 a fingerprint of someone who's kind of
11:45 familiar in a way that suggests there's
11:47 something deeper at play
11:50 if each block has the same mass it's not
11:51 too hard to see what's going to happen
11:53 they transfer their momentum entirely
11:55 with each collision you end up getting
11:57 three total clacks now if we increase
11:59 one of those masses by a factor of 100
12:02 it gets more interesting because once it
12:04 hits that block it retains a lot of its
12:06 momentum and it ends up taking a lot
12:08 more collisions to turn it around it is
12:09 a legitimately hard problem
12:10 I'll tell you that it's a hard problem
12:12 to figure out but I'm just gonna tell
12:14 you the answer because the pattern is
12:15 what's going to be interesting here all
12:17 in all when the dust settles it ends up
12:20 being 31 total collisions so we had
12:22 three and then 31 if we up it by another
12:25 factor of 100 to 10,000 most collisions
12:28 happen in a very big unrealistic burst
12:30 and it's dependent on the idealism of
12:32 the situation and what I love about it
12:34 is you get a beautiful dramatic pause
12:35 before the final because you remember
12:38 our pattern was three then 31 and then
12:40 finally it's going to be 3 1 4 and you
12:42 might not see it I wouldn't blame you
12:44 it's a very surprising result but it
12:45 turns out if you keep playing this game
12:48 and you upped by various powers of 100
12:50 what ends up happening and again I want
12:52 to emphasize this depends on the
12:55 idealism of the situation the total
12:57 number of digits in the collision are
13:01 the same as pi 3 1 4 1 5 9 2 and at this
13:04 point it does not matter if the physics
13:06 is idealized if you have a soul you have
13:08 to know why
13:17 right it's a one-dimensional situation
13:19 there's no circle I don't see a circle
13:22 and pies digits are counting something
13:23 that is a very weird thing for pi to do
13:27 that's not what it does so what follows
13:29 is a detective story tracking down the
13:31 circle and you're not shying away from
13:33 the math to get a satisfying answer to
13:35 ministry you dive right into that math
13:37 and you learn what you need to learn but
13:39 it's not because it's useful it's
13:42 because the story has drawn you in now
13:44 what about that weird sphere problem I
13:47 will admit that maybe most of the
13:48 popularity there has more to do with a
13:51 mildly clickbait t title that I gave it
13:53 you see I called it the hardest problem
13:56 on the hardest test which is actually
13:58 kind of the point you see there's this
14:00 contest given to some of the most
14:01 ambitious math students in colleges
14:03 around the United States and Canada it's
14:05 called the Putnam it is famously hard
14:08 you know the mean score on this is 2 out
14:12 of 120 it's a very hard test and it's
14:13 given in these two parts each with six
14:15 questions number one is hard because
14:17 it's the Putnam and they get
14:18 progressively more challenging so the
14:20 pint that you know you get to five and
14:23 six it ends up being real it ends up
14:25 being crushing let's be honest and this
14:27 problem that I talked about earlier the
14:29 sphere probability tetrahedron situation
14:32 showed up as number six on one of these
14:35 tests so the video is not about the
14:36 problem per se you do see how to solve
14:40 it but it's a story about how you dear
14:42 viewer whoever you are whatever your
14:45 background in math you're not actually
14:46 that different from the top students
14:48 because what we can do is walk through
14:50 step by step the problem-solving tactics
14:53 that could lead you to find the clever
14:56 insight to answer this question that is
14:58 you know maybe a stretch to call the
15:00 hardest problem on the hardest test but
15:01 it's positioned as the hardest problem
15:04 on a famously hard test and in the same
15:06 way that people watching Star Wars I
15:08 think get a little buzzed within them by
15:12 thinking what if I had the force I like
15:14 to think that people watching something
15:15 like this get that same buzz thinking
15:18 hmm what if I were to solve the hardest
15:20 problem on a Puttnam test
15:21 yeah it's a fiction it might be a
15:23 fiction but that's exactly what pulls
15:25 you in and I know I've been a little bit
15:27 focused on my own channel here but I
15:28 guarantee if you look at any of the most
15:30 successful math outlets out there they
15:32 succeed by leveraging some component of
15:34 story maybe the most popular math
15:36 channel on YouTube numberphile great
15:38 channel one of the best things that it
15:40 does is it exposes the humanity and the
15:42 character of different mathematicians if
15:45 we look at stand-up math by Matt Parker
15:47 he leverages comedy and wit in order to
15:50 talk about very technical topics but in
15:53 a very laughable way a personal favorite
15:54 of mine is a channel called
15:56 looking-glass universe and when you
15:58 watch a video you almost hear in the
16:00 narration the smile behind each word and
16:03 the whole channel is a sort of
16:05 scientific omage to Lewis Carroll and
16:07 Alice in Wonderland so you want to talk
16:08 about incorporating fiction into science
16:13 this exemplifies it but even then even
16:15 if you buy me that there's some
16:17 storytelling component to be had in math
16:20 that it can be genuinely entertaining I
16:21 know that some people are going to be
16:24 thinking yeah but when am I ever going
16:28 to use that math surely the stuff we
16:30 should teach our students isn't that
16:32 playful puzzle stuff it's the useful
16:33 stuff that's the reason we emphasize map
16:36 and put it core in the education system
16:39 who cares about puzzles but here's the
16:42 thing about math even if it's not useful
16:44 even if it's almost trying not to be
16:46 useful it has a way of coming back
16:49 around do you remember our friend Hardy
16:51 from earlier well one of the reasons he
16:53 was not only okay with but like weirdly
16:55 proud of the fact that his worth had no
16:57 applications is that he had just lived
17:00 through two world wars so at that time
17:02 utility and morality were not exactly
17:04 synonyms and shortly after the quote we
17:07 saw earlier he is what he writes he says
17:09 no one has discovered any warlike
17:10 purpose to be served in the theory of
17:12 numbers or relativity and it seems very
17:15 unlikely that I know any will do so for
17:17 many years now in hindsight we can
17:20 almost laugh at this because relativity
17:22 is critical for most physics to include
17:26 GPS GPS guided weaponry Anniston number
17:28 theory I'm sorry Hardy as pure and
17:29 platonic as your primes might have
17:31 seemed that's the backbone of modern
17:34 cryptography so even when he's trying
17:35 to make it useful it had a way of coming
17:37 back around and you know those block
17:39 collisions I would have put money on the
17:41 fact that you would never need to know
17:43 the solution to this problem you would
17:45 never actually need to apply the fact
17:47 that you get a circle out of this and
17:49 yet a couple months after I made it a
17:51 quantum computing researcher came up to
17:53 me and pointed out a discovery he made
17:58 that the math behind that is identical
18:00 to not similar to but identical to the
18:02 math behind a very famous quantum search
18:06 algorithm so bizarrely tracking down the
18:08 circle in that detective story puts you
18:09 in a better position to understand
18:12 quantum computation I wouldn't have
18:13 guessed that
18:15 that's what math does it shines a light
18:19 on unexpected connections so what makes
18:23 people engage with math well honestly I
18:25 think the most compelling answer is
18:27 neither the usefulness nor the story but
18:29 understanding the bizarre way that they
18:31 intertwine with each other you know the
18:33 easy half here is that sometimes the
18:34 best narrative is rooted in a really
18:37 good application but much more
18:39 counterintuitive and just as true is
18:41 that some of the most useful math that
18:43 you'll ever find or that you can teach
18:45 has its origins and someone who is just
18:48 looking for a good story thank you very much
18:48 much [Applause]
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