What Makes People Engage With Math | Grant Sanderson | TEDxBerkeley

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 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 [Applause] you

Channel: TEDx Talks

Views: 842,699

Rating: 4.9870763 out of 5

Keywords: TEDxTalks, English, Education, Education reform, Math, Teaching

Id: s_L-fp8gDzY

Length: 19min 1sec (1141 seconds)

Published: Fri Mar 13 2020

Reddit Comments

[u/DifferentFlatworm8]

very nice and very true

[u/orenmn]

Fantastic. Wow.

"And at this point, it does not matter if the physics is idealized. If you have a SOUL, you have to know WHY!! Right?!" ~ Grant Sanderson

[u/HoleMax]

It's nice to see the face behind the voice.