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43: Interview II with Author Ben Orlin (Change is the Only Constant: the Wisdom of Calculus in a Madcap World)
399 Plays5 years ago
Ben Orlin has been a guest on the show before. He got famous with a blog called 'Math With Bad Drawings", which is what it says on the tin: he teaches mathematics using his humble drawing skills. His last book was a smorgasbord of different mathematical topics, but he recently came out with a new book 'Change is the Only Constant: the Wisdom of Calculus in a Madcap World', which focuses more on calculus itself.
This episode is distributed under a CC BY-SA license. For more info, visit creativecommons.org
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Transcript
Introduction to Ben Orlin and his new book
00:00:00
Speaker
Ben Orlin has been a guest on the show before. He got famous to the blog called Math with Bad Drawings, which is what it says on the tin. He teaches mathematics using his humble drawing skills. His last book was a smorgasbord of different mathematical topics, but he recently came out with a new book, Changes the Only Constant, The Wisdom of Calculus in a Madcap World, which focuses more on calculus itself.
00:00:27
Speaker
I'm Sophia and... And I'm Gabriel. And you're listening to Breaking Math. And of course we have on with us Ben Orlin, author. Welcome. Yeah, thanks so much for having me. Yeah, this is very, very exciting. Your second book. This is just really cool. I love that first book. It was so much fun.
00:00:43
Speaker
And I got to say, I was excited to see that you chose calculus as a topic. I've had a conversation with my brother who got his bachelor's in applied math. He actually has talked about wanting to be a Calc teacher
Why focus on calculus?
00:00:57
Speaker
specifically. This is the topic that made him fall in love with math. One of the questions I want to ask you is why you chose this topic specifically and also as a follow up if there were other topics you considered but ultimately chose this.
00:01:11
Speaker
Yeah, I think it was calculus from the start for me. Actually, I think I knew even before writing the first book that when I did the follow up, it was going to be about calculus.
00:01:18
Speaker
And the reason is that calculus sits at a very special place in the center of the American math curriculum in a way that it doesn't actually in every country. So I taught in the UK, and there, as in most countries, the math curriculum is just sort of integrated. Everything sort of sloshes together, and you'd learn a little bit. You might learn to differentiate polynomials. In fact, we taught that to 9th and 10th graders, but sort of before they'd seen anything in the way of limits, it was just a very different sequence than we do in the US.
00:01:44
Speaker
Anyway, in the US, calculus sits on this throne right there at the end of secondary school or the start of university. And it has, particularly has a reputation in the US as being this really fancy, exciting, scary subject. And I like, there are things I loved about it and things that I thought,
00:02:04
Speaker
the reputation of calculus gets wrong. And so I thought it would be an interesting topic to engage
Comparing math education: US vs UK
00:02:09
Speaker
with. Wow. I actually had no idea that math was taught substantially different from place to place. That's actually fascinating. That makes me, oh, so many questions. I guess one question would be, do you have a preference for how it's taught, say, in the US versus in England?
00:02:25
Speaker
I think no winds up being the answer. It's funny because there are these very visible differences like what order do you teach topics in and what's included in the curriculum and those feel like they should be the determining factors like those just feel like such big factors from the outside. But then when you actually get inside a classroom the things that wind up mattering
00:02:44
Speaker
are much more local. It's much more about, well, first of all, what's the relationship between the teacher and the students? Do they have a shared vision for what the students are learning? But also, what are the moves that the teacher is making? What did the discussions look like in the class? What specific problems are the students working on? So it ends up being the things that make math education work or not work, I think turn out to be very specific more than global.
Where to find the book
00:03:07
Speaker
And Ben, actually before we go deeper into the book or the process of creating the book, I'd love to ask you for our listeners, where can our listeners get a copy of your book and where can they read more about you and your work?
00:03:22
Speaker
Yeah, no, yes, yes, good point right before we get too far into the weeds. So the book is called Change is the Only Constant, and the subtitle is The Wisdom of Calculus in a Madcap World. And you can get it pretty much anywhere books are sold. I think they still have some, we did a signed pre-orders through a store called Majors and Quinn, M-A-G-E-R-S, and Quinn, Quinn like the name. That's in Minnesota, they ship anywhere. I think they've still got some signed copies left, or you can get it anywhere books are sold, you can get it at Amazon, Barnes and Noble, Target, wherever you want it from.
00:03:52
Speaker
Cool. Yeah, that's the world over, huh? Anyone in the world can, because they're very good at distributing, correct? Yeah, that's right. Yeah. They should be able to ship internationally too from majors in Quinn. Oh, fantastic. Next question I want to ask you again for our listeners, who specifically is
Making complex math accessible
00:04:07
Speaker
your target audience? Who would you recommend read your book?
00:04:11
Speaker
Yeah, so the book, I think, similar to the last one, actually. So Math with Bad Drawings, I wrote very much with a curious nonfiction reader in mind. I was picturing someone who's probably an adult, maybe has some vague memories of math in school that may or may not be fond. A curious, intelligent person who isn't necessarily, let's say, steeped in mathematics, and somebody spends a lot of time thinking about mathematics, usually, and so wants a fun experience of the subject.
00:04:41
Speaker
So I think the new book is similar, I think, in that regard. It's meant for a very, yeah, I would say an intelligent, curious, non-fiction reader with a sense of humor. Yeah, I noticed that your book was actually probably simple enough for a third grader to read it and get quite a bit out of it. Some of the stuff, I think on page 80-some where you
00:05:06
Speaker
start talking about like dimensionality is like something that'd be a little bit more difficult to get across to like a young person from the book but I mean it I was pretty like in awe of like how well it explained concept and also I laughed on page 44 when you talked about the second derivative of happiness in your experience in England would you like to talk about that at all?
00:05:27
Speaker
Oh, sure. Yeah. Yeah. Well, first of all, yeah, thanks. I'm glad that it strikes you as accessible. That's definitely something. That was really fun with the first book, actually. I was getting a lot of emails from parents who were saying, you know, I got this book for myself or I got it for my teenage kid. And then my 12-year-old or my 10-year-old got a hold of it and was enjoying it a lot. That makes me really happy to hear, I think, certainly the fact that I'm a teacher and enjoy working with kids, hopefully, I guess, winds up inflecting the books a little bit.
00:05:53
Speaker
Yeah, so I certainly hope that the calculus book can work that way, too. I think the pictures help. The pictures are a nice place for the adult reader to rest their eyes, hopefully, and then to make things a little more vivid and easy to follow. But I think also have to, hopefully, the nice side effect of engaging younger readers, too.
Practical applications of calculus
00:06:10
Speaker
So then the one, you mentioned the second derivative of happiness, right? That was what you were bringing up? Yeah. So that was actually a fun, I mean, in some ways that conversation that that grew out of is sort of emblematic of the book. This is while I was teaching in the UK. And I had a friend, James, who was an English teacher, and he's just a very curious guy. He's very happy to talk about pretty much anything.
00:06:32
Speaker
Um, and so he hadn't taken math since, you know, junior, senior year of high school. Uh, but we got to talking at some point where he was describing the shifting of his happiness. He was sort of, he was trying to find language to describe. You were just sitting around chatting, having tea because, you know, just the UK, this is what you do. Um, and he was trying to find some language to describe the changes in his happiness over time. So well, things have been getting better, but you know,
00:06:57
Speaker
They weren't so good a couple weeks ago and now they're getting better, but maybe are they getting better faster? And he was sort of straining for language that I recognized because I was about to walk out the door and teach calculus in a few minutes. The language that I recognize is derivatives. He was trying to describe rates of change.
00:07:13
Speaker
Um, and so that wound up being a, uh, that's I think chapter three in the book, uh, just sort of unpacking that conversation and using the derivative, which is usually used to describe stuff. We have data for, you know, you use derivatives to describe, um, physical motion or to describe inflation and an economy, um, or things like that, where we have some.
00:07:32
Speaker
real computations we can do. But applying those same ideas of derivative is just a rate of change. It's how fast something is changing, whether it's growing or shrinking. Applying that to something much fuzzier, like emotions, like how happy you're feeling. Yeah. And when you mentioned that, it strikes me that calculus has been around formally for something like, what, 350 years? And still, we don't have most people using general calculus terminology.
00:08:00
Speaker
Yeah, yeah, that's right. It's strange because the derivative really isn't so elusive a concept. You can make it really tricky if you want, right? So the way the derivative is often introduced is you say, okay, we want to know how something is changing at a particular moment, right? Like we want to know how fast a car is going right this instant. And then you immediately get into this lovely little paradox where you're like, well, how fast is the car going right this instant? Well, let's do a freeze frame on the car to see how fast it's going.
00:08:26
Speaker
Well, if you do a freeze frame, the car's not moving anymore, so it's very hard to say how fast it's going. The speed loses its meaning when you look at a single moment. Anyway, to a certain degree, that's the way I present it in the book, and that's often the way it's presented to students. It's this difficult concept that comes, the only way to understand it properly is through the careful development of the language of limits.
00:08:49
Speaker
And so I think that's why the language of calculus hasn't particularly penetrated and spread and isn't a universal inheritance for students, because we often hide it behind this kind of paywall of algebra and limits and some sort of tricky conceptual frameworks. But as you're saying, just a rate of change and how is the rate of change changing, which is the second derivative, those are actually pretty handy just everyday ideas.
00:09:17
Speaker
You had mentioned a second derivative. Real quick, I wanted to share some, I don't remember if we had discussed this on the previous episode. If we did, we can edit this out. But there was a fun fact I learned about derivatives. And I'm sure by now you may be aware of this fact. So of course, if we talk about position as our original, what we're talking about, what am I going to say here? If we start with position, right? It's a function of time. Position is a function of time. Then the first derivative of position is, of course, speed.
00:09:48
Speaker
Velocity. Velocity. Thank you. Thank you. Good Lord. Early in the morning for me. Sorry. As a former teacher, I should have got that. Velocity, of course, for our listeners being positive or negative, speed only being positive. Yes. Yeah, exactly. Okay. And then the derivative of velocity is, of course, acceleration. And then this is where things get a little funny. The derivative of acceleration is, what do you guys know of it? Jerk. Jerk, correct. And the derivative of jerk is?
00:10:14
Speaker
See, this is where I lose. I can't go past jerk, right? So jerk is our third derivative. So I don't know the name for the fourth derivative. Snap, crackle, pop, or the next three. Yeah, correct. That's literally what it is. So that shows that mathematicians do have humor as well. Now, in all of my studies of physics, and they're not entirely extensive, they are as much as a graduate student of engineering would study, I've never even used jerk, for instance. So I'm not sure why those exist. I mean, conceptually, they, you know, like, like,
00:10:42
Speaker
However fast, whatever level you're on, however fast that is changing. So I guess it could go on arbitrarily. But it is true that the terminologies chosen are snap and then crackle and then pop. So fun little trivia there. Right. Now I like that. It seems like in, at least in a calculus one course, if you're going to be taking more than two derivatives, you might as well be taking infinite derivatives.
00:11:04
Speaker
you may be talking about something like sine or cosine where you have this, the derivative starts repeating after some number of derivatives. Oh, you know where I saw the third derivative recently? I'm not going to remember the details of it, but I was reading the book, which is a collection of quanta pieces about mathematics. So I'm blanking on the name right now. I'm sorry, what pieces about mathematics?
00:11:27
Speaker
So, Quanta is a publication that's maybe five years old or so, five or six years old, which does a lot of really interesting reporting. They have some excellent writers who write for them. It does a lot of reporting on math and science, and particularly focuses on trying to do really honest
00:11:46
Speaker
rich math and science reporting. I think almost every publication out there, the New York Times will try to write pieces about if there's a really groundbreaking once or twice a decade kind of mathematical development, they might try to write a piece about it. But there's very few publications actually making an effort to cover news and research mathematics. And Quanta is really the best and to some degree the only source for that kind of news.
00:12:11
Speaker
Anyway, there's a handful of pieces in there that are about, I'm sure I'm going to botch this in the telling, but are about a sort of distribution that shows up in a lot of settings. Sort of anywhere you have a system that's kind of loosely connected, do you remember where, you know, sort of like the way buses might interact if you have bus drivers trying to be careful not to have their bus arrive too soon after the last bus.
00:12:36
Speaker
And there's some physical systems like this. Anyway, somewhere deep within this chapter, you wind up with a third derivative being the thing you need to describe sort of the differential equation that drives these systems.
00:12:48
Speaker
Interesting. And I did like about your book was the chapter where you go...
Visualization and real-world models in calculus
00:12:54
Speaker
I have a couple of questions relating to this in a second, but it's the whole thing where it was like the chain rule and not just the chain rule, but you were saying like the machine that creates pictures out of formulas and I would create like a grid for multiplication and for squaring, it would do some other thing and...
00:13:15
Speaker
Yeah, this chapter was inspired by a short story by a writer named Margaret St. Clair who wrote, I've only read a few of her stories, but she wrote this collection of stories in the maybe late 40s, early 50s that were about this sort of couple of the future called Una is the girl and Jic is the guy. Jic I think is a great name for a guy who you know is not going to be 100% sympathetic, just like Jic. I don't know, I'm not on board with a Jic.
00:13:43
Speaker
Anyway, so it's this sci-fi stories about this futuristic couple and they're sort of playing out these like domestic
00:13:50
Speaker
disputes and little squabbles. The stories are largely about appliances of the future. Anyway, and so one of the stories is about Jic, who's always trying to teach Una the math that he's enjoying and is curious about, brings home this mathematical visualizer machine, which is supposed to help you see any kind of mathematical idea. You put an equation and it'll show you what it means.
00:14:16
Speaker
Uh, which is sort of a funny story to read in 2019 because we sort of have that. I mean, we have Desmos, we have Wolfram Alpha. Like we really, we actually have some, some tools in a way that St. Clair couldn't have necessarily, uh, well, she did foresee, but could never have experienced herself when she was writing these stories. Um, we actually do have math, magical math visualization machines.
00:14:36
Speaker
And so anyway, what I read about in that chapter is the visualizations that they come up with to understand these calculus ideas. And what's strange about it is that these aren't visuals that show up in all that many calculus classes. We tend to do it just purely based on symbols and formalism in calculus. And so it's strange that even though the technological environment has advanced tremendously, we're teaching it kind of the same way that St. Clair was thinking of in, you know,
00:15:06
Speaker
1950 or whatever it was. Now we have a few questions that are just like, um, just kind of more open-ended questions, uh, based on those sort of things that are in your book. So, um, the book deception on, uh, the biography of a fad was the name of the chapter.
00:15:29
Speaker
Yeah, that's right. Where you talk about exponential growth and logistic growth, exponential being like compound interest and logistic being like a chemical reaction. First question is, could you explain just real quick what those are? And second of all, Moore's law, do you know how that might relate to which way you would classify that and which makes the most physical sense?
00:15:56
Speaker
Yeah, so in that chapter, what I wanted to do was sort of explore a really simple mathematical model for how a fad might work. And so basically the model that you come up with is what's called the logistic growth curve, where basically the idea is it starts out growing quite shallowly, but actually at a pretty high percentage rate. So you're starting from almost zero, but it's actually growing pretty fast if you look at it as a percentage.
00:16:20
Speaker
And then it continues to grow and grow and grow once you look at the graph. And it's kind of growing steeper and steeper and steeper. At some point, you hit a rate of maximum growth. And then it starts to level out.
00:16:30
Speaker
So that to me is a pretty good, I mean, you know, you think about what a fad is and that's what a fad is, right? It starts out small, it's not popular. It gets popular, it hits some point of like maximum hype and then it starts to level out and fewer people are popping on board to the fad. So I think you could draw graphs like that for, you know, Rubik's cubes were huge in the 1980s or like Beanie Babies in the 1990s or, you know, it doesn't have to be toys, sort of anything, you know, the musical artist or whatever you pick.
00:16:58
Speaker
So, I sort of liked that as a, and there's a simple differential equation to describe this, so I liked logistic growth as kind of a simple model of how fads work. Right, so you brought up Moore's law, which is, well, that's a good time to second actually. The trick with logistic growth, of course, is that at the beginning it looks a lot like exponential growth. It looks like the growth is getting faster and faster and faster the way that exponential growth does. And the problem, right, I guess what's tricky about it in practice is that exponential growth sort of can't last forever.
00:17:27
Speaker
Right. Like almost by definition in a universe with finite resources, things that look like the growing exponentially just can't. Um, so like for the first, I remember doing this with students in like 2010, 2011, the, I would show them data from Facebook, their number of users from like 2004 to 2008. Um, and it fit an exponential growth curve almost perfectly. It was an incredibly good model actually, because Facebook had gone from a few million users to tens of millions to at that point, I think around a billion.
00:17:55
Speaker
Um, so it looks almost perfectly exponential. And of course it hasn't been since then they've, you know, leveled off at two, two and a half billion. Cause you know, there's only so many people with, with internet access. Um, so everything, even if it looks exponential for a while has to run into constraints. Um, just so we live in a finite universe.
00:18:12
Speaker
Yeah, I always think that's interesting. Yeah, I don't know why logistic and exponential is always fascinating me, but I enjoyed that chapter quite a bit. And the second chapter I wanted to ask you about has to do with limits. And why does d of x squared kind of tend to disappear if you do calculus using just d of x's and multiplying it out straight?
00:18:40
Speaker
Right, yeah. So this is a good, right, maybe it's good to give some backdrop on how calculus is usually taught, sort of how it's structured, which is, you know, what are the problems you're looking at in calculus? Like we talked about before, it's rates of change. It's a lot of things happening, you know, an action that's unfolding over a tiny instant, you know, a fraction of a fraction of a second, or it's a physical process that's unfolding over a tiny, tiny bit of space or something like that.
00:19:06
Speaker
And so the way calculus is universally taught is you develop this idea of a limit, and it has this fancy notation that goes along with it. And then what that lets you do once you develop that notation and play out the consequences of it is it lets you plug in algebraic formulas and manipulate them very carefully. And you should get the right answers even if you don't totally know what limits are or why they work or what's going on.
00:19:31
Speaker
So I definitely didn't want it in a book full of cartoons, you know, that's meant for a general readership. I definitely didn't want to be trotting the reader through a lot of limit notation. So what I was hoping to do was come up with, you know, when I wanted to explain a concept to come up with a good visual language for that.
00:19:48
Speaker
The one that's out there is obviously not something I devised. It's been part of calculus since it was developed, is to think about little shapes, little rectangles. For example, take a square. If you've taken calculus, the derivative of x squared is 2x. That's one of the first facts you learn in calculus.
00:20:10
Speaker
And so what does that mean? Well, it means if you take a square and say it has side length X So X could be 5 or 6 or whatever at 5 by 5 square 6 by 6 square And you imagine that square growing just a little bit imagine X getting a little bigger So it goes from 5 to 5.001. Mm-hmm. And then the question is how much is the square growing by?
00:20:32
Speaker
And basically, I'm picturing it growing to the right and up. So I'll have a little bit of strip to the right and a little strip at the top where it's growing. And then up in the corner, there's this tiny, tiny little extra square up in the upper right-hand corner where it's grown as well. But that's so much smaller than the strips that you don't even really need to take into account in your calculations. Yeah, it always seems like you'd be subtracting a strip from a square, which is just a square.
00:21:02
Speaker
Yeah. Yeah. I think that's another way to think of it. Yeah. Which is weird because you're dealing with like the individual little like layers of square as it is. So it's always fun to me just to realize what level of infinitesimal that you're dealing with and how it all. Yeah, it's fun.
00:21:21
Speaker
Yeah, there's a great, so this is a book that kind of makes the rounds on the internet sometimes from the early 20th century called Calculus Made Easy, I think it's Sylvanius Thompson, I think is the author. And so it's this textbook from, yeah, I think the first edition was 1910 or 1913. But the funny thing about it, it reads beautifully now, it's actually totally accessible and in some ways better than, you know, modern calculus textbooks that you'll see. And one of the very first chapters is called On Different Degrees of Smallness.
00:21:49
Speaker
with the point being that sort of like we're trying to have this conversation in terms of geometry, but you can think of it in terms of say time, where if you're talking about millennia, right? If you're a historian or archeologist who thinks in terms of millennia, then one year is almost a negligible, it's like a rounding error, right? If you're a, I don't know, modern political historian, years are pretty significant, but one week is kind of a rounding error.
00:22:18
Speaker
Yeah. Right. And if you're planning, I don't know if you're like planning out your year, you guys are planning, you know, which, which guests to have on which podcast episodes and what, what topics you want to cover. Um, like a week is not a rounding error. A week is pretty important, but you know, an hour is a rounding error, you know, when you schedule the podcast for two or three. Anyway. So depending what scale you're looking at, different things look, look big and look small. Um, which is of course, not a, not a shocking truth, but turns out to be one of the kind of key ideas that's, that's in the engine of calculus.
00:22:47
Speaker
Yeah, and speaking of the engine of calculus, you went into one optimizing problem, the one with the optimal size city being made out of strips of ox. If I remember correctly, the ox was cut into 60 strips and the strips were used to outline a city. It was based on some real thing.
00:23:09
Speaker
Yeah, yeah. So this is a story, this is one of those ones that kind of circulates as folklore and math departments. I don't remember who I, I mean, sure I heard it from my wife first, but I don't know where she heard it from. But this is a story that's actually in the Aeneid by Virgil, or at least is sort of alluded to in the Aeneid. And the idea is that an exiled princess named Alyssa, I think in the original versions, shows up on the Mediterranean coast.
00:23:34
Speaker
the northern shore of Africa, and somehow manages to negotiate. I don't quite understand what these negotiations would look like. Manages to negotiate for as much land to keep for herself as she can cover with an oxide, which I've never been in a situation where that was close to happening to me. I don't know what the alternatives were, like as much land you can cover with some- Like I did two-thirds of my transactions are oxide-based.
00:23:58
Speaker
That makes sense. Yeah. So yeah, I spent some time in New Mexico, but I haven't been there in a little bit. So that must be what it is. And usually I'm in Santa Fe, not Albuquerque. So, so no, no, no, no, no. Next time in Albuquerque. So anyway, so there went up in this, in this oxide situation where Alyssa has a piece of oxide and she gets as much land as it can cover.
00:24:17
Speaker
Um, so, yeah, who knows what language this negotiation was unfolding in, but she decides to interpret it a little loosely as rather than how much land can she cover with an oxide. It's how much, uh, land can she encircle with an oxide. So she's going to cut the oxide into little strips, you know, like, like thread almost like rope.
00:24:36
Speaker
and then arrange the rope so that she can cover, she can enclose more land. I guess the idea being that if she were to cover, it sort of makes sense. It would be very weird if she arranges it in a giant circle. It would be very weird to give her a circular strip, but she doesn't get the land inside the circle. That's still there. So anyway, it would be a strange shaped country.
00:24:58
Speaker
Anyway, so then the problem is basically, and this is something that, a simpler version of this you do in every intro calculus course, is what's the best you can do with this rope?
Exploring optimization in history
00:25:08
Speaker
You've got some amount of rope, and what's the best area? What's the biggest area that you can close off using that rope?
00:25:14
Speaker
Yeah, and I thought it was interesting that the problem, I mean, it doesn't have anything to do with the calculus of variations, but I thought that maybe we're on optimizing problems, wanting to know if you have any future books that might have calculus of variations in them.
00:25:31
Speaker
Yeah, this one, actually, both of the optimization problems that I discussed in this book sort of do lend themselves a little better to calculus of variation. So this one, the one with Alyssa and the rope, the version I discussed in the book is mostly just assuming she's going to do a rectangle, because that's the one that an introductory calculus student can do, and that's the one that's a little easier to kind of graph and make sense of. So you sort of think about different rectangles she could arrange it in.
00:25:56
Speaker
And if you play around with it, and I've done experiments like this with middle schoolers, a long thin rectangle is a very bad idea because you're spending all your rope on the long dimension. The long dimension is not helping you very much. Every extra little bit in the long dimension is not buying you very much extra area. So you really should be investing more in the short dimension of the rectangle.
00:26:19
Speaker
So long skinny rectangles are bad, and it turns out the best rectangle, not a huge surprise, is a square. And then where it gets interesting is where you, and this is again what you tend to do in a calculus course, is if you, instead of building it out in the open space, you build it along the water, where since one of the walls is now made of the beach, you only need to build three walls, two coming away from the beach and one parallel to the beach.
00:26:42
Speaker
And that actually totally changes the solution. A square is no longer the best you can do because you're sort of spending extra rope building away from the beach with those two walls when it would be easier to just build longer along the beach because you only have to use one little scrap of rope to build along the beach. Anyway, that's still just using rectangles, though. So if you want to talk about all shapes, then you need calculus of variations. Which in the book, it wasn't a semicircle.
00:27:11
Speaker
Yeah, it's sort of a sleight of hand. I give the solution that's the rectangle solution. And then I think in a sentence or two, I say, well, and you could actually do even better, which would be a semicircle. I don't actually go through that, because it would take calculus of variations. So that was one of the pleasures of getting to do a book like this that's full of cartoons, is you can make all kinds of leaps that you couldn't make in a class. If I did that for some students, they'd be like, wait, where did the semicircle come from? Show me the equations. How did you do it?
00:27:39
Speaker
Technically, it would be too hard that students wouldn't have the techniques to take those derivatives and to solve it in this more general form of calculus. But in a book, the idea makes sense, so all I really need to do is lead the reader through the concepts.
00:27:55
Speaker
So, as a former middle school math teacher, I never taught calculus, but I know that calculus itself, in the American system, is split up into several skills that are spread throughout Calculus I and Calculus II and Calculus
Key calculus components taught in courses
00:28:08
Speaker
III. There are so many things. Obviously, there's the fundamental theorem of calculus. There is something that we will talk about called the chain rule. There's, what else? It's been a while since I've done Calc I. That's it. That's it. It's just those two things, actually. I don't know how they drag it out so much.
00:28:24
Speaker
So we, you know, of course, we got real inspired reading your book and we thought, well, let's try to think of our own clever analogies to describe some of these components of calculus for somebody who has never been in calculus. And I'm always going to use, you know, my nieces and nephews who are between the ages of three and 14 right now, and none of them have had calculus yet.
00:28:44
Speaker
So, you know, I think with chain rule, first I have to explain to them what it is and for our listeners as well, but then explain our analogy and how it illustrates it. And actually, I shouldn't say ours because, Sophia, you thought of all of this.
00:28:58
Speaker
Yeah, so I just thought it'd be fun because I know that you come up with a lot of... The happiness thing was really what inspired me to do this. So we have f of g of x where f is the sound wave frequency on a record and g is the number of rotations and radians of the record.
00:29:19
Speaker
And then f of g of t, of course, would be the frequency that the needle makes when it's sweeping the record at that point. And you know that its derivative must be proportional to the derivative of the speed of the sweeper, because if the pitch is constant on the record, then if the speed doesn't change, then the pitch doesn't change. So yeah, that checks out. And it must be proportional to the derivative
00:29:46
Speaker
of f at g of t, which would be the sound at the position that the needle would be at at time t. Because if the speed that it's moving around is constant, then it's only the etchings on the record that contributed the final pitch. So then you get f of g prime of x is equal to g prime of x times f prime of g of x.
00:30:09
Speaker
Now, real quick, I wanted to explain, again, what we didn't say quite yet was for our listeners who had never even heard of the chain rule. The chain rule is when you have a function, like a function of a function, like Russian dolls almost, right? You know, like, and again, as you said, so like, if you start off with the function g of x, you know, g of x for those who are in algebra, they know that it means you do something to x, you know, it could be anything at all.
00:30:34
Speaker
But then the Russian doll part happens when you have another function of that function So you do something to the process that's doing something to x and then when you take a derivative of that That requires the chain rule. I'm just I'm trying to think about how I'd
00:30:51
Speaker
Oh yeah, and maybe, and this is one question that I have been with your experience, like making things really accessible, like especially with all your work on calculus, how might you improve that analogy, putting on the spot?
00:31:06
Speaker
I like the analogy a lot. Yeah, I got to think it through more, but I think having a concrete physical system where we're talking about where we've got two functions that are very clearly meaningful. And so we can think of the chain rule as... I think where the chain rule gets tricky, as with a lot of things in math and in calculus particularly, is when it's very decontextualized.
00:31:28
Speaker
And so you're learning it as a rule about symbols, basically, rather than a rule about objects or physical systems. A lot of what happens in math class is you've got people who enjoy learning rules about how to manipulate symbols.
00:31:44
Speaker
and people who don't enjoy that and one can enjoy it for a variety of reasons or not enjoy it for a variety of reasons but math class winds up separating people into those who enjoy these these symbol manipulation procedures and those who were not on board for it so I think having physical illustrations for
00:32:01
Speaker
why those rules are true or why at least they might be true or seem intuitive to try to build some intuition for them is definitely a big part of teaching. Nice. I think what we'll probably do is send you the written form of what we just told you about explaining the chain rule using sound and a record player. Also, as you had mentioned earlier, it's very important to say that the record wouldn't be like a song, it would be a monotone. No, it would be a song. If it were a monotone, then it demonstrates that is proportional to G Prime.
00:32:31
Speaker
Okay, okay, got it. Yeah, that one certainly takes some sussing out, but it's a fun analogy. Now, how about we try, I'll go ahead and toss you this one. If you can explain an analogy with a UV substitution, go for it.
00:32:46
Speaker
uh right substitution being uh right we're kidding we're kidding right okay we spent so long trying to we spent a long time trying to think of a story or an analogy or something that would explain you dude we failed it's that's not that's not possible to do on a well i shouldn't say that i should challenge you we haven't proven that it's possible
00:33:07
Speaker
That's correct. Thank you. Thank you. For our listeners, if anyone can think of a cool story or an analogy to explain you substitution to a middle schooler, please send it to us. We'd love to hear it.
00:33:20
Speaker
Yeah. I think one way, this doesn't capture the whole breadth of the substitution, but I think even in like Algebra 1, you run into situations where relabeling your variable makes things easier. Yeah. So if you're looking at, if the equation you're trying to solve is 2x plus 1 plus 5,
00:33:42
Speaker
groups of 2x plus 1 plus 20 groups of 2x plus 1 equals 17 groups of 2x plus 1 minus 4 or something. If you keep having the same thing appear over and over, it would be easier just to take 2x plus 1 and call it y or call it t or something. To some extent, u substitution, this doesn't capture the geometry of it, but you can think of it as just relabeling. Relabeling is something we do across mathematics.
00:34:07
Speaker
And a lot of these can get very complex. Like, I mean, even just in regular algebra, like you've seen the derivation of the cubic, right? The general cubic. At some point, yeah, I certainly couldn't reproduce it. At one point you replace one of the variables with W cubed minus three over W cubed or something like that. It's pretty insane. Or one over W cubed.
00:34:36
Speaker
Now it's time for find the calculus, a game show where we tell you random things and you have to tell us how calculus relates to them. You win as many points as you win. Okay, infinitesimal quantities of points, I presume. Yes. Oh, yes. Yeah, exactly. Right. Yeah. I mean, the jigs up, but yeah, I guess we'll continue, but no, it's kidding. But okay. A kayak that has to avoid whirlpools.
00:35:02
Speaker
Oh, interesting, a kayak that has to avoid whirlpools. Right, I mean, whirlpools, if I were, I feel like I'm not, my multivariable is not at the top of my brain right now, because the book is mostly about single variable calculus. But a whirlpool is essentially a sink, right? It's like, if you think about this two-dimensional world of the water, the surface of the water, the whirlpool stuff swirls around it and then gradually narrows in, you know, sort of, you know, think of it almost like a black hole in the whirlpool.
00:35:31
Speaker
So, that's it. Obviously, actual whirlpools are vastly more complicated than that. But that's the first thing that comes to mind for me is basically you're navigating a vector field where every point in the water is pushing you a certain direction. And that's definitely a calculus problem. Awesome. Now, how about a garden infested by termites? Garden infested by termites. Well, here, clearly, we're talking about population growth.
00:35:59
Speaker
And so how do we model population growth? Well, with some differential equations, and we hope that our termite infestation has already hit the logistic growth curve, has already maxed out as opposed to having more exponential growth left to do. Osmosis. Osmosis, interesting.
00:36:18
Speaker
Right. So osmosis, we're talking about concentrations when I like to think back to my high school chemistry. So we're talking about where you have more salt on one side of a membrane than the other side, but the salt can't move across the membrane. So instead the water moves across the membrane to reach equilibrium. And I don't know the differential equations that you would write down for this.
00:36:41
Speaker
And to some extent, this is a cheating answer because everything in the universe can probably be described with some differential equation or another, whether you can solve it or not. But certainly, there should be laws, equations governing how the exchange of materials across that boundary.
00:36:59
Speaker
And one maybe thing to remark upon is that the differential equations that you have in your book are very simple ones and obviously they get way more complex, but I always struck me how much complex you get even out of the simple ones.
00:37:14
Speaker
Yeah, thanks. Yeah, yeah. Definitely in the book, I try to find stuff that's ... I know a lot of your listeners obviously are people who are already quite invested in math and curious about math. And so I try even though I'm writing to be accessible to an audience that isn't familiar with calculus, maybe never even took calculus. I mean, very much who I want the book to be accessible to. I think for those who have taken calculus and do know calculus, there should be lots of interesting new different connections.
00:37:43
Speaker
Yeah. Evil Knievel. Evil Knievel, right? Clearly, as he's going down the ramp, clearly, it's just all equations in his mind. He's just calculating derivatives and intervals left and right. I mean, projectile motion is one of the elegant early applications of calculus.
00:38:03
Speaker
What you can do with calculus is you can start from the idea that an object flying through the air, as always, we ignore air resistance. Gravity is the only force acting on that object. There's a constant acceleration due to gravity as it accelerates downward. Then from there, with some data, you can figure out the trajectory over time.
00:38:24
Speaker
Awesome, now we just have two more and we might cut a few of these out just for whatever, but I think it's good to have data more than not have data. Baking souffle.
00:38:35
Speaker
Baking a souffle. I would love to know how to bake a souffle. They're really delicate, right? They collapse if you make any noise. Or is that just cartoons? I think they collapse if you change the temperature too much in the oven. It has to be a very constant temperature and the volume increases for a very long time and has a certain softness to it.
00:38:57
Speaker
Until it hardens up. That sounds good. That sounds right. I guess come to think of it, my experience with souffles is 90% through cartoons, 10% through an episode of British Bake Off that I saw in the last year. What do they have to do with calculus? I would say souffles are delicious as is calculus.
00:39:20
Speaker
Two points for creativity or? Yeah, you know, you get a whole point. Wait a minute. Isn't there a calculus problem like Gabriel's horn or something like that where one thing goes to zero, but then something else goes to infinity where the actual relevance goes to zero, but the creativity goes to infinity. So I'll take that. Yeah. Right. Right. And the name named in your honor. So yeah. And then if you just take like a direct Delta function or is that it blown where that still turns into one point after all?
00:39:47
Speaker
Although I do think that Gabriel's horn is the function that when it's rotated around what is put in a lathe around the x-axis, it has finite volume but infinite surface area. OK, got it. Yeah, Gabriel's horn is actually one of the last chapters in the book is about Gabriel's horn, which is this cool shape that has the feature. It's sort of this infinitely long, spindly looking thing that stretches, is 3D shape stretching all the way down the x-axis forever and ever. And it has the weird property that it has finite volume,
00:40:17
Speaker
but infinite surface area, which is to say, since it has finite volume, one could imagine filling it with paint, you know, because that's what volume is, right? Paint fills up some volume, but because it is infinite surface area, you could never paint it. Yeah. Wow. That's crazy. Which is a weird, anyway, one of the chapters in the book is sort of about trying to unpack that paradox and explain why it's not quite as paradoxical as it sounds.
00:40:40
Speaker
But if they want to find that, they can find it in your book and they can find that
Ben Orlin's online presence
00:40:45
Speaker
where? Anywhere, yeah. So change is the only constant. The Wisdom of Calculus in a madcap world. Ben Orland is me. I'm the author and also the illustrator. So there's a lot of very bad cartoons in there. What's your favorite cartoon in the book? Oh, good question. Let me grab the book. So one that I remember enjoying was drawing. There's a chapter about Sherlock Holmes.
00:41:07
Speaker
there's this story where he tries to solve this problem involving a bicycle and gets it wrong and turns out actually by thinking about tangent lines, you can solve the problem. Anyway, Sherlock Holmes is really fun to draw because Sherlock Holmes is basically a person holding a magnifying glass and with a weird hat on. Oh, no, holding a pipe. I'm sorry. Yes, that's how I was drawn is with a pipe. So pipes are pretty fun to draw and his weird hat is fun to draw. And so like I only do stick figures. I have very limited drawing skills. So I would say drawing Sherlock Holmes is probably my highlight.
00:41:38
Speaker
Let's see. We asked you, do you want to plug your Twitter, your website? Sure. Yeah. Yeah. So my, my blog is math with bad drawings, just math and bad drawings.com. You can follow it on Facebook. There's like 20,000 people following on Facebook. And there's, I post cartoons there. Facebook's algorithm basically refuses to show you links to the blog, but we'll show you cartoons. So if you follow it on Facebook, you'll see some cartoons. And if you follow me on Twitter, I'm just Ben Orlan on Twitter, B N O R L I N.
00:42:06
Speaker
There are also, I post cartoons and links to blog posts and try to share other people's work. If you're not on Twitter, there's actually a really lovely math and math education community on Twitter, which Twitter has its drawbacks, but also has introduced me to a lot of really wonderful writers and creators and interesting people. Absolutely. Awesome. Thank you so much for being on the show. Yeah, thank you so much. Ben, it's always a pleasure. Yeah.