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Life Imitates Math

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Life Imitates Math

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Life Imitates Math

Life Imitates Math

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In his new book, The Calculus of Friendship, math professor and writer Steven Strogatz looks back on his 30-year correspondence with his high school math teacher. Can calculus, differential equations and chaos theory help explain the complex nature of human relationships?

JOE PALCA, host:

From NPR, this is SCIENCE FRIDAY. I'm Joe Palca. Up next, how life imitates math. In his new book, Steven Strogatz looks back on the 30-year correspondence he maintained with his high school math teacher.

Now, for some of us, high school math and the teachers that taught it conjures up feelings of dread and anxiety, of sweaty palms and standing at a chalkboard trying to solve for X, but in this book, and in his new series of columns in the New York Times, Steven Strogatz makes revisiting the math of our high school days a pleasant experience.

Strogatz is a professor of applied mathematics at Cornell University. His book is called "The Calculus of Friendship: What a Teacher and a Student Learned About Life While Corresponding About Math." And he joins me from the Cornell Campus. Welcome back to SCIENCE FRIDAY, Dr. Strogatz.

Dr. STEVEN STROGATZ (Professor of Applied Mathematics, Cornell University; Author, "The Calculus of Friendship: What a Teacher and a Student Learned About Life While Corresponding About Math"): Well, thanks very much, Joe.

PALCA: Nice to talk to you, and if everyone wants to join this conversation and hopefully doesn't want to complain about math, because that's not what we're here to talk about, the number is 800-989-8255, 800-989-TALK.

And I'm curious. I mean, I've been looking at these New York Times columns, and I'm wondering: What are you aiming for here? Are you trying to convince everybody that math is a good thing or a cool thing or a fun thing, or are you just having a good time? Or what's going on?

Dr. STROGATZ: It's a little bit tricky for me to say. I'm not exactly sure what I'm doing myself. I'm trying.

(Soundbite of laughter)

Dr. STROGATZ: You know, maybe it's an attempt to get people see, the thing is it's not remedial. I'm not trying to re-teach math per se. It's not tutorial. I'm trying to give people a better feeling about what math is, why mathematicians do it, why we and also people who love math, why we love it.

And I think maybe if I could just sum it up this way, I was just having lunch with a neighbor down the street, and she put it to me this way. She said when she reads the columns, they make her want to like math.

(Soundbite of laughter)

Dr. STROGATZ: I mean, in other words, she still doesn't like it, but they made her want to like math.

PALCA: But one of the problems, and tell me how you struggled with this in the book and to some degree in the column: Okay, so we all know integers, and we know the signs of multiplication and division, but pretty quickly, you get to sigma and epsilon and delta and, you know, integral functions, and they just look like well, they look like Greek for want of a better term...

(Soundbite of laughter)

PALCA: ...but they're very confusing to people who haven't learned about them. So how do you tell them what's behind those symbols?

Dr. STROGATZ: That's the question. I'm going to try to do it all sorts of different ways. I mean, this is a column for adults, for one thing. It's not really for kids. So I can make use of people's knowledge of philosophy or pop culture or stories they've heard in their life or history.

So to me, math is connected to everything, and by trying to pitch it that way, I think maybe that can draw people in.

PALCA: What kind of response have you been getting?

Dr. STROGATZ: Well, to me it's a shock. You know, the Web can be a cruel place.

(Soundbite of laughter)

Dr. STROGATZ: With very opinionated people. Everybody knows more than the person who's writing. So I was expecting the usual, let's say, lively discussion.

PALCA: Yeah, okay.

Dr. STROGATZ: And in fact well, I'm just really taken aback with how appreciative people are. There's a kind of feeling of gratitude that comes from the comments that I really didn't anticipate.

PALCA: I think that there's a certain willingness of people to complain and say what's wrong, but the fact that you're getting positive comments is really telling, because I think, I think what you've accomplished is you've sort of brought people into a world that they normally are excluded from, and you've done it in a way that doesn't scare them, and I think that's really something to be proud of.

Dr. STROGATZ: Well, thanks, Joe. It's an interesting point, though. This word you're mentioning, exclude, I think that's part of what the thanks is about, that people do feel excluded from math, and some of them don't care. I mean, they weren't interested anyway.

But it a lot of the people seem to have a real hunger to understand, and they don't like the feeling of alienation, and they would - they're curious about what's this all about, and they like the idea that they might have a second chance to appreciate it now.

PALCA: Well, that's an interesting point. Where do you think people go off the rails when it comes to math? I mean, if you're teaching math in college, you're already talking to people who have selected to be interested in math for a certain fraction of their life ha, ha, but...

(Soundbite of laughter)

PALCA: Sorry.

Dr. STROGATZ: That works with me.

(Soundbite of laughter)

PALCA: But, you know, when do people get when does this feeling of exclusion start, because I like math, but I got through calculus, and then I was done. I mean, nobody asked me to do any more, and I didn't particularly have to do any more, so I didn't do any more.

Dr. STROGATZ: Well, there's I would say everyone hits the wall somewhere. That is so for some people, it's long division. You know, it's interesting to me the way you put it earlier, that for you, it was something about in the range of the Greek letters. Once you got to the sigmas and epsilons, it sounds like you started although you say you like calculus. But a lot of people don't get that far.

For many people, it's X. You know, as soon as they took algebra, and now we're not talking about a number, like three, but we're talking about a mystery number that's hidden in a black box that's called X. That is very abstract, and it bothers for a lot of people, that's it. Then they're dead after they see X.

Other people, it's the proofs and the long names, like the Pythagorean Theorem. Some people do fine in high school, and then as soon as they hit calculus, they say that was it for me.

So what might come as a little surprise to some of your listeners is that I shouldn't really speak on behalf of all mathematicians, but let me just say for my own self that there are walls that I hit.

You know, I'm a professional. I've been studying this subject my whole life, and there are parts of math that people can explain to me as much as they want, and I never understand them. So I have my wall. I think everybody has a limit.

PALCA: Well, it may I'm wondering. Do you think it might be that mathematicians by nature are not necessarily considered to be the most gregarious, outgoing group of people?

Dr. STROGATZ: Well, there is that knock. I don't know if that's so far. I mean, it is true that some of us are more interested in ideas than people, you know, or are happier with solitude than in company, but certainly there are very gregarious mathematicians. There are some who are, you might even say sexy or attractive. You know, so we remember Russell Crowe, you know, in "A Beautiful Mind"? He can clean up very nicely.

PALCA: That's true, that's true.

Dr. STROGATZ: And John Nash, the man he was portraying, was apparently a very dashing figure as a young man.

PALCA: Right, but you do tell the joke of how you can tell the difference between an extroverted mathematician and an introverted, and the answer?

Dr. STROGATZ: Yeah, the answer is that the extroverted mathematician stares at your shoes when he's talking to you.

(Soundbite of laughter)

PALCA: I love that. I think that's really a great way to put it. All right, well, we're talking about math and people's attraction and avoidance. It's one of those love-hate or fear-and-interest maybe we should describe math as, like, you know, bungee jumping. You know, you take a chance, you jump off and see what happens.

But anyway, you are welcome to join the conversation, 800-989-8255. And let's take a call now from Charla(ph) in Oklahoma City. You're on the air, Charla.

CHARLA (Caller): Hello. When I heard it was Math Day, I was thrilled. I love math, and I am a home-school teacher, and I teach at a co-op, and so I always try to get kids interested in math right away. And of course, their questions -their first comment, before any questions is: Why do I have to learn this?

You know, and so one of my answers has been to them that, like in algebra, you always have your unknown, you always have your X, and in algebra, we always try to get what we know on one side of the equation and what we don't know on the other.

And so I tell them, you're going to use that for the rest of your life, whether it's when and where you decide to go to college, what kind of career you're going to have. You need to always start separating what you know from what you don't know to make a good decision.

And so I think that sometimes helps them at least get started, you know, and then the other comment I wanted to make was, I always - I tell them that each and every one of them can understand mathematics, but there's no one who really cannot understand mathematics. It's that we all, like teachers are like transmitters and kids are like receivers, and sometimes, I think that some math teachers just don't transmit on a wavelength that that receiver gets it. But there is someone out there who can transmit to their on their wavelength.

PALCA: Okay...

CHARLA: So I think that if we you know, because, like, even in the home-schooling community, you look at different curriculums, and some of them are, you know, more kinesthetic learners, you know, have to learn by with manipulatives and putting them all together, which I can't stand. You know, I'm a traditional math person, but...

PALCA: Well, I guess you have the luxury in home-schooling in being able to tailor the message a little bit, but I think the point is interesting that there are different techniques that work for different people, right?

Dr. STROGATZ: Sure, that's right. I mean, I wanted to go back to Charla's point that her students often ask here: Where am I ever going to use this, or why am I learning math? And so she made the point that, you know, you may in life, it's often helpful to separate what you know from what you don't know, but I would also add a few other reasons that math is appealing and valuable.

There's some people would say it's important just to teach logical thinking in general. That could be one argument. Another is that some of the most fascinating professions you could have in this world require and benefit from a real good understanding of math, whether you want to go into computers or any engineering or science field or finance or biotech or high-tech with computers.

I mean, if you decide early on that you hate math, and you can't do it, then you're really cutting yourself off from many interesting careers. But I don't want this to just be about careers, because in my heart - and I think in most mathematicians' heart - the real reason that we do math and that we hope our students will learn to love math is the same reason that we want people to love music and poetry and great movies - that this makes your life richer. I mean, these are beautiful things.

You don't usually hear people say, why do I need music? Well, we know why we need music. It's good for our soul. It makes - gives us pleasure. It makes us feel things. And if math is taught right, it can do the same things.

PALCA: And - well, it's interesting you say that because there - I mean, clearly - and people have begun to realize this in books and, I think, in art, there is beauty in math. I mean, in fact, there was this diagram that went along with the most impossible story that I've ever tried to tell about math, was is this thing about Lie symmetries, and these multidimensional symmetries that you - but there's a beautiful picture to go with it. So it makes you kind of wonder, boy, if I could understand that, maybe I'll know where that picture came from.

Dr. STROGATZ: Uh-huh. Okay. That's some pretty deep stuff, Joe...

PALCA: Yeah.

Dr. STROGATZ: ...Lie symmetries. That is a very advance math.

PALCA: Well, it's very odd, because I still feel that at some point - and this is the challenge. That's why I'm so interested that you're doing this in a popular setting. Because the challenge is you go down a path, and then all of a sudden you want to explain something and you wind up having to wave your arms because there's so much technical - I mean, somebody who was very involved in Lie symmetries tried to explain it to me, and then basically said, it's really complicated and it has to do with symmetry. And that's about as far as we could go.

And that's why I'm saying if you're trying to do this for a popular audience, how do you - I mean, the problems you take on, you know, the monk wandering up the hillside and coming down, whether theyll cross at the same time and - at some point, can you prove it? Those are sort of easy questions to visualize. Are they deep math?

Dr. STROGATZ: Well, there are - I think that for virtually every big concept in math, there is some homey way of looking at it, there really is. So, like back on this question of symmetry, I'm planning to write about symmetry a few weeks from now and I'll be talking about flipping your mattress. You know that, yeah, you're supposed to flip your mattress. A lot of people don't bother to do it, but it's supposed to help you get more even wear out of the mattress. And so, once you start thinking about it, you realize there's four configurations of the mattress that you can put the - like you could interchange the head and the foot or you can flip it over...

PALCA: Mm-hmm.

Dr. STROGATZ: ...or you can interchange the head and the foot and flip it over. I mean, if you think about it really the mattress is a rectangle. It's got a top, a bottom and a head and a foot. And so, just visualizing it, there are four ways your mattress could be arranged, assuming that you've got, like, a bed frame and you can't put it sideways. So then the question is what's the best way to get even use out of your mattress? And that turns out to be a problem in what mathematicians called group theory, which is the math of symmetry.

And - now here's the part that I think is spooky and magnificent, which is that this question about mattress flipping is related - as Ill try to explain when the time comes - to the symmetry of water molecules, I mean, the deep understanding of what makes water so remarkable and also why water is a greenhouse gas.

I mean, earlier we were talking about possible climate change in connection to polar bears. The idea that you can connect climate change, water molecules and mattress flipping through this abstract lens of group theory, that's the kind of transcendent, beautiful unity that comes when you understand math.

PALCA: We're talking with Steven Strogatz about the beauty of math. You can join us at 800-989-8255. I'm Joe Palca and this SCIENCE FRIDAY from NPR.

I hope you're going to talk about rotating tires too because that's - I mean, another symmetry we all have to deal with.

Dr. STROGATZ: Absolutely.

PALCA: Okay.

(Soundbite of laughter)

PALCA: Let's make another call now and - take a call and go to Richard(ph) in Syracuse, New York. Richard, you're on the air.

RICHARD (Caller): Yeah. Thank you for taking my call.

PALCA: Sure.

RICHARD: I've had a lot of math over my life, and it - somewhere along the way, it kind of occurred to me that math is more a language than anything else. We go through our life, sort of intuitively, doing mathematical things and working within the laws. But, unless we have mastered the language of mathematics, we're unable to converse with people about it. And so, we just kind of see these mathematical things and sort of brush them aside or take them for granted, but we never really discuss them. And the language is what gives us that ability to discuss it. I just wonder what's your thought about that?

PALCA: Okay. Thanks, Richard.

RICHARD: Yeah.

PALCA: Go ahead, Steven Strogatz.

Dr. STROGATZ: Yes, math is certainly a language and it's also more than a language. There are truths that math can express about itself that go beyond anything that I know of in language. I mean, there's a famous unsolved problem in math, to this day, no one knows how to prove this - that you can take any even number and it's the sum of two prime numbers. Like, for instance - let's see if I can do one in my head.

If I take 32, that's an even number, it's 19 plus 13. Those are both prime numbers, right? They can't be divided by anything except themselves and one. So - and people have checked even numbers out to billions and trillions and it seems to be true, that you can always find two prime numbers that will add up to any given even number, but nobody knows a proof. Now, if that statement is correct, that's a fact that is out there, it's more than something, you know, in language.

This is - in other words, math has content beyond its just linguistic aspect. So it's a language but it's more. It's really - I dont know what it's like. It's an art. It's a science. It's a game. It's a language. It's math.

(Soundbite of laughter)

PALCA: So, you know, I had - actually, again, I was putting this question to a mathematician who was interested in prime numbers and finding the largest ones, the Mersenne primes was one of the...

Dr. STROGATZ: Yes.

PALCA: And he was saying, well, you know, he goes to the National Gallery in Washington - not the national - I mean, the Natural History Museum in Washington on the mall, the Smithsonian, and they have the Hope Diamond. And nobody goes in there and says, well, what's so interesting about a big diamond? Who cares?

(Soundbite of laughter)

PALCA: And it was his point about primes. He said, they're just they're interesting. I mean, they're the I loved that he said they're the jewels of mathematics. And I thought...

Dr. STROGATZ: Oh, that's nice.

PALCA: ...what a nice way to put it. We have to talk some more about this book. We have to take a break in just a minute, but in 20 seconds, before we take a break, why did you write it?

Dr. STROGATZ: This book about...

PALCA: ...you correspondence with your math teacher.

Dr. STROGATZ: Well, it's partly a tribute to a magnificent teacher, and I want people to think about the teachers in their lives and maybe even consider thanking them for all they did.

PALCA: Yeah. Well, it seems as if you got that chance - and he's okay with this book, I presume.

Dr. STROGATZ: Mm-hmm.

PALCA: All right. Well, that's nice.

Dr. STROGATZ: Yeah, he's...

PALCA: No it's a lovely book. And again, there are symbols in there. So pick it up, don't be scared...

(Soundbite of laughter)

PALCA: ...there's some math in there. Anyway, we're talking with Steven Strogatz about math, and relationships and the interesting things that the math world has for mathematicians and those of us who are just - mathematicians and those who love them, let's put it that way. We'll be back after a short break. Stay with us.

(Soundbite of music)

PALCA: From NPR, this is SCIENCE FRIDAY. I'm Joe Palca.

We're talking this hour about math, relationships and the places where they meet. My guest is Steven Strogatz, he's a book mathematician - a book mathematician, yes - a book author and now an author and a columnist for The New York Times. You can see his columns on Monday.

And before we leave, we started to talk about your new book about your relationship with your math teacher. I think one of the things that happens is the book in the book is, you can clearly see the delight that each of you takes in working out problems. And it was a tie that I mean, the relationship changed, you were the student and he was the teacher and then you became the big math professor and he was the high school teacher, but the delight in the problems stayed all through your - stays through your lives.

Dr. STROGATZ: Uh-huh. It's true. It was it's I should speak about him in the present tense, he still alive, Mr. Joffray, is his name, Don Joffray, and he's going to be turning 81 in March. So we have stayed in touch since I was in high school. And it's a it was for all these years, a relationship based on a mutual love of math problems, but it also became something much deeper as time has gone on. And we've gotten to know each other, but it took awhile.

You know, when my wife first heard that I had at that time something like 25 years of letters with my teacher, she said, oh, that's so nice. You must know so much about him, you must know everything about him. And I just had to shake my head and say, well, you know, we really just write about math problems. And she said, oh, that's such a guy thing...

PALCA: Uh-huh.

Dr. STROGATZ: ...you know, that is how could it be that you would actually, there is something to that that men can play basketball each weekend with each other and not know how many kids the other guy has...

PALCA: Yup.

Dr. STROGATZ: ...but on the other hand, we did gradually deepen this relationship to something we're we finally did open up to each other. But truly it is based on what you said, Joe, that it was about the delight in posing problems, thinking about them. And so it's a book that's kind of a collection of letters, but also more than that because they try to explain how calculus, which is the study of change, can be understood through the story of what is really emotional change, a transformation in a student's heart.

PALCA: Yeah. It's a nice metaphor. And I think it's it does take some work to read the letters if you're not a mathematician, but it's still as easy to understand the emotions because those are what I think everybody would understand.

Dr. STROGATZ: Mm-hmm.

PALCA: Let's take another call and go to Josh(ph) in Berkeley, California. Josh, you're on the air with SCIENCE FRIDAY.

JOSH (Caller): Howdy? Howdy, Joe? Thanks for having me.

PALCA: Sure.

JOSH: I just had to comment - you guys are talking earlier about why some people are afraid of math. I, myself, am a math student at Berkeley, I'm a math major. And when I was young, I failed math in high school, actually. And I thought about it a lot, since I'm almost 30 now.

And I think the reason might be because a lot of mathematicians in high school seem to teach math like a history teacher would teach history, that they stood up and just wrote a whole bunch of facts from the board, and that's it. And I think it's such a vibrant and beautiful subject, and sometimes that's not really conveyed by high school math teachers.

PALCA: Hmm. Is that your experience at all?

JOSH: Yeah.

Dr. STROGATZ: I understand what Josh is saying, and I think that's what many people's reaction is that they see the authoritarian side of math. And you often hear it said, there's a right and a wrong answer, and math is great because of that, or but for some people it's that's its worst aspect that...

PALCA: Right.

Dr. STROGATZ: ...they feel there's no room for creativity or for the expression. But of course that's a misunderstanding, because even if there is a right answer there can be many creative and imaginative ways of getting to the right answer, including wrong ways of getting to the answer that are really interesting.

PALCA: But Josh, why did you stick with it? What kept your attention?

JOSH: Well, actually it's (unintelligible) I failed math in high school. I never thought I'd be a mathematician, but I've always loved physics, but I never thought I could do it. But then (unintelligible) community college later on and our teacher there are really kind of put me under her wing - her name is Juliet Stefanov(ph) - she took me under her wing and she explained it to me and she really give me some attention and I just it like lit a fire in me, and I - it's my favorite subject now.

PALCA: Well, that's cool, all right.

Dr. STROGATZ: Joe, if I could just jump in with one thing, I mean...

PALCA: Yeah. Sure.

Dr. STROGATZ: ...maybe you want to jump in with another caller, but I something that Charla(ph) said earlier and that Josh has said now I wanted to react to it so Charla mentioned that she sees at her wall when she's teaching home schooling as a transmitter and the student has a receiver...

PALCA: Mm-hmm.

Dr. STROGATZ: ...and Josh has talked about, you know, the teacher as like giving facts at the board for the students to take. Mr. Joffray was not anything like either of those models. And that was the point for me, about him, that he always acted like we we're in this together, that we're on the same team, he didn't stand above his students.

He was in love with the puzzles of math, the beauty of it. And he sort I mean, when you did math with him in his classroom, it was more like we're mountain climbers and we're on the same team. We're trying to climb this mountain, the mountain of math together. And some of us are better climbers than others and can tell what we see at the summit, but we're all on this team and we're all helping each other along. And so there was no top-down feeling of the teacher telling the student. It was all of us trying to figure out this magnificent puzzle.

PALCA: I think though the issue is that somebody - we were talking earlier about the language that you need. There are fields of endeavor where someone who is a novice can jump in and make a contribution. But people who are novices at math feel like they have nothing to contribute in the discussion of, you know, of even mildly complex math. Is that...

Dr. STROGATZ: There is some truth to that.

(Soundbite of laughter)

PALCA: Yeah. Yeah, yeah.

Dr. STROGATZ: Yes, it is. I mean, there are rare opportunities for the amateur to contribute. It is - has become possible in recent years with the advent of personal computers. That is, people have discovered interesting things in computers that professional mathematicians hadn't seen, and occasionally an amateur mathematician or a beginner will think of some very ingenious proof or solution to some old problem. But it is true that it's very, very unusual.

PALCA: Okay. Well, I commend people to your column and to your book, and I hope we can continue the discussion about math. I still, sort of, feel like I'm knocking on the outdoors, outside of an interesting world that I can't quite see into. But maybe that'll change as the columns appear. And I think I look at them with the time they require to understand to some extent.

(Soundbite of laughter)

Dr. STROGATZ: Well, thank you, Joe.

PALCA: Okay. Steven Strogatz, thanks very much for joining us. Steven Strogatz is a professor of applied mathematics at Cornell University. His latest book is called, "The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math."

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