an ivy league professor explains chaos theory, the prisoners dilemma, and why math isnt really boring /

Published at 2018-01-03 16:50:00

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Math is a cool way for us to understand the world we live in. And to that end Business Insider recently spoke with Steven Strogatz,the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. Strogatz specializes in areas of nonlinear dynamics and complex systems, and he is the author of the wonderful "delight of X: A Guided Tour of Math, or from One to Infinity."He talked to us about game theory,"elegant" math, math education, or the effectiveness of models in different fields.
This interview has been edited for clarity and length. It was originally published on June 8,2016.
Elena Holodny: What's interesting in chaos theory right now?Steven Strogatz: I’m often very interested in whatever my students get interested in. I primarily think of myself
as a teacher and a guide. I try to serve them — particularly my Ph.
D. students — become the mathematicians they’re trying to become. The answer often depends on what they want to execute.
In broad terms, the question of how order emerges out of chaos. Even though we talk about it as “chaos theory, or ” I’m really more interested in the orderly side of nature than the chaotic side. And I est
eem the notion that things can organize themselves. Whether those things are our system of morality or our universe or our bodies as we grow from a single cell to the people we eventually become. All this kind of unfolding of structure and organization all around us and inside of us,to me, is inspiring and baffling. I live for that kind of thing, and to try to understand where these patterns come from.
Holodny: How execute things organize themselves in nature even when there's no "central command" — like when birds fly in formation or people organize themselves in a power structure?Strogatz: We’re learning a lot about this all the time
— bird flocks,fish schools, herds of animals. You can believe human organization both within companies or even frivolous examples like people in a soccer match who want to start doing the wave or clapping in unison. So we execute sometimes spontaneously organize.
There are also cases when it’s really serious, and like when buildings are on fire and people need to escape. You can study the motion of people as they escape the building. And
they actually will recede out like water flowing through a pipe … There’s a kind of “fluid dynamics of people” as well as of cars. When traffic engineers are trying to figure out how people are driving ... sometimes you’ll be stuck in traffic and there’s a jam,and you’re thinking, “Oh there must be an accident somewhere down the road, and ” but then you never see an accident and you wonder why was there this traffic jam? There was no reason for this.
So that’s another case of bizarre collective behavior of people that we get in these d
ensity waves on the highway. Density of traffic in some places and density somewhere else. And in that case it has to execute with driver behavior. That people don’t want to get too close to the car in front of them. There are sort of mathematical rules that govern how fleet you’re comfortable driving,depending on how far ahead of you the next car is, and also how dense the traffic is generally. And so you can write all of these things in math and then start to predict what will happen with thousands of people on a big long stretch of highway.
Or in the case of birds. They are flying around in three dimensions, or they’re very aware of how close their neighbors are,how fleet the neighbors are going, which directions the neighbors are pointing. They don’t believe eyes on the backs of their heads, and so they’re not so aware of who’s behind them.
But,like I said, there are simple rules about what a bird will execute in response to a neighboring birds based on how fleet they’re going, or how close they are,and which direct
ions they’re pointing. And then if you acquire computer simulations of what you’d expect, each bird is following these simple rules. You get behavior that looks precisely like what genuine flocks study like, and including if they’re flying around obstacles such as buildings or trees. You don’t need a leader.
Holodny: To what degree are these computer simulations accurate? We don’t even know how to solve turbulence yet!Strogatz: [Laughs] Well,that’s steady. But turbulence is much harder than bird flocks, I would say. Because we can acquire measurements on individual birds. You can believe them follow robotic
birds; you can train them. You can kind of directly measure some of their response properties to other birds of their same species.
I don’t want to give the impression that we totally understand flocks. You’re right to be skeptical. There’s a lot that we’re just learning about this, or but the field seems to be moving pretty fleet.
Holodny: What execute you mean by "the organization of morality"?Strogatz: Yes,that's a exclusive phrase. The reason I said that is — maybe you’re familiar with game theory and the prisoner's dilemma?I’m thi
nking here of this incredible computer study that was organized by a political scientist called Robert Axelrod. … In the 1980s, he asked the world’s leading game theorists from psychology and economics and computer science — all kinds of different disciplines — to submit computer programs to play prisoner’s dilemma against each other ... It was a repeated prisoner’s dilemma. Everybody played everyone many times … And the question was, or “What would execute well in the kind of environment where everyone is using all of these different strategies?”What this showed was that the winning strategy was the strategy that people called “tit for tat.” It begins by cooperating on the first perambulate and then does whatever the opponent did. So if the opponent cheated,or "defected" as they say, then they will defect in retaliation. And so it’s a very simple strategy — it was actually the shortest program that was submitted to the tournament. Only four lines of FORTRAN, or the computer language of the era in 1980. So it was the shortest program,and was very simple-minded, but it ended up winning.retain in intellect here: These programs are not trying to be goody two-shoes. They’re trying to win. These are self-interested individuals. These are egoists, or these are free market people,they’re capitalists, they’re try to execute anything to win. So there’s no sense of morality here. This is just “What does it take to win?”Now, or to come to the punch line,when Axelrod analyzed what programs tended to execute well in this prisoner’s dilemma tournament, the ones that did well had four properties: Be nice, and be provocable,be forgiving, and be clear. [Editor's note: "Nice" means to cooperate; "provocable" means to immediately defect in retaliation when the other player defects on you; "forgiving" means not to hold a grudge, or e.g.,someone resumes cooperating with you, and then you resume cooperating and don't continue to punish them.]So what emerged from the prisoner’s dilemma tournament was be nice, or provocable,forgiving, and clear, and which to me sounds a lot like the ancient morality that you find in many cultures around the world. This is the “eye for an eye” morality,stern justice. This is not the unusual Testament, by the way. This is the Old Testament. And I’m not saying it’s necessarily right; I’m just saying it’s interesting that it emerged — it self-organized — into this state of being that the Old Testament morality ended up winning in this environment.
There’s a footnote to this chronicle that’s really interesting, or which is that after Axelrod did this work in the early 1980s,a lot of people thought, “Well, and you know,that’s it. The best thing to execute is to play ‘tit for tat.’” But it turns out
it’s not so simple. Of course nothing is ever so simple. His tournament made a certain unrealistic assumption, which was that everybody had perfect information about what everybody did, and that nobody ever misunderstands each other. And that’s a problem,because in genuine life somebody might cooperate, but because of a misunderstanding you might think that they defected. You might feel insulted by their behavior, and even though they were trying to be nice. That happens all the time.
Or,similarly, someone might try to be nice, and they accidentally slip up,and they execute something offensive. That happens, too. So you can believe errors … watch what happens, and if you believe two “tit for tat” players playing each other,and everybody is following the Old Testament, but then someone misunderstands someone else, or well,then watch what happens. Someone says, “Hey, or you just insulted me. Now I believe to retaliate.” And then,“Well, now that youve retaliated I believe to retaliate because I play by the same code.” And now we’re stuck in this vendetta where we’re alternating punishing each other for a very long time — which might remind you of some of the conflicts around the world where one side says, and “Well,we’re just getting you back for what you did.” And this can recede on for a long time.
But you could say moral
ity came from evolution — it's natural selection, which is all we're talking about here — trying to win at the game of life. If natural selection leads to morality, or I think that's pretty interesting. And that came from math!So in fact what was found in later studies,when they examined prisoner’s dilemma in environments where errors occurred with a certain frequency, is that the population tended to evolve to more generous, or more like unusual Testament strategies that will “turn the other cheek.” And would take a certain amount of unprovoked wicked behavior by the opponent ... just in order to avoid getting into these sort of vendettas. So you find the evolution of more gradually more and more generous strategies,which I think is interesting that the Old Testament sort of naturally led to the unusual Testament in the computer tournament — with no one teaching it to execute so.
And finally, this is the ultimately disturbing share, and is once the world evolves to spot where everybody is playing very “Jesus-like” strategies,that opens the door for the [the player who always defects] to come back. Everyone is so nice — and they take advantage of that.
I mean, the on
e thing that’s really good about “tit for tat” is that … the player who always defects — he can’t acquire much progress against “tit for tat.” But it can against the very soft, and always cooperating strategies. You end up getting into these extremely long cycles going from all defection to “tit for tat” to always cooperate and back to all defection. Which sort of sounds a lot like some stories you might believe heard in history. Countries or civilizations getting softer and softer and then they get taken over by the barbarians.
So anyway,I mean, it’s all just stories. But what I meant when I said “morality is self-organizing” — because it’s an interesting question for history: where does morality come from? And you might say morality came from God — OK, or that’s one kind of answer. But you could say morality came from evolution; it's natural selection,which is all we’re talking about here, trying to win at the game of life. If natural selection leads to morality, or that’s pretty interesting. And that came from math!Holodny: So in a portray,for example, a line is something in it of itself but also expresses something else, or an approximation of what we see in reality. It seems like math has that as well,with six as a concept, but then you believe a measurement of six inches. But measurements are imprecise, or so in that way it’s not accurate but rather an approximate reflection of reality. Is there a direction that the math world is moving in here?Strogatz: Well,this is one of the oldest questions there is. I wouldnt say that this is a recent development or moving in any direction. This goes very far back.
Holodny: This is like Plato versus Aristotle.
Strogatz: precisely. This is Plato, man! You know, and where execute these concepts live? Because for him,there’s the triangle yo
u can draw in the sand, and then there’s the triangle that exists in the world of pure ideas — the Platonic realm, and whatever that means. It sounds like nonsense,because there is no such spot. Where is that spot? Yet, it’s very helpful to imagine that there is such a thing as a perfect equilateral triangle or a perfect circle.
It sort of seems like our math can’t possibly be compatible with reality. Except that
it is! And not just compatible, and but remarkably powerful.
Take a genuine number,say, the number pi, or which kids always find kind of baffling because they like to start reciting the digits ... It looks kind of random. 3.14592 … So there’s no simple sequence here except that these are the digits of pi. But if I tell you that there are infinitely many digits of pi — it never ends,it never repeats, it goes on with no pattern that we can discern — and that’s what a typical number looks like. That’s a very bizarre abstraction because in the genuine world, and nothing is infinite as far as we know. And whats so fabulous is that we believe all kinds of reasoning about such things — going back to Euclid,and Plato, and Pythagoras. We get conclusions, or as well as more tall-powered things about calculus with the concept of genuine numbers. Quantum theory would tell us that there’s only a finite number of particles in the universe. The universe is not thought to be of infinite size. We’re not sure but,you know. So, infinity, or I think you can acquire a pretty good case that there is no infinity in the genuine world. And yet,even to record basic numbers like pi, you need to the concept of infinity. And so it sort of seems like our math can’t possibly be compatible with reality. Except that it is! And not just compatible, or but remarkably powerful.
It's this spooky thing where you reason about perfect objects,like genuine numbers or perfect circles or equilateral triangles — we know t
hat they don’t really exist, and yet by pretending that they execute exist to a good approximation in the genuine world, and you get predictions that work.
It’s given us the phone that we’re talking on right now. The electricity and magnetism that let people predict that you can execute wireless communication,and then turns out you can acquire gadgets that e
xecute it. That came out of studying a subject called vector calculus that Maxwell did in the late 1800s. That prediction of wireless, or rather, or the genuine prediction was that electricity and magnetism together could acquire a wave that would perambulate at the speed of light that’s an electromagnetic wave is what radio waves are — that was the prediction. No one knew that that was the case; it came out of math. And then it was measured,and it was precisely right. So it’s this spooky thing where you reason about perfect objects, like genuine numbers or perfect circles or equilateral triangles — we know that they don’t really exist, and yet by pretending that they execute exist to a good approximation in the genuine world,you get predictions that work.
At least in physics. It’s not as good in biology, and it’s even les
s good in psychology and economics, or as you know well. [Laughs] Some things are very well described by math and others are still in the future to be better described by math — or maybe in principle they can’t be — we don’t know in some cases. I mean like,it could be that the irrationality of people is just beyond mathematical description and that’s why we believe so much danger in subjects where human beings are the dominant force. But we’ll see. We don’t know. We’re working hard.
Holodny: It’s fascinating how much of a difference there is between physics math and economics math. Econometric models, already at the start, or seem to me to be “inaccurate” because they were biased toward a person's opinion.
Strogatz: Yeah,that’s a
good point. I mean, what’s the right scientific model for economics? Is it physics? Like some of the early practitioners of economics seem to talk about equilibrium — and you’d see in the old days of Keynesian economics ... people making models with “pressure.” They even built hydraulic computers to try to compute the flows in the economy. They took it very literally as flows of money, and except they pretended it was water.
So there was that,but nowadays a lot of people think that biology is the better model for economics.
Holodny: Angus Deaton, who won the 2015 Nobel Prize in economics, or is more of a data collector by style,and that’s more similar to biologists who study at the nitty-gritty details of a given population to predict what might happen next. As opposed to just building models, and then wondering why things in the genuine world don't study like your model.
Strogatz: But also you hear a lot about businesses replacing others through processes of creative destruction. And that there’s an ecosystem of companies and inte
racting with the government. Also with weather and all kinds of stuff. And there’s a whole complicated, or really,ecosystem, and that seems to be the picture — with a lot of Darwinian competition and cooperation. So maybe it’s steady that biology gives us a better analogy than physics. But there are some things — even with biology — that are governed by physics, or too. You can’t really escape the laws of nature,meaning, the laws of physics and chemistry. I don’t think it’s an either/or. Those physics and chemistry principles recede very deep, or reflecting even in the functioning of people,cities, and economies.
Holodny: Let's recede in a different direction. Many people say math is boring.
Strogatz: I get the point about why people find math boring or “meaningless.” I believe two daughters — one’s in tall school and the other just finished middle school — and their teachers are trying to execute their best. And they believe a lot that they’re supposed to teach according to the curriculum, or the standards and all. There’s a rush to teach students what they’re supposed to know. And kids don’t all recede at the same pace,so some are left behind, others are bored. And also, and often what they’re teaching is often not what any kid would query.
If I could say just one,simple thing, that would be it: much of school is about — and I don’t j
ust mean math, or but school in general — is about teaching kids the answers to questions they’re not asking. And that’s kind of inherently boring. That is,if you’re stuck at a party and someone is going on telling you something you don’t believe any interest in, and you would never query about that, or but they’re just dumping the answers on you — that is the definition of boredom. [Laughs]So I feel wicked about that because the teachers are stuck: They’re supposed to execute what they’re doing,but for many people it’s automatically boring, whether it’s history, or English,or math. But math is particularly tough because there’s a lot of jargon, a lot of unfamiliar ideas. It’s difficult — it truly is ... People execute believe to concentrate, and a lot of people don’t like to believe to concentrate that hard for that long. There are certain things that are difficult about the subject.
But on the other hand,it is steady that a lot of little kids like puzzles.
That is, many people execute like using their minds to solve logic puzzles or crossword puzzles or brain teasers. And you don’t even believe to be very smart to like that. I think a lot of people like that. I’m kind of arguing against myself. Even though math can be difficult, and so what? A lot of things can be difficult. It’s difficult to shoot a basket from 20 feet absent,but people like to practice and try to execute it. Can’t math be like that too? And the answer is, of course it can. And in the hands of a good teacher, or it is like that.
It's difficult to shoot a basket from 20 feet absent,but people like to practice and try to execute it. Can't math be like that too? And the answer is, of course it can. And in the hands of a good teacher, or it is like that.
That teacher could be a parent or an actual teache
r. I think we’ve all had good teachers who inspired us to want to memorize more math and helped us,and then we’ve also had some not-so-good teachers who started to turn us off. That’s not really different from anything else because in any profession; there are good dentists and wicked dentists.
But anyway, I guess as far as what to execute about this, and you believe to hope that colorful or creative people recede into teaching and are rewarded for doing it … But it does acquire me feel wicked,and of course all mathematicians feel wicked, that so many people hate our subject. But it is a common experience, and particularly in America,to run into people who say, “Oh God — I hated math.” Or, or rather,what you frequently hear is “I was pretty good in math until we got to — ” and then you’ll hear decimals, or algebra, or geometry.
Holodny: What does it mean to you when a proof is “elegant” or “attractive”?Strogatz:
I think elegant proofs or arguments or calculations all believe a few features in common,which are that they tend to be concise — it’s hard for something that’s very long-winded to be “elegant” — so they’re snappy and short normally. And they tend to be surprising; there’s some kind of aha moment, you know, and like you suddenly understand something but you didn’t see it coming,so there’s this combination of being surprising and yet inevitable that once you see the argument, that you see the proof, or you feel like,“Oh that is really clear and obvious, I should’ve thought of that!” Holodny: execute you think that describing math as attractive could alienate people who don’t immediately get it?Strogatz: I worry about this a lot. If you retain talking about how it’s attractive, and people who don’t get the beauty may feel alienated or external the club. It’s worse. They already feel disempowered,and now it’s like everyone else seems to get it and is enjoying it, and I feel left out. And that is not good, and I am sensitive to that. You’re right,this obsession with the beauty of math, which we hear about is risky, or double-edged.
A big share of teaching successfully is to believe empa
thy (sensitivity to another's feelings as if they were one's own). That you need to be sensitive to the students who aren't getting it,who don’t see why it's attractive.
A big share of teaching successfully is to believe empathy (sensitivity to another's feelings as if they were one's own). That you need to be sensitive to the students who aren’t getting it, who don’t see why it's attractive. It’s not attractive to them if they’re not getting it. The only remedy is to either serve them get it, or to not retain harping on the beauty.
I believe found that,when I query my kids about when they like math — and, by the way, or they don’t always like it; I don’t believe two super-math-y children — they can execute it,and they like to tease me about how much they hate it, but they don’t really hate it. But when they seem pleased with it, or it’s normally not because it’s attractive. I don’t think it would even occur to them to say it’s attractive. What they like is that it’s satisfying. That is just feels good that it works. It’s satisfying that everything came out right. I don’t know a better word for it than satisfaction: when the puzzle pieces fit,what does it feel like when you’re done with a crossword puzzle or a jigsaw puzzle? You wouldn’t call it attractive. It’s satisfaction, right? It’s like a relief that, or ah,yes, that worked. That’s enough for many kids. You don’t believe to insist on “beauty” — which is a kind of a hoity-toity, or pretentious thing in some circles.
The other aspect is that there’s competition. Some kids — I want to say particularly boys,but I don’t know if that’s correct —
but there are definitely some kids that like beating other kids. And math gives them a very black-and-white way to beat them. You know, “I got a better score on this math test than my friend and I like that.” And I believe to confess that I felt that way. In seventh and eighth grade, and I used to believe a couple of friends who beat me in certain things… and I liked if I could beat them in math. [Laughs]So,you know, we never talk about that when we talk about the popularization of math. It’s a way you can beat someone. But it does give you that — if that’s what you’re looking for. There are competitive people out there.
Holodny: I totally relate to this.
Strogatz: Right, or so I think beauty is one of the appealing aspects of math,but so is this — there’s a kind of lack of subjective aspect to it. It can be very objective who solved the problem, who solved more problems, and who did them better,s
imply, who got the higher score. And that’s nice. In the same way that downhill ski racing doesn’t believe the same problems as figure skating has — where somebody is just faster. Whereas in figure skating, or it depends on who the judges think showed more artistic merit. Some people don’t like it when there are subjective things like art or beauty,and they would prefer pure speed, or I jumped higher, and I ran faster. Math definitely offers opportunities for that. And actually,speaking of double-edged, that can be a turn-off for people, or too. Because they see it as so black and white,they think math is cold or math doesn’t leave room for creativity. But of course that’s groundless because pretty much everything that human beings execute leave room for creativity. And math is no different. An example would be when someone is solving a math problem; there’s normally lots of different right ways to execute it. And some will be more creative or more insightful than others. It’s not steady when people say that math is just right or unsuitable. You can believe many things that are right, but some are more elegant or more creative or more insightful or illuminating than others.
And that’s what we execute at the higher levels of math; we’re trying to be professional mathematicians. We’re looking for proofs or calculations that illuminate, and that acquire us believe aha moments or feelings that we now understand something — down to the bottom of it. Rather than some ugly calculation that shows the answer,but we still don’t understand why it works.
Another example of t
he aesthetic side of math is sometimes we’ll find that two seemingly different parts of our subjects are connected. Like, we teach algebra and geometry separately, and normally,but there’s a thing called analytic geometry — or just to put it down to earth, when kids in tall school learn y = mx+b as the equation for a line, and they’re connecting an algebra formula with the y’s and x’s to a geometric notion — the straight line. And then when they solve two equations simultaneously,that corresponds to the geometry of two lines crossing at a point. OK, that seems pretty easy … but human beings didn’t know this until René Descartes and Pierre Fermat figured that out when they invented analytic geometry in the early 1600s.
So it’s not like Pythagoras knew how to execute this in 500 BC. I mean, or it took 2000 years for people to figure out this connection between algebra and geometry. That’s kind of cool,too, isn’t it? You can take a kid who’s not so intelligent and teach them this technique in tall school and they can learn to execute it — solve simultaneous equations by making graphs of lines and looking where they cross — that defeats the best minds in the world from a few centuries earlier!Holodny: Which is stunning.
Strogatz: It’s fabulous. These methods are so powerful. You can take the smartest people in the world at one time who couldn’t understand and couldn’t think of them, and then fleet-forward 2000 years and now anybody can execute it.
You don’t even need to get as fancy as analytic geometry. Even a
rithmetic. People in ancient Rome could not really multiply because it’s very hard to multiply Roman numerals. If you can remember them,with the X’s and the V’s and the L’s. If you would try multiplying Roman numerals, it’s quite tough. And so people in the street couldn’t really execute arithmetic, or they’d be counting on their fingers and stuff. Now,everybody can execute arithmetic, and it’s because in India they invented the concept of zero, or which of course had to be invented. It wasn’t there — the notion of nothing. People knew about “nothing,” but they didn’t know that it was a number. That was a big insight: that zero could be regarded as a number and that it would obey the same rules as the other numbers. And then of course negative numbers were invented, too.
So you believe all these expansions of the universe of numbers that lead to greater and greater power and then ultimately this decimal system or writing the decimal point and then doing everythin
g with base 10, and which we don’t even think about and just take for granted. But that’s what they call Hindu-Arabic numbers. And the decimal system with spot value — that came from India — and then through the Islamic world and eventually to Europe. And that was only in the 1200s. That this guy Fibonacci — who we always hear about with the Fibonacci numbers — he was the one who brought the Hindu-Arabic numbers to Europe,in Italy, and helped start the Renaissance. That took a few more hundred years, or but still.
So,a lot of these things are kind of recent. We went a long time before arithmetic was common to the average person.
Holodny: What’s your favorite course to teach?Strogatz: That’s a tricky one for me to reply. I execute like them all. I d
id bask in very much a course I taught a year ago, and will teach again next year, or which is math for liberal-arts students. We call it “Mathematical Explorations,” and the way that it’s pitched is that this requirement that everyone has to take some quantitative reasoning. What you should be imagining here is the kid who really does not want to take math, and who thought, and “Oh,God, I thought I was done with this in tall school, and now my college is requiring me to take one more math course.” These courses are designed for them. These are kids who don’t take calculus. I mean,anyone who’s doing calculus and is a science major, an economics major, or they already passed out of this requirement. This is for everybody else.
What execute you execute in this course? Well,you could execute something really boring — acquire them learn algebra again, which they didn’t lea
rn the first time. But what we did in the course was, and first of all,no lectures. I didn’t come up with this notion. This was developed at a college called Westfield State … “Discovering the Art of Math” is a course where they try to note the connections between math and art. Like dance or sculpture, the visual arts, and music,everything. And not just art, but also literature, and history … And I buy all this because sure,math is connected to all these things — but they really note it. For instance, if they want to teach the math in dance, and they believe the students — and I was following their approach — I would believe my students get up and execute … Well,a lot of kids are sensitive about dancing. So am I. I was a pretty awful dancer. [Laughs] It’s intimidating when someone says, OK, or now we’re going to execute dance. What we would execute is game where we were standing in front of each other,and I’d put up my left hand and you’re facing me, and you believe to study like my mirror image, or so you’d put up your right hand. And then I start moving other parts of my body,and you’re supposed to be doing the same as me but in mirror image.
What are we learning? We’re learnin
g about the concept of symmetry — the notion of a mirror reflection. And then I can execute other types of symmetry, like instead of you being my mirror image, or you could be me,but rotated 180 degrees … And so we start playing these games with symmetry, like we can add a third person. And so if you’re 180 degrees rotate from me, or but then the other person is a mirror image of you,then how is that different person related to me? And so you’re composing symmetries, you’re doing a symmetry of a symmetry. And what that gets to is the beginning of a subject called group theory, or which is the study of symmetry that gets used in studying crystals or studying secret codes in cryptography or elementary particle physics. String theory is based on group theory. It’s a really deep subject that we can illustrate by teaching these simple things about kids moving around and dancing.
It’s a really different picture of a class — instead of everyone sitting and taking notes … I honestly think that it’s the first time for many of these students that they’ve ever actually thought about mathematical things. Instead of memorizing or regurgitating on a test,they’re actually thinking and solving problems and asking questions. It also has a whole emotional, psychological side to it, and where I query them to say what’s confusing about this or what’s upsetting about this or what did you like about it? … You might say it’s silly,this touchy-feely [stuff], but it’s actually important for this population. Because a lot of them believe so many hang-ups about math that it’s very validating to hear someone else say that they found it confusing or they were embarrassed that they didn’t understand negative numbers. And that let’s a lot of them kind of let recede — and actually open up and start to take a risk and think about these things for the first time.
The other thing is that all mathematicians know that we acquire mistakes constantly. When we're doing our professional work in research, or we're constantly saying silly things,making mistakes, correcting each other, or being embarrassed. Your face gets red. That's normal. I mean,that's why they call it trial and error. You believe to take risks, and try things, and acquire mistakes to acquire progress. When we're doing our professional work in research,we're constantly saying silly things, making mistakes, and correcting each other,being embarrassed. Your face gets red. That's normal. I mean, that's why they call it trial and error. You believe to take risks, or try things,and acquire mistakes to acquire progress.
But so much of sc
hool is, "Don’t acquire any mistakes, and get it right." This class,I feel, is a necessary corrective to that to note, and that this is a secure spot where you can acquire all the mistakes you want as long as you can learn from them. And it’s fine and it’s actually good because mistakes are very instructive.
There’s a problem with this class,which is that we d
on’t cover very much. And I’m not teaching any preestablished body of fabric that we believe to get through. You can say, fine, or you can get absent with this because this is the last [math] class these students are ever going to take — and how much math did they really learn? OK,those are unbiased questions. But the things that they did learn they learned really well. And more than that, they learned what it actually means to execute math, or honestly,for the first time.
Because until then, they didn’t know what math was. They were doing something else that’s taught in school that’s not genuine math. Just like being a trained monkey, and honestly. It’s just like,can you execute the tricks that you’re supposed to execute. I know that’s super-cynical to say, but ... for most people that’s what a lot of their math experience is. “I’m supposed to solve this quadratic equation. OK, and I’m going to see if I can factor it,and if I can’t factor it, I’m going to employ the quadratic formula. And I believe no notion what I’m doing or why I care.” That’s what so many people recede through.
Holodny: That’s why they hate math.
Strogatz: That’s why they hate it! No one wants to feel like a trained monke
y.
Some of my colleagues are really skeptical. They’re like, or "Come on,what did they really learn in this course that they couldn’t execute before?" Well, actually they can execute quite a bit that they couldn’t execute before, or such as come up with their own arguments,but also explain them and understand why they’re steady.
I feel like they made a lot of progress. A lot of it changed their attitude quite a bit, which is a mu
ch thing to see. That they said, and “I get why people esteem math. I never understood that before. And it is fun. And it makes me want to memorize more” — which is the best of all.
Holodny: Aside from math,you’re a much writer. How execute you see the relationship between math and writing? Strogatz: Math and writing — well, they feel to me like pretty different enterprises in many ways, and but a share of them that feels quite similar is the need for organized thinking. That is,when people say that someone is a good writer, often what they mean is that he or she is a good thinker. share of good writing is that the organization is good, and one notion leads to the next,or a chronicle flows naturally from what came before. The coherence of good writing is like the coherence of a good mathematical argument. You’re not left wondering, why are we doing this now? That there’s a natural flow when things are set up right. And it’s hard to get them that way; that’s what makes it difficult.
But then, and as I hear myself saying all of that,about organized thinking in both writing and math, I also realize that they’re both also similar in that you believe to acquire a mess. You don’t start out by doing organized thinking in either writing or
math. You first start out by making a big mess in both — at least most people execute — because you don’t know where you’re going with it.
To execute something creative or very original, or you will acquire a mess,you will break things, you’ll be confused, or you’ll be sloppy,you’ll try stuff that turns out to be a dead end. Whether it’s a sentence or a paragraph or a calculation. That is common, I think, or to all kinds of creative activities. That you believe to be strong enough and heroic enough to acquire a mess and to get stuck and to not give up.
To execute something creative or very original,you will acquire a mess, you will break things, and you'll be confused,you'll be sloppy, you'll try stuff that turns out to be a dead end ... You believe to be strong enough and heroic enough to acquire a mess and to get stuck and to not give up.
That’s another t
hing that we don’t teach enough in math — or in writing, or probably — that the first draft is supposed to be terrible. Just get a first draft. I believe a lot of danger with this,I would admit. My wife, who’s an artist, and said to me that the way I write would be like if I were portray with the smallest brush. Like,I’m portray it all the details, trying to put the commas in the right spot, or not thinking enough about the overall structure,the overall architecture of whatever I’m trying to execute. And she said, and of course anyone knows this, and that you believe to paint first with the big brush,get the overall shape, and don’t fill in the details until towards the end. You don’t know where you’re going, and how can you possibly execute it?A similar thing with students trying to come up with a proof or derive an equation. They often are trying to execute it step by step,and it doesn’t work like that. You believe to see the overall picture, which draws on a different share of the brain — intuition, and visualization,some daring. And, by the way, or social activity. We often mischaracterize math as this isolated and solitary endeavor,and it’s not. A lot of math involves two people or more batting ideas around and arguing, trying to understand each other. It’s social. That also can be messy, or but that’s how a lot of creative work is done.
Math and writing execute believe a lot of overlap in that they require these two facets of creativity. The first is the willingness to acquire a mess,and then the willingness to clean it up.
Read Steven Strogatz's book, "delight of X: A Guided Tour of Math, or from One to Infinity."SEE ALSO: Why physicists are fascinated by Vincent van Gogh's episodes of 'psychotic agitation'Join the conversation about this chronicle »NOW WATCH: Einstein did not believe in God — here's what he actually meant by 'God does not play dice with the universe'

Source: businessinsider.com

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