My guest today, Sarah Hart, has taken on a mission that many might think would be impossible: To make math fun and interesting to everyday people.
Sarah Hart holds two academic positions. First, she’s a professor of mathematics at Birkbeck College at the University of London. A pretty standard academic appointment. But her second appointment is more unusual. She’s the professor of geometry at Gresham College, a position established in the 16th century that to this day upholds its original mission to provide free lectures to the public. I stumbled onto one of her lectures online and I had to watch them all. I can’t wait to talk with her today because I’ve been on a crusade to make math education more engaging, and she’s someone who’s actually figured out how.
Steven LEVITT: So I’d love to talk about your efforts to popularize mathematics, which is how I became aware of you. And you’ve given this series of wildly popular public lectures that people can easily find online, and the ones that first caught my attention were on the relationship between math and music. And as little as I know about math, I know far less about music. My mother forced me to take piano lessons when I was young and I had neither talent nor interest, but I do think if my piano teacher would have explained music the way you do, I think I might’ve loved it.
Sarah HART: Oh, thank you. Well, maybe you weren’t ready at that age. Maybe it was—
LEVITT: So you have a little keyboard with you. And one thing you do in one of your lectures is you just play a note, say a C note, and then you play a note an octave higher and then you play the two different C notes at the same time. Could you just do that now?
HART: Yeah. Let’s try this out. Okay, so that’s the C, and then, as you say, you can play another one an octave higher.
LEVITT: Okay. Those sound really different to me. So, play them together.
HART: That’s them together. So here’s a C. Here’s a high C. And here they are together.
LEVITT: So, the crazy thing is, it really sounds like you’re playing one note. Honestly, my entire life, I don’t think I’ve ever had anyone do that simple thing you just did, which is say, “How do you know two C notes are the same? Well, I’ll play you them separately and together.” It’s a great example of how, when you show people something instead of telling them, it’s so much more powerful.
HART: Exactly. What we do when we’re learning something or experiencing something is we need to actually see it happening or to hear it — we need something to grab onto, to hold on to. And in mathematics, that’s super, super important, because I could say to you, “Well, here are the laws of music and write down some equations,” and that is not going to speak to you in the same way that me saying, “Okay, those two things sound the same.”
And you can think about situations where a group of people are maybe singing something together and you will be just naturally doing that exact phenomenon. The men will be singing an octave lower than the women, usually, and you don’t even think about it. If you’re singing happy birthday at a child’s birthday party, you’ll just instinctively do that. And you’ll all feel that you’re singing the same tune, and to that extent you are, but you’re not because you’re not all singing the exact precise note. You are singing notes octaves apart and these simple relationships that hold between the frequencies of those notes and what makes them sound, to our ear, the same thing when we play them together.
LEVITT: Yeah, of course. It’s really math that explains what makes you notice. So, why do all C notes sound the same to us?
HART: It’s because when we hear a note at all — if a note has a pitch at all — it’s because what’s happening is the air is vibrating and it’s vibrating regularly, periodically, and that vibration will have a certain frequency. So when we think of the pitch of a note, it will be caused by vibrations of air and they vibrate in our ears and that frequency terms the pitch. Now, the notes that are an octave apart, they have this very beautiful relationship. That low note there — the higher one is exactly twice the frequency of the lower one. So it’s this lovely, simple mathematical ratio of the frequencies.
Now, that alone isn’t enough to explain why they sound the same, but that’s the first step along the way. And this was noticed even back as far as Pythagoras, who did some experiments with stringed instruments, and he realized that there was this relationship between the lengths of strings and the sounds that they made. Now we know that’s related to the frequency. There are other nice notes that don’t quite sound the same as each other, but they sound nice together.
If you multiply the frequency, not by two, but by one and a half — three over two — those sound quite pleasing together. And there are other ones. If you multiply by four over three — so again, very simple relationships — you get “Here Comes the Bride,” or something like that. So they’re very pleasing relationships between certain pairs of notes — these things called intervals — and they all have these nice whole number relationships between them, or very simple fractions.
LEVITT: So, does the pleasing come because of these whole numbers — they tend to have their peaks and their valleys more or less at the same time?
HART: What eventually took hundreds of years to realize this, but what’s really going on with these frequencies is that when you take a string and vibrate it, you pluck at the string and it vibrates at some frequency, but there are other vibrations that can fit into that string. So, if you imagine a string, it can have a wave moving along it — or you can fit two whole waves or three whole waves or four whole waves in that length. And, so, what you get are these nice whole-number multiples, they’re called overtones.
And the exact composition of this will depend on the musical instrument you’re playing. Violin sounds different from a flute, sounds different from a piano, but you will get some kind of combination of these whole number multiples of a particular frequency. When you double the frequency, you’re getting that frequency that’s twice as high, you’ve already got a little bit of that as an overtone in the lower note. So your mind is somehow prepared for that. It’s already a little bit there. So when you add in that higher note, you’re giving your brain something it’s already prepared for and that’s why they sound so pleasing together.
LEVITT: So, here’s something that puzzles me. You’ve already explained that a high C vibrates twice as fast as a C one octave lower. In between those notes on a piano though, there are 12 keys and those keys, if I’m using the right language, are each a semitone apart. Is there a reason why we divide an octave, where “oct” means eight into 12 pieces? I mean, eight pieces would make sense or even 16. How do we end up at 12?
HART: So, this is a really fascinating question. And the reason, again, it goes back, probably, to the Greeks. And we don’t— This is the Western classical tradition. It’s not true in all musical traditions, but the reason we divide into 12 is because originally the octaves were seen as nice things — double the frequency. And then these intervals, which I showed you before, where you multiply the frequency by three over two, which we now call a perfect fifth — it’s called a perfect fifth because if I just play you the white notes I’m playing on the piano. One, two, three, four, five, right? So that’s why it’s called a fifth, but that’s begging the question for later.
Now, Pythagoras discovered that if you start at a very low, for example, C, and you go up by octaves and you keep going if you go up by seven octaves, you’ll have doubled the frequency seven times, and if you’ll just do that in your head, you’ll find that you’ve multiplied the frequency by 128. Okay. Remember the number 128. Now, if instead you start on this low note and instead of going up by octaves and doubling, you go up by these fifths, which are a nice pleasing interval. You start here, you do a fifth, you do another fifth, you do another fifth, and you keep going and keep going.
What happens is after you’ve done this 12 times, you get to this seven octaves higher note that we did when we went up in octaves. So, in other words, 12 fifths on a piano equals seven octaves. Now, that means this 12 number, if you just go up and down octaves, you can fit all of those 12 different fifths that you did — they give you the 12 different notes in an octave.
Now, that is nice, but there’s this puzzle at the heart of a piano, because if you actually think, what is that interval of a fifth? You’ve multiplied by three over two — one and a half. If you multiply by three over two, and you do that 12 times you’ll get like three to the power of 12 divided by two to the power of 12. That is not 128. It’s 129.7 and a bit. Twelve-fifths are not equal to seven octaves. So what on earth is going on?
LEVITT: Yeah, what is going on?
HART: Well, the thing that’s going on is that actually when we’re playing music, we don’t tend to require a seven octave range and 12 fifths. So this approximation that is used — and that’s what it is — it was good enough. And you can never, actually get a whole number of fifths into a whole number of octaves because fifths involve the number three, this three over two thing, and octaves just involve doublings.
And however many times you multiply threes together, you’re never going to come out with an even number. It’s just never going to be possible. So, because of that, there’s actually no number that you can divide an octave up into such that everything works out nicely in terms of fifths and octaves. So because you can’t do it, then you say, “What’s a good approximation?” And 12 is a good approximation.
LEVITT: So I’m surprised. The way you’re describing it, I would have thought that the musical — I don’t know the right word — canon that the ideas behind octaves would have far preceded the math and the math would have come later to make sense of it. But it sounds like you’re saying the development of music was highly integrated into mathematical thinking from the very beginning. Is that right?
HART: Absolutely. These very early experiments about the lengths of different strings making these lovely ratios; it was all bound up in this idea of how everything has a wonderful universal harmony, and these numbers, one, two, three, four, and their ratios, this is how the universe was created, believed the Pythagoreans. It was all mixed up into one idea, and even there was this belief that actually mathematics — well, and I believe this — underlies everything.
So, if you’re thinking of music, that’s number turned into sound, and astronomy is number as it goes through time and space. And arithmetic is number just studied for itself. And geometry is number as it relates to space. So you’ve got all of these different understandings of number and pattern and music is just a part of that.
LEVITT: So I might be wrong, but I’m guessing that the idea that mathematics applied to astronomy probably held us back hundreds of years in actually figuring out what was going on because things were not so perfect, as we really hoped they would be when the earth — oops, it wasn’t really the center of the universe.
HART: Although, it has to be said that theology got in the way as well, a little bit, because there were writers amongst the Greeks who had suggested that the sun was at the center and the earth was going around it. So it wasn’t just this new idea from Copernicus and Galileo. It had been suggested previously, but then others had intervened and said, “No, no, the earth must be in the middle because — religion.” So there are fixed ideas.
You get people like Kepler. I feel so sorry for Kepler, because he was so adamant that there had to be some mathematical reason why the orbits of the planets were the distances they are from the sun. And he drew this amazing picture, fitting the five platonic solids — nesting them inside each other. And at the time there were six planets known, and so you put a platonic solid in between each one and that’s why the ratio of their orbits — and of course, now we know, no it doesn’t. And there’s more planets than that anyway.
So, yeah, a lot of good minds perhaps spent time trying to fit things into the theory that they had, wanting things to be circles when actually they’re ellipses. Because we’re humans and humans get ideas that they want to be right and mathematicians are humans as well. That could happen to the best of us.
LEVITT: So, when you look at a piano, there are white keys and there are black keys. But then there are these 12 semitones and it doesn’t seem obvious to me that there’s any particular reason why some keys are white and others are black, but I always thought somehow that because there were fewer black keys and they’re physically smaller that maybe the black keys were somehow less important. Is that right or wrong?
HART: It depends whether you are wanting to just play a tune and you don’t mind what key it’s in because the white keys give you the key of C major. So they allow you to play a scale. So early instruments or early music, you didn’t have to necessarily be in tune with lots of other things. You could think of a song and you would play it in whatever key you wanted. You didn’t have to be playing with an orchestra or anything. So if you just want that basic scale, you’ve got the white notes, but then—
LEVITT: But let me stop you there for a second, because I just want to make sure this is clear. If you start at note C, you can play the full key of C major by only playing the white notes — the white piano keys. However, because of the way a piano is set up, if you want to play the key of, say, B Major or D Major, you would just start with a note of B or D. But then you’d also have to play some of the black piano keys, right?
HART: Yeah. So that’s why we have the black notes, really, is to allow you the full range.
LEVITT: But why do we like C so much? I mean, why on the piano does C major get preferential treatment by being the only major key that just the white piano keys play?
HART: It’s not that there’s anything special about C per se, it’s that you’ve got a scale, a thing you can play just on the white notes. It wouldn’t have to be C, that’s just an accident of the first person to do this decided that was how they were going to tune it. Although you don’t have to tune it. You could tune it to something different and you could get the same pattern of things just shifted up or down.
But with the black notes, it allows you to not just play C, but you can go up a tone and play D, or another tone and play E, or F, and so on. So you can play all of the scales and you can play the minor scales and you can do everything you want to do on a piano. But my instinct is there’s this kind of compromise between the flexibility of having everything which you want, and then the ease of being able to play some things more easily.
And also, if you were to evenly space all the notes then you start to get into problems of being able to reach. So, Rachmaninoff could reach 12 notes or something, but most of us can only reach maybe one more than an octave. The pianos become uncomfortably big if you don’t squash the notes up in some way.
LEVITT: How much of what sounds good to us is just familiarity versus the dictates of math? So, my wife loves hardcore metal music — crazy music from bands with names like “Five Finger Death Punch.” And when I first met her, I couldn’t even listen to it, but now I kind of like it much to my surprise, just having heard it a lot.
HART: Yeah. So, I would say the octave is probably the only thing that is universal in that sense. That sound that when you play them together, those sound similar to everybody. But then once you get beyond that — this division of the octave, which really comes from the, perhaps, obsession with or liking for these perfect fifth intervals. Actually, there are other ways to think about things and if you favor different intervals or gaps, then you might get different answers to these questions.
So, yes, we can say that mathematically speaking, we understand because of the mathematics of frequency and the laws governing frequency that there are some things that perhaps to our ears might sound more resonant, but that doesn’t completely answer the question, because if we talk about these small ratios— We like a ratio of four over three or something, but why don’t we like seven over five, which isn’t that much bigger, but it sounds horrible? And there are also intervals that a few hundred years ago would have been absolutely not allowed. So, the fourths and the fifth that we mentioned, these were approved of for 2,000 years in Western music.
But when Monteverdi comes along — who’s written, you know, some amazing music in the Baroque period — he was including things like intervals of a third, which is this one. Those were referred to as licentious modulations by the church, just going to drive the kids crazy and put them off their worship. So, at different periods of time different things have been in favor. Even something like—
You’d think half an octave would be a lovely sound. So, what do I mean by half an octave, really? It’s halfway up, musically, so the frequency — it’s got to be not halfway along, but root two of the way along to get that geometric mean. That thing is called a tritone — half an octave. And you don’t hear that very much until maybe the last century, I suppose. But, of course, you hear it all the time nowadays, if you have ever watched T.V., because it’s the first two notes of The Simpsons theme tune. But until that came along you didn’t hear much of that.
LEVITT: The 20th century classical— What we call classical composers of the 20th century, they really did rely heavily on mathematical foundations it seems. But, is it not true that in the end, the marketplace didn’t really buy those ideas? My impression is that people listen much more to Bach and Beethoven and Wagner than most of the composers of the 20th century, who were doing such fascinating things trying to bring math into music. Do you think that maybe math wasn’t enough, and something more was needed?
HART: Well, response one is: are these the licentious modulations of the 20th century? So there are composers who have tried, as you say, consciously to incorporate mathematical ideas, and I think there are various levels of success. And one potential pitfall is something that can happen in mathematics as well. You box yourself in. So, you can prove amazing theorems but no one cares because you’ve got too much structure and not enough room for creativity and playfulness.
So, if you are really over-specifying every single thing in your music and saying, “Right, I will have an algorithm for what the next note is. I will have an algorithm for what the length of the notes are. I’ll have one for whether they’re loud or quiet.” You can take it to an extreme that no one wants to listen to it, including the composer. So, I think what happened in the 20th century to some extent is that ideas got pushed to their limits, and then — it’s strange because it’s almost as if it sounds random at some point — if you put so much structure in that you have absolutely no control of what’s going on, then it’s almost indistinguishable from just random sounds. And then, you got to really think, “What is the purpose of this? What am I doing this for?”
Whereas, the more successful music that involves structure is where there’s enough freedom still to find the constraint invigorating rather than repressing. So you’ve got people like Hindemith whose “Ludus Tonalis,” which is like tonal games, was playing with these ideas of having different keys. And he had some cool structures in there. This is a whole kind of series of pieces all linked up together, and working through all the different starting notes, and the last one and the first one, they were the rotation of each other mathematically, which is cool and difficult to do, but that did not stop him from writing a beautiful piece of music. If you listen to these they are full of structure and they’re fantastic, but they aren’t off-puttingly sterile, which is always the risk.
LEVITT: Have you ever tried composing music?
HART: Well, yes, but only in the sense that teenagers write poems. Yes, to be truthful, but no, to be sensible.
* * *
LEVITT: Hey, Morgan, how’s it going?
Morgan LEVEY: Hey, Steve. So this listener question comes from Ben and he wants to know if you have any experience with what’s known as a “sophomore slump,” meaning something you did well once, but are having trouble succeeding at a second time around. Like how musicians can have trouble producing an album that’s as innovative as their first, or authors run into writer’s block when trying to produce a second novel. Have you ever felt creatively stuck in your academic work or other parts of your life? And do you have any advice for moving beyond it?
LEVITT: So the sophomore slump is a complicated issue because there’s so many different forces that can be contributing to it. So let’s just take SuperFreakonomics, the second book that Stephen Dubner and I wrote. There’s every reason why that should have been a huge success. We had just written Freakonomics. We knew how to write a book, but honestly it didn’t sell nearly as many copies as Freakonomics and maybe it wasn’t as good or maybe something else was going on.
But what’s often at the heart of a sophomore slump is that the first time around you got so incredibly lucky, off-the-charts kind of luck to do really well in your first one, whether that was our book Freakonomics or whether that’s a hit album from a musician, that really you just can’t expect to get as lucky the second time. So even though a lot of factors help you, there’s a big boost from having been successful the first time, it’s hard to do as well the second time.
LEVEY: But I want to challenge you on something because SuperFreakonomics was still a best-selling book. And I think what Ben is asking is he’s having a harder time even accomplishing the task the second time around. Whereas you and Stephen Dubner were successful in writing a book. Now, it wasn’t as successful as the first book, but you still wrote a successful book.
LEVITT: We did. What was hard and what maybe Ben is running into is that the first book you write, you have a lifetime of experiences to draw upon. And the second book you write, you got as much time that’s passed since you wrote your first book. Your good ideas are gone in some sense, and I think that’s what happens to many people.
So one thing I say to people who are running into a sophomore slump is, “Well, maybe you shouldn’t write a second book. Maybe you said everything you had to say in the first one. It’s time to do something different.” But the second thing I say is, “Look, it’s inevitable. If you’re someone who had success with anything the first time around chances are part of that was skill. And a lot of it was luck.”
LEVEY: Well, thanks for writing in, Ben. I hope that relieves some of the pressure off of writing your second book — good luck. If you have a question for us, you can reach us at email@example.com. Steve and I read every email that gets sent. And we look forward to reading yours. Thanks.
LEVITT: So, people listening today might wrongly get the impression that you only mess around at the edges of math, but you do hardcore math as well. So one of your most cited papers was published in the Journal of Algebra in 2007, and it’s called “Commuting Involution Graphs for Sporadic Simple Groups.” So let me just read the abstract for people.
“Let K, which is less than or equal to G, which is less than or equal to aut of K where K is one of the 26 sporadic finite simple groups. And let T, a member of a G, be an involution and X equals T G. The commuting involution graph, C of G and X, has X as its vertex set with two distinct elements of X joined by an edge whenever they commute in G. And for most of the sporadic simple groups, we compute the diameter C of G and X and give detailed information about the elements at a given distance, from a fixed involution T.”
I mean, seriously, that seems really hard. I don’t even know what those words mean. How long would it take for you, an excellent communicator, to explain that paper to me in a way that I could understand? A year of lessons?
HART: So that word “group,” if you have done an undergraduate degree in mathematics, you will know what a “group” is — it’s a particular kind of algebraic structure. “Simple” — that’s one of the biggest misnomers in the history of the world because it’s a kind of structure that really is the building block of all of these other things that are called “groups,” whatever they are. So “simple,” really means you can’t break them up any further, but unsimplifiable really would be a better word for these things.
And then “sporadic” — they’re the weirdest ones. They’re the duck-billed platypus. They’re the ones that just don’t fit into anything else. They’re there, they’ve been found, but they are really unusual and strange. So, this paper is about groups, but then it’s the hardest ones and then it’s the strangest types of these structures. And so what we were doing is trying to really construct some pictures in order to get a better handle on some of the structure that we’re trying to analyze. But probably after a year, you’d definitely fully know what the abstract meant. Then we’d look at section one.
LEVITT: So you currently hold the title of Gresham Professor of Geometry. It’s a position that was first established in 1597 and held previously by some quite famous people. You are also the President of the British Society for the History of Mathematics. So, I’m decidedly not a historian of mathematics, but the names of a few other former Gresham Professors of Geometry were familiar to me, and one of those was a gentleman named Roger Penrose who got the Nobel Prize in 2020 in physics and he worked on black holes, and I remember reading one of his books when I was young and impressionable, and really being affected by it. I’ve never met him. Have you ever run into him?
HART: I’ve been in the same room as him enough times to perhaps say I’ve met him. But, yeah, his book — so I don’t know if it’s the same book that we have both read called The Emperor’s New Mind.
LEVITT: Yeah. Exactly. That’s exactly the book.
HART: That book blew my mind as a teenager when I read it. I already liked maths by that stage, but he talked about these things called Turing machines, which are theoretical computers. Alan Turing, famous mathematician, I won’t say invented computers, but he did a lot of the theoretical background to it. And these machines are really the most basic thing you could imagine that could perform a calculation.
So it can look at a number one bit at a time. So, zeros and ones it can look at, and it can change a zero to a one or a one to a zero, and then it can do about two other things. And those very, very simple machines that Roger Penrose wrote about in his book — it turns out that if that machine can not do a particular calculation, then no computer can do that calculation.
In other words, those tiny, simple building blocks are enough to build up the whole edifice of anything you could have an algorithm for. And it’s so mind blowing. This powerful observation about just a few simple instructions can give you everything you could ever compute is amazing.
LEVITT: It is amazing.
HART: And that really just blew me away. And it’s the elegance of that thinking, that mathematical logical framework, that’s one of the things that makes mathematics so amazing for me, and really got me very excited about it and off I went to university.
LEVITT: You’re at Gresham College, founded by Sir Thomas Gresham, and he’s the same Gresham behind Gresham’s Law in economics, which is essentially that bad money drives out good. Now, I had no idea until I began reading about you and about Gresham for this interview that he lived in the 1500s — 200 years before Adam Smith, he was coming up with these economic laws. We usually think of Adam Smith as the founder of economics.
HART: And this guy, he was such an amazing entrepreneurial spirit, I mean it basically seems like he invented the stock markets in Britain — the Royal Exchange is basically him. He became very rich. He was a financier and altruistically he left in his will — or maybe hoping for rewards in heaven off his altruism — this provision for a college to be founded and there would be seven professors and the subject titles are geometry, astronomy, music, rhetoric — which, I was trying to explain to my daughter what rhetoric is, and I got far as arguing, but — jurisprudence, medicine, and theology.
So those were the seven subjects that he thought would encompass what you needed to know at that time. The thing that set it apart, was that the professors give lectures, public lectures, to anyone who wants to come. So Gresham is pretty unique, the Gresham College, because it’s not a standard university. It exists just to give these free public lectures. So you can’t get a degree from Gresham College or anything like that. It’s just, “Here we are. We’ve got knowledge and we want to give it to you for free.”
LEVITT: Now, one thing that is missing from the list of names of past Gresham Professors of Geometry are any women. It took 423 years for them to finally get around to naming a woman to the position. You’re the very first. Have you found it difficult being a female mathematician because the field has so long been male dominated?
HART: Well, there have been some challenges because, of course, if you are going into an area where you’re different in some way, then occasionally there will be people who think that you don’t belong there. I think it’s easier than it used to be, of course, because 50 years ago they would openly tell you you didn’t belong, and maybe 20 years ago they might think it, but wouldn’t say it out loud.
My experience has been broadly positive. I was very lucky that when I went to university as an undergraduate, I had two tutors in my first year and one of them happened to be a female professor of mathematics. I didn’t know how unusual this was at the time. Later on, much later, I looked it up and at that time there were only two women professors of mathematics in the whole United Kingdom. And my tutor was one of them. So, I had no idea that she was so very unusual. And that helped to make the fact that she was just there getting on with it, no one treated her any differently. She didn’t treat us any differently. That was a great initial starting point.
And, of course, I’m very privileged to be able to say, “I just got on with it,” because I know that some people face much more challenges; if you’re a woman of color, then it’s even harder still and you feel even more that you’re perhaps the only person in the room who is like you, but the positives have so far outweighed the negatives. You do get the occasional dinosaur who’ll say, “Women can be good, but they can’t be as good as the best people,” and then you just give them a withering stare. But I hope that just by doing what I do and being a woman in mathematics, I can help make things a little bit easier for the next generation that are coming through.
LEVITT: So I looked up the data on the gender of people getting math Ph.D.s in the U.S.A. and the latest data I could find were for 2015/16, and for pure math, it’s still only about 25-percent female. Why do you think that is? It’s especially strange because in the U.S. women are now 60 percent of college graduates and actually 40 percent of math majors in college but just not pursuing math as a profession.
HART: I’m not perhaps an expert in why people make educational choices, but this is just my feeling here — that socially women are perhaps brought up to be more accommodating somehow and polite, and those things that don’t go well with focusing for 22 hours a day and not leaving the house and not changing your clothes and stuff. And there is perhaps this slight aura of some sort of macho thing in pure maths of, “We do really hard problems and you’ve got to just go all in.”
And perhaps the things that women are brought up with our social conditioning to be like. We are forced to learn some other skills, which might mean there are more avenues open to us and subconsciously you’re drawn maybe away from those areas of pure mathematics a little bit. But that’s all very wishy washy. It doesn’t appeal to my sense of wanting to give you a proper researched answer. So this is just my feeling.
LEVITT: That fits with my own advice, which I don’t think is gendered, but I try to talk every one I can out of getting an economics Ph.D., because I think for almost everyone, it’s a bad choice. I hate to say this, because maybe some of the people I’ve encouraged to get an economics Ph.D. are listening — the only people I encourage are the people who have such poor social skills that they have no chance of succeeding in a more demanding social setting.
I’ll tell you a story. I suspect it’s true in math because it’s true in economics, but at least when I was younger, there was a sense that if you were too normal, you couldn’t possibly be a good economist. And one of my colleagues — who’s been a very successful economist — and he was incredibly normal and he also had the burden of being very good looking. The older faculty — my older colleagues, they would literally say, “He’s too normal. That guy can’t really be a good economist. He doesn’t have any quirks.”
HART: I’ve been told I’m too normal. If I meet people, if I’m at a party or something and they ask what I do, and I say, “I’m a mathematician,” I get this kind of, “Oh, you can’t be a mathematician — you don’t fit in the box of mathematicians that I’ve got in my brain.” However much we are rational beings and we know rationally that there is nothing a mathematician particularly has to look like, our subconscious minds are hearing that narrative.
And so, if you don’t fit into, I guess, old eccentric white guy with a beard, it just might put up the tiniest of little barriers and even if you say, “Look, with a one percent less chance each year of a person proceeding onto the next stage of a mathematical journey because of that thing — those one percents accumulate and you get a drop-off rate. There is real opportunity loss there if we are perpetuating this idea that you have to be a weirdo to be an academic; you’re losing the people that could contribute in new ways to your field. And we lose all that brilliant talent and skill that is going elsewhere and doing other things.”
LEVITT: Absolutely, or at the very least we should just recognize that you don’t have to be white and male to be a super weirdo. There are plenty of really weird people of color. So maybe if nothing else, we can start with just making math and economics safe for weirdos of all types.
HART: Everybody can be a weirdo. That’s the rallying cry.
LEVITT: So, over a decade ago, I did some academic research with Roland Fryer on the gender gap in math performance for younger kids in the U.S.A. And in those data, we found that girls and boys performed equally well when they entered school in kindergarten, but already by fifth grade, the girls had lost 0.2 standard deviations relative to the boys. It was true across every racial group, every region, every socioeconomic level and family structure, and it was especially pronounced at the top of the distribution.
And we completely failed in understanding why. We could look at some explanations — it didn’t seem to be lower expectations by teachers. It turned out, interestingly, and probably surprisingly to you, that girls who had moms in math related occupations lost just as much as girls whose moms weren’t. Parents reported spending as much time on math with sons and daughters. So, it was really a frustrating project because we got such discouraging results, and yet we couldn’t offer any answers at all.
HART: The only thing I can even begin to think is that it’s like unconscious bias that you do to yourself, maybe. A long time ago I read that a study had been done where they gave girls and boys some tests to do, and then they repeated this and the only difference was at the top, you had to tick whether you were male or female and doing that made the girls do worse. So just becoming aware of their own gender, temporarily.
LEVITT: So, interestingly again, with Roland Fryer, and also with John List and Sally Sadoff, we went and tried to replicate those results here at the University of Chicago and we absolutely could not. So that research in psychology goes under the name of “stereotype threat.” I became very interested in that because the implication was that girls forgot they were girls until you made them remind themselves they’re a girl — that made no sense to me.
So we started out doing subtle things where we would give math tests and we would have people circle their gender — no impact. Then we just got more and more extreme because in the original results, the more extreme you got, the more extreme the impact was. And so we would have Sally Sadoff — my cousin and an economist and my co-author — she would go in front of the room of students and she would say, “Look, this is a test where girls have—” well, women, these were really women, these were M.B.A. students, mostly, “—women have done very poorly on this test before.”
And the more we said, the better the women did on the test. The women were crushing the men by the end. And I really don’t know what to say. But let me tell you the other thing we did in our paper, which I think is much more interesting; this is the research I did with Roland Fryer. We looked at these cross-country math tests that are given. These are the ones where the United States always does terribly and there are headlines in the papers, bemoaning how bad math education is in the U.S. — and there were interesting findings related to gender.
The first was that, indeed, across the planet, girls tended to underperform boys in math, but there was also a very strong relationship overall between broader measures of gender equality in a country and how well the girls did relatively on math. So countries like Norway and Sweden and Finland that have a lot of gender equality, girls did just as well as boys did and that makes a lot of sense with your story of implicit biases. Okay, here’s the thing that was really crazy. There was a second set of countries where girls dominated boys in math: Iran, Jordan, Bahrain, and Egypt. And the girls didn’t outperform the boys because the boys did badly. The girls were really, truly off-the-charts good at math.
And the only real explanation we could come up with is that those were countries in which they had single-sex schools, and it really made us wonder whether maybe teaching girls math away from boys could have huge benefits. Maybe much broader than in these countries. And we tried without any success to convince a bunch of different school districts in the United States to let us do some female-only classrooms, and in the end they all said “No,” but I still think it’s possibly an experiment we should be doing because I’d love to know the answer to that.
HART: Yeah. Here’s an N=1, but I did go to a girls school.
LEVITT: So, what I find odd is that you are such an amazing communicator and you have such a broad set of interests and knowledge about music and literature and history, and you seem to be an extrovert. I have a hard time reconciling the person I’ve been chatting with locked up alone in a room for weeks or months, focused on something as abstract as commuting involution graphs. Does it seem odd to you these two different sides of yourself?
HART: Well, of course, I live in my own brain so it doesn’t necessarily seem odd, but I think it is a little bit unusual. I’m pathologically interested in things. So if you talk to me about anything at all, I just find everything interesting. So actually I need to be shut in a room to stop me wandering off and going, “Oh, I wonder why door handles open in that way.” And that’s why it’s actually good for me to be in a room, thinking carefully about one thing for a long time.
LEVITT: Alright, last question. What can we do to get kids, especially girls, excited about math and science and keep them engaged as they progress in school?
HART: One way I think is to meet them where they’re at. One of the things I do by going to schools is, I do talk about art and music because you might not know that you’re mathematical, but you really like drawing perspective drawings. That’s maths. Or you really like the beautiful symmetry of an Islamic design on a wall of a mosque. That’s maths underneath that.
Mathematics is sold as being — it’s all doing hard equations. There are hard equations, but that’s not the essence of it. It’d be like me saying to you, “Oh, economics is all about money.” It’s so reductive. We encourage people to be rounded individuals who read and who do cultural things. Maths and science, they are part of culture, too. So I think it’s just part of that whole thing.
LEVITT: So I think I do know the answer to how we could keep so many more people interested in math. We just need to clone you and get you in every classroom in the world, and I think our problems would be solved.
Obviously, I was joking about cloning Sarah. But seriously, why aren’t more classes taught via recordings of superstar teachers? Sarah Hart could tape high school geometry lectures that would inspire many students around the world. Mayim Bialik could teach neuroscience.
Imagine, if when you attended high school, you didn’t have the random set of teachers you had, but rather you were taught chemistry, social science, and humanities by the winners of a teacher’s version of American Idol. Don’t you think you would have learned more and enjoyed it more? There might still be live teachers in the classroom, their role would just be changed. Is it really such a crazy idea?
One-hundred-and-thirty years ago, people were entertained by live music produced at home. But with the arrival of recording technology, society moved to a different model, one in which superstar musicians and entertainers emerged. Why didn’t the same shift happen in education? I don’t know. But maybe it should have. At the very least, it’s a question we should be asking.
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People I (Mostly) Admire is part of the Freakonomics Radio Network, which also includes Freakonomics Radio, No Stupid Questions, and Freakonomics M.D. This show is produced by Stitcher and Renbud Radio. Morgan Levey is our producer and Jasmin Klinger is our engineer. Our staff also includes Alison Craiglow, Greg Rippin, Joel Meyer, Tricia Bobeda, Emma Tyrrell, Lyric Bowditch, Jacob Clemente, and Stephen Dubner. Theme music composed by Luis Guerra. To listen ad-free, subscribe to Stitcher Premium. We can be reached at firstname.lastname@example.org, that’s P-I-M-A@freakonomics.com. Thanks for listening.
HART: So I’m like, “Okay, why is a centaur not an insect?” And I thought, this will stump her all the way to school. But unfortunately she came straight back with, “Well, do they have an exoskeleton?” And [laughs] I have to admit that they didn’t.
- Sarah Hart, professor of mathematics at the University of London and professor of geometry at Gresham College.
- “Online Lecture: The Mathematics of Musical Composition,” by Sarah Hart (Gresham College, 2020).
- “An Empirical Analysis of the Gender Gap in Mathematics,” by Roland G. Fryer and Steven D. Levitt (NBER Working Papers, 2009).
- “Exploring the Impact of Financial Incentives on Stereotype Threat: Evidence from a Pilot Study,” by Roland G. Fryer, Steven D. Levitt, and John A. List (American Economic Review, 2008).
- “Commuting Involution Graphs for Sporadic Simple Groups,” by C. Bates, David Bundy, Sarah B. Hart, and P.J. Rowley (Journal of Algebra, 2007).