 So a thing that mathematicians like to talk about, and dare I say, even sometimes complain about, is the way that mathematics is presented in the media, the way that mathematics is thus seen by the general public. There's a few cliches that are returned to again and again, like, oh, what is topology? Oh, it's like where a donut is the same as a coffee cup. This is by Crayon, if you have played with this kind of deep learning image generated from text generator. I thought I would make one for this talk. So this is what you get if you ask it for a picture of a mathematician turning a coffee cup into a donut. And I guess we could have a whole, another discussion about why the mathematician is like a slightly deformed, like maybe white guy with a tie. I'm not sure that that's somehow. Anyway, there's a, there's some story to be told there. But, so this is, you know, this is one thing, like, okay, where the coffee cup turned donut people, it's somehow, if you're a number theorist, like me, maybe sort of people are like, oh, somehow they know something about like prime numbers that are gonna like compromise your credit card or something like that, right? I mean, these are the sort of things we see here. This is a thriller novel called Prime Number by a guy who actually ran for governor of Wisconsin last time and then I found out he had a sideline writing math number theory themed thriller novels. So, when I talk about outward facing mathematics, I wanna talk about sort of one partial remedy to these complaints, which is that I think there are ways for mathematicians to tell our own stories. Unmediated through like other people talking about us. And by mathematicians, I want everybody to understand because this is a broad group, that I mean that very broadly. I mean, like people who are doing mathematics should talk about it, whatever the professional format in which they do math. And that can take a wide range of forms. Just as a preview, today I'm mostly gonna talk about like writing, because that's what I know. That's like what I know best. But I wanna just start by sort of really making it clear that there's a lot of different formats in which people promote math. And one very famous one, I mean people have these amazing YouTube channels, Three Blue, One Brown is one of them. I meant to write down Grant Sanderson, is that his name? Okay, I should have written down his name on the slide but I forgot too, but this is like a fantastically popular, I know it's sort of hard to read these screen clips, but the key number is 4.60 million subscribers and this is like a very broad reach kind of thing. We have a podcast, like this great podcast if you've heard it by Evelyn Lam. Who I think is here in Utah, is that right? Anybody know? Okay, and Kevin Knudsen. Even, I'm trying to stay relevant. Even if you search for a hashtag math talk there is like a lot of it. If you guys know David Zurich Brown, he turned me on to math talk the last time I was visiting him in Atlanta. So it's fascinating to sort of see like what people do with these like extremely short videos, like a lot of which are made by kids, some are made by teachers, but some are made by students. And again, look at the number 325.1 million views on this hashtag. So maybe one thing I wanna sort of start with, as we talk about sort of how outward facing mathematics can look before I zero in on the sort of part of it that I know the best. I kind of always think of it as like, there's this big curve of reach versus time. I don't know if these small letters are readable from the back. So I'll tell you what they say if those, now that I look at it, that writing might be a little bit too small. Is it? I'll tell you what it says. Roughly speaking, there's lots of ways we can communicate mathematics. And there's an inverse relationship between how many people we can reach and how much time and effort we can ask of people. And I think everything is kind of a raid along that line. So for instance, if you go on national TV, you are reaching like a huge amount of people. I know you guys are all modern people who are on the internet all the time, but people still watch TV like in gigantic numbers, like much more than almost anything else you can do. But the amount of time you have is like incredibly limited, right? You can ask for a few seconds of people's attention. And then as you go down, probably a lot of you guys like me are on Twitter and post there, and that in principle has like a pretty broad reach. And you get technically 280 characters. I still keep mine to 140 because I think that's the correct way to do it. But I mean, you have a, but even so, what you can write in 140 or 280 characters is like people can think about it for like a minute, at least you have more time. And then in the middle are kind of like, the stuff we'll mostly talk about today, like magazine articles and books, things where an article you can ask for, you know, a half an hour of somebody's time, like a book, you can really ask for like, you know, 10 hours of somebody's time. And those have smaller reach than social media and TV. And then in some sense, like farther along the line, one thing I want to emphasize in today's talk is that I see this stuff about mathematical communication as an extension of the teaching mission that I think most of us in this room have, like whether we're grad students who are like training to teach or experience, high school teachers or educators at colleges, like most of us are teaching math to people. And I want to emphasize that that is another form where if a student is in our class, we get a lot of their time. And we can ask a lot of them, but maybe we only see like, between like 20 and 300 of them, depending on the nature of the class. And of course, the ultimate example is PhD advising, which some of us do, which is where like you get somebody for like five years, but only one of us, you know, just one person, but that's sort of the ultimate point on the curve. But I think somehow like this is a good depiction of the range of how mathematical communication can look. So maybe let me sort of give one kind of, because one reason I'm giving this talk here is because people do like ask me a lot, like how to start doing this kind of thing. There was a lot of interest, especially from earlier career people, about becoming more visible as a mathematician. So maybe just one important principle about that is there really isn't any official path, which is bad in the sense that it's hard for me to tell you what to do, but good in the sense that nobody can really stop you from doing it. There's sort of no official channel. There's no degree. There's no credential. You can just start in whatever way you see fit, whether that is blogging, blogging still exists, whether it's social media, whether it's home brewing your own YouTube videos. I'm thinking about how much I want to tell my own story. That's what I have in my notes to do right now. All right, let me briefly do it. So honestly, for me, I always wanted to be a writer. I took lots of writing classes when I was in high school and when I was in college. And I even, as Rafe mentioned, Rafe said he read my novel. Did you really? You're part of a very small group of people. Well, I know it's a very small group of people who purchased it. Maybe more people write it in the library or something. I don't know. So even to the point of going to grad school and creative writing after college and being like, I'm gonna be a novelist. This is what I'm gonna do. And it turned out that for me, that life was depressing and it stunk. And I miss math every day and I wanted to go back and do math. I will say, side note for the undergrads, I definitely recommend trying something else and then you'll realize how much you like math. Or you won't, I mean, by either way, you learn something useful about yourself. So somehow, sort of accidentally, over time, this writing thing kind of worked its way back into my life. And I'm happy to talk more about that. The only problem is that all of my experiences sort of learning how to do this in like the 90s and the 2000s, the media landscape has changed so much that I'm not sure how relevant it all is. But I'm happy to talk about it a little bit more later, like my journey into this. But one point I wanna make now, just to sort of emphasize something I tried to indicate by putting the two things on the same curve is that I do see writing about math and I would say by extension, any form of hour-facing math, any form of math communication has sort of the same thing as we're doing in the classroom. Not exactly the same, but it's on a continuum that includes it. And let me say two things that I mean by that. So one part of it is the empathy piece. So one thing I hope people will agree with this that like a thing that we have to do as teachers, that's absolutely essential. And that I think is a challenge, especially when we're starting out, is there's something that we know very well and we have to imagine ourselves into the mind of somebody who does not yet understand it well. That is an action of empathy, right? That's an action of sort of imagining ourselves into another person's situation. And it is not easy. It's something we have to learn to do and get experience with. It's something, you know, I always say this thing I learned from a Robin Gottlieb who I know some of you guys have worked with too. I've talked to some other people in the room about sort of the master teacher, Robin Gottlieb. And she was the one who taught me like, don't say stuff is easy in class. It's easy for you because you know how to do it. But it's not easy when you don't know how to do it. Almost nothing in math is really easy when you don't know how to do it yet. So this, this act of empathy is, is crucial to our classroom teaching and it's crucial to writing too. I mean, one of the main things we do when we write for an audience is try to imagine ourselves into the mind of the reader, put ourselves in their position and try to give them what they need to keep on paying attention to this thing that we've offered them, which is exactly what we're doing in the classroom. But the other aspect, which is maybe a little bit contradictory, I'm just gonna be a little provocative and call it marketing. Because what are we doing again? What are we doing in the classroom? It would be great if we were so clear and so interesting and so wonderful that all students had to do is just like listen to us and they would walk out like knowing everything we wanted them to know. But it's not true, right? I mean, in our interactions with our students, we're asking them to sort of spend some time and do some thinking and do some work beyond what's happening when they're right there with us. And so a lot of the work we do in teaching, I would call it marketing. We're trying to make it seem like it's gonna be worth it to do that extra work. And I think the only way we can do that is to really convey our own excitement about the material. So these two goals are, I mean, they're complementary to each other. The first goal is like, imagine ourselves into the other person and try to think about what they want. But you have to do that while also being kind of authentic and true to yourself. Like I think one thing I've learned about writing is that you can't write about something just because you feel, if your motivation for writing is, I feel like people should know this, it's not good enough, it's not gonna work. Like you have to actually care yourself. If you don't internally feel that excitement, if you just feel like it's important, people should know, like it would be good for people if they knew this, your kind of lack of intrinsic motivation is gonna come out on the page the same way, can people agree with me here? Like in the classroom, right? If they're teaching something and you yourself don't actually care, are there any teachers here who are so great that you can teach a good class when you don't care? I'll bet there are some who are that good. I'm not that good. Mostly what we're trying to do, in addition to giving the other people what they need to convey what we're excited about. And if we can feel that excitement like while we're writing or while we're teaching, I'm pretty convinced that that's a sort of necessity too, just as much of a necessity as the empathy piece. I mean, I guess for me, this is why when I write a book, I always like, I try to be organized. I write this sort of very careful outline of everything that's gonna be in the book and then I sort of send it to the publisher and they're like, okay, that's good, like write that. And then I start researching and learning stuff and then I sort of throw out all of that and then I write about like the new stuff that I'm learning in my research because that's what I'm excited about, right? When you're learning something new, that's exciting. That's what I put down on the page and I think that's better because I stuck with the original plan, which is maybe stuff that I already knew and sort of thought that would be good. I mean, maybe one way to think of it is like, especially if you're writing in a short form, what you're gonna convey is not very much. It's a little bit like the trailer for a movie, right? A trailer for a movie doesn't tell you everything that's in the movie. Like the job of the trailer is not to get every plot point of the movie. It's to convince you that it's gonna be worthwhile actually seeing the movie. And I think that's what we're doing especially when we write shorter pieces. We're not gonna explain everything we can't, not in that amount of space, not at that point in the curve with how much time somebody's giving us. What we can do is make it seem like, hey, if you want to, it would actually be worthwhile to learn more about this. And that's the marketing piece. Okay, where am I in my slides? So a few more general thoughts and then I'm gonna talk about this writing workshop that I led this semester and like what I learned there. So one thing I've learned doing this is that especially those of us who are teachers, we get very used to being in rooms with people who are there to talk math with us because they have to be. That happens a lot, right? If you're teaching in college, like most of our credits are for assigned, required courses. So one thing I've been surprised about just in this path of like doing a lot of like writing books and writing articles and newspaper and stuff is that people actually really wanna read this. That's important for two reasons. One, it's actually easier than you think to place articles and sell stuff because there actually is demand. And I think the people whose job it is to publish books and like edit newspapers are aware of this. But it also means that you can do stuff, but you might not think it's gonna work. So for instance, this was the header for a piece I wrote in Slate back when Mi-Tang Zhang first released his paper about bounded gaps. And sort of to my surprise it was the most, I've written a lot of pieces for Slate over the years. It's like I have a very long relationship with them. This was the most popular and I never expected that. I never expected that a piece about number theory, which I usually don't write about would be the most popular thing. So yes, we can have from our classrooms, sometimes we can get, we just don't know how much hunger there is out there to learn about math. On the other hand, we're still fairly weird people, right? In like how much we care. So let me also say a thing about this particular article, which at least for me was a good test case and from which I learned a lot is, I think what I didn't do here is really try to explain anything about what Zhang did. I think there's, I think it would be super hard to do that in the context of a thousand word piece aimed at non-technical readers. A thousand words is really short. That's like about three typed pages if you have a reference for that. So all I did in this piece is try to give some impression of why people thought bounded gaps was true. Like what is the, like how would it be something that people would even have an opinion about? Yes or no? And I think a general lesson about any kind of short form communication whether it's a YouTube video or a short article and this can be kind of hard for us is that you can kind of do just one thing. You kind of are gonna pick one idea and you're gonna get in, do it and get out. And that's, you know, we love to write like 75 page papers and get everything in. And this is a habit of ours as those of us who are researchers that we kind of have to work our way out of if we're gonna sort of do things that are gonna have any kind of reach. We have to sort of pick an idea and be like, even though there's so much other good stuff, my job right now is to convey this one single idea and if I successfully do it, that's a success. And I'm gonna talk more in a little bit about sort of academic habits to unlearn. And actually maybe this is a good time which is good because it's my next slide. Oh, so one other thing I was gonna say here is just that for those who have done it, I think of it as a little bit like, who's given like a 20 minute research talk? Like a group of people, right? And it's hard, right? It's hard because you wanna see so much stuff. And I'm sure none of you have ever done it but you've probably seen people who like try to give their 60 minute talk in 20 minutes, right? Don't do that. I know none of you guys have ever done it. But the point is when somebody says give a 20 minute, or the new hotness, five minute talks, if you guys been to a conference with Lightning Talks where it's like five minutes, I mean, you don't try to say everything, you're like, I just gotta pick one thing. And if I get out having said that, that's good. It's like that. So I mean, it's a different kind of skill from like writing everything. Okay, so one question I'm asked all the time and I just wanna address it directly is people say, but don't you have to dumb it down in a kind of honestly, slightly negative tone of voice? Maybe an accusatory tone of voice. No, I truly don't think so. In fact, by the way, I don't even like, I put it in quotes because I kind of wanna object to the terminology even as I mentioned it because I think people ask me because people who are not in math are not dumb, right? They don't know certain things that we know but they know lots of other stuff. I mean, it's, so I don't even like the phrase but I would also say somebody might say like, okay, if you were gonna like write about, I'm trying to be a good example. Okay, let somebody was gonna say, okay, you gotta write about like Peter Schultz, like write about perfectoid spaces in a thousand words. Like how could you do that without quote, dumbing it down? Well, I would just say, well, I wouldn't do that. Like that wouldn't be a good choice of a thing to write a thousand words about in the Washington Post. You gotta pick, I mean, you gotta pick the things to write about that actually can be, have this feature that you can pick out one idea that you can write about in a limited amount of space or again, like talk about in your video or write about on your blog or whatever. We get to choose which things we decide to communicate and I do think there's, yeah, there's honestly, there's some parts of math that are probably not suited to this kind of communication and I think we avoid this issue by just like choosing which things we're gonna communicate because I promise there's plenty. Right, there's plenty to do. Okay, so now I wanna pivot a little bit and talk about a new thing I tried this year which was super fun, which was I told you that I started out as a novelist. How many of you guys have ever taken like a creative writing course in high school or college? Look at decent number. Okay, so many of you know, there's like a very common style if you're doing like poetry or short fiction of like the workshop where the class is not really a class where you're told stuff. It's a class where you sort of get together with your draft work and everybody like really like digs into it and critiques it and tries to understand what's working and what's not. And I thought because I know a lot of people have an interest in sort of like doing this kind of communication to the general public I thought it would be interesting to try to transport that mechanism, the mechanism of the writing workshop that I learned from back when I was trying to write very sensitive and deeply felt short stories into this new context. So I did it this past spring, I did it on Zoom because I wanted to get a kind of pretty broad participation and I felt like it was probably the last semester that people would be willing to do a Zoom course. So I was like, let me do this now. And I picked eight students and we had a great time. I don't wanna tell you sort of some of the stuff we learned because I felt like I learned a lot. I thought I knew a lot about how to write about math and science for the public. But I got some good new lessons for this that I wanna share with you. But let me just start with like a slogan. Because a fundamental question right here, like what is good math writing anyway? And honestly for me, this is gonna be a little bit ideological. I basically think good math writing is good writing. I think there is so just understand where I'm coming from. I think what makes math writing good is not that different from what makes a short story good or what makes a news article in the New York Times good. I think there are some sort of basic principles of effective, not just even effective but lively English prose that are pretty general which is why I thought this format might work. And let's see if I can get you guys to agree with me. Okay, oh here is my little logo and you can search this and look online if you wanna sort of see the solicitation for this workshop. Okay, so here's I think the most basic thing. Now the rest of this talk, which is not gonna be so much longer, is just gonna be about some of the lessons that we found ourselves talking about again and again during this workshop. So again, maybe I should say something about the format. So again, eight students, eight weeks, each week a student has submitted a draft of a piece in advance and we just like spend an hour going through it word by word. Well first of all, everybody had an advance, right? So we had all like written voluminous comments on the piece. This is not all math by the way, as you'll see. Like it's basically all the sciences but like I feel like the principles should be the same. And then we just really got deeply into it and I gotta say guys, it was super fun. Part of my ulterior motive here is I'm trying to convince other people to lead workshops like this so I'm emphasizing how fun it was but I'm not just marketing like it really was fun. So but here's a principle that we found ourselves coming back to again and again that you should ask every single word in your piece what work it's doing. This is certainly something I learned when I was like trying to learn how to write fiction and it's not the way we customarily read. It's something that has to be learned and it's something that has to be practiced. I mean, we're all like academic types, right? We read pretty fast and when you read fast you're not really reading each word. Reading each word is slow but when you're editing, you're on your own or other people's stuff. This is exactly what you have to do and you have to be like a little bit merciless about the words. Let me give an example. And by the way, I did present this and we got in a good fight about it. So feel free. Everything I said probably there was like almost nothing that we didn't workshop that somebody didn't disagree with. So if you think my edits are bad, that's cool, we can talk about that. But so here's a sentence. This was actually from somebody who's working machine learning that appeared in a draft. You can't really escape the proliferation of these models. This means like AI models, not like fashion models. And your everyday life. So we talked about this sentence and what did we say? Well, the first thing we said is like, anybody wanna nominate a word they think is doing like less work in this? Okay, in the back. That's my first one too. Okay, I think that's, I agree. First place to look is the word really. I'm not against the word really, it has its place. But here it's just kind of acting as a sort of a soft intensifier. It's hard to say how the sentence would be materially different without it, except that it would take a little bit shorter to read. So let's just do it. Let's see how it looks without that. Now when we think syntactically about this sentence, you can't escape the proliferation of these models. Well here's the thing. We're saying that the models are proliferating a lot. There's a lot more of them, but it's not like their reproduction that you're trying to avoid, right? You're trying to avoid the models themselves. And so this isn't what happens a lot. Like the writer is trying to convey a lot of information, but in a way that's syntactically like a little bit confused. The writer is trying to say, you can't escape these models and also these models are proliferating a lot. But it's not really what it means. So I would say, if we wanna talk about the proliferation, let's do that somewhere else and let's just say what we're here to say, which is that you can't escape these models. And then finally, where are you when you're failing to escape these models? Like where else would you be except in your own life? So maybe I would even go so far. You can't escape these models. That's what the sentence is saying. And this is much punchier. Now let me say, I'm gonna stop here for a couple of comments. One, I do think in the context of this particular essay, this was followed, because I was talking about this with somebody else, they were like, but maybe it means like, as opposed to like specialized situations like you're in jail and you're trying to get paroled and like the model is deciding, okay, so in this particular case, following this was some examples that were everyday life. So I'm like, that's already doing it. You don't need to say it in your everyday life because what's much better is like examples from your life showing the context in which a specific is generally better than an abstract. I also wanna say, I was talking to my son about this, he's 16, that I guess there's a Chrome extension or something called Grammarly. Does anybody use it? Anybody? Oh, okay. Anyway, my son was saying like, oh, it just always tells you like make every sentence shorter and admit needless words. So that's true, but I do want to, because I'm gonna talk about sort of concision in this talk, but there is a place for like long complicated sentences too. So he was just accusing me of being like a human Grammarly and just like making everybody's sentences short and boring. No, there's a whole nother thing about like the music of the paragraph and short sentences alternating with long. I think I won't get it to a here, but the point is even in a long sentence, every word should be doing work. The sentence is that don't need to be long, shouldn't be long, and that makes room for the sort of long, complicated sentence that are actually doing something complicated and have to be long and complicated because that's the work of the sentence. Okay, so that's an example of the kind of thing we do, but this kind of stuff I would say is identical with what we might do in a short story. So, and we did a lot of stuff in our workshop that was identical to what we would do in critiquing short fiction, but maybe I wanna, for our time together, I wanna focus actually a little bit on some things that are distinctions between our academic writing and this kind of popular writing, specific academic habits that I felt like I was noticing that my students had to unlearn. So, one of them is this, that I think in popular writing, it's really important for the author to somehow be feel present in the piece, for the reader to feel like somebody is there talking to them. And I bring this up because it's the exact opposite of what in mathematical writing we're trained to do. Like, we're sort of trained to present it as if like we are sort of floating heads made of ectoplasm who like sort of barely physically exist who are conveying this knowledge to you to the point that, you know, as you guys know, we don't even use the pronoun I in our math papers, right, we say we, who, me and who else? I mean, you know, it's, but it's exactly because we're trying to distance it from being like a personal statement by us. And yet in popular writing, the exact opposite is required. So here's an example of a kind of thing. I didn't have a good example from class, so I'm gonna use variables here, but there were definitely, people say things like this. Like, one might think of X as a form of Y. This, if we weren't talking about writing, this probably would not even stand out to you at all because it's a totally normal thing to see in an academic paper. But it is exactly not, I mean, right, like one, like who? Some unidentified being, and then by the way, and we're not even, and this unidentified one is not even thinking of X as a form of Y. They just might do that. So these verbal strategies are a way of distancing ourselves from actually being like, I'm here saying something. But again, as academics, I'm sorry, but we do that, right? It's our habit to sort of try to scurry away as fast as possible from the idea that I'm standing here before you telling you something. And that's the exact opposite of what we wanna do. I mean, side note, I actually took a great class in college on sociolinguistics that had a whole unit on politeness. Has anybody ever studied the theory of politeness? So there's a whole thing that existed every culture, by the way, and in English this is what we do. We're like, you know, can you pass this all? Like, well, obviously they can, but there's like a sort of a presumption against actually asking someone to do something. Instead, you sort of ask a, would it be possible for you, like could you? Like, so this is kind of a similar strategy, a sort of politeness strategy for academics, but we gotta get past it. We shouldn't say this in an article for the general public, we should say you can think of X as a Y. That's stronger. Or even, if you dare, think of X as a Y. That's a little more aggressive, right? It's a different vibe, and which of those is better actually sort of depends on the overall, I guess like, how comfortable you feel you've made the reader. They should be pretty comfortable before you're like ordering them around, like you do in the second version of this. But both of these have more of a feeling of like, I'm there with you. The first one, by the way, also, we're not even gonna talk about this today, but just as the authors should be present in the piece, ideally the reader should feel present there too. The reader should feel like they're there being addressed, and one sort of, I don't know if you consider it a trick or a technique is to actually explicitly have the second person appear. Okay, so that's one thing. The physical felt presence of the author is something that, where we learn as academics to avoid, and instead we have to embrace. I think another thing is that when we're writing a math paper, we never really ask ourselves like, what's the goal of this piece? Cause it's pretty obvious, right? We have a theorem and like in the abstract, we say we prove this theorem and then the goal of the thing we're writing is to convince the reader that we actually proved it. What I found, you know, reading my students papers is that for popular writing, it's actually not so obvious, even while you're writing what it is you're trying to do. Are you trying to explain some concept to the reader? Are you trying to persuade the reader to take some course of action? Are you trying to unconvince the reader of something that you think the reader might already think? Those are sort of three categories. They're pretty different. And I found that often because as academics, we're not really trained to ask ourselves, what is the actual goal? What's supposed to happen as a result of this piece? Because it's sort of somehow so obvious. Sometimes the piece itself seemed to sort of not be sure what it was trying to do. And we often had to sort of go back and figure that out in the course of our discussion. And so finally, okay, I guess one more thing, and this maybe is pretty expected, is that of course as mathematicians, we use a lot of language that's internal to our own discipline. This example is not from math. This is from one of my biologists. So this is an example of writing that in some sense is perfectly clear and the words are all doing work, but it's just not gonna work in the New York Times. I think if you look at it, you can kind of see like somehow it doesn't feel like popular writing. I wouldn't say it's jargon, right? It doesn't say like the MB2C64 repository. That's me pretending to be a biologist. I don't know any actual biology words, but their papers are filled with words like that, right? But somehow there are specific words, like a medical scientist would certainly say therapeutics and not like medicines, because I think therapeutics means something more specific, but I think we could say medicines. That would be okay. It's not so much that the word is wrong, but that it sort of signals something we don't want to signal about who's supposed to be reading this. Another thing, and this is actually really common in math, is that there are words that are regular old English words but that we use in a discipline-specific way. So I think here an example is like, two examples that struck me were like suspend and community. I mean, those are both words that we sort of think we know what they mean, but here they're being used in a specialized sense and that can trip up a reader. So I mean, I think just one thing, sort of basic thing we have to unlearn is just our natural tendency to talk the way we talk to the people we know and just be aware. And one way to do that is in a workshop like this where I had people from a bunch of different sciences, a bunch of different sciences, and so when the biologist would write something like this, then the computer scientists and me and the statisticians and et cetera, et cetera, would sort of be like, oh, I didn't expect to see that word in that way. That said, one thing I don't want to do, and this is kind of important, is convey the impression that general audience writing should not contain any specialized language. Because one thing I've learned is that like, people love learning a new word, but I think you can do one. Remember I always said a piece should do one thing? I mean, I think if there's one concept that has a specialized word in our language and you just want to explain that, that's great, but then that has to be the goal. Like the goal is to sort of explain this concept and then in the end, I hope we all agree with this, like learning stuff is like pretty pleasurable and people like it and giving people the feeling that they came away knowing one more thing than they knew is an absolute seller. Like people like that and people will want to read your stuff if you can do that. So I'm just about done actually pretty early, we're gonna have plenty of time to talk. Let me just close by making one important point because I think sometimes when I talk about this stuff, especially when I lead with this sort of short litany of complaints, it sounds like I'm saying like, oh my God, like math journalism sucks so bad, like why aren't we doing this instead? Like we who actually know. That's not what I think actually. I think at least for me, my goal is that there should be a multiplicity of approaches and a multiplicity of places from which people can learn math. So maybe I should have said multiplication out of subtraction, I didn't even think of that. But all I'm saying is like, I think it would be actually terrible if like the journalists who do an incredible job writing about math were replaced by us. I think they should instead be supplemented by us. I think to be honest, look, I mean, to be honest, there should be ways for other people to tell our stories in a way that's maybe not like biased towards our particular concerns and ways of seeing the world. But I think it shouldn't only be that. I think people, mathematicians, and again, I'm gonna emphasize I said it in the beginning, but I'm gonna say it again at the end, mathematicians interpreted very broadly at every stage of their mathematical enterprise. I want them to be empowered. I want us to be empowered to tell our own stories directly to the public as one channel of communication. And I hope if people wanna talk about running further workshops like this, I thought it was great. And I think I do believe in it as a way to sort of train people up to do this and feel free to come and talk to me about doing that afterwards. So with that, I'm gonna stop and take questions which I really hope people will have because I would love to hear what people think or what questions people have. So thank you guys so much. Okay, are there questions? Comments, please. Yeah, over there. 5%, I don't know. Oh, so the question was like of the ideas that I have, which I took to mean like ideas about math and not like all ideas. Like how many translate into actual like research works? No, it's very low. I sort of took an estimate and said like 5%. And actually it's an interesting question because I think that's a good example of something that I think we don't convey to the public about how our profession works. Like how failure-rich it is. Like how much of it consists of like failing to do things again and again. I mean, in some sense, like what you read about in Quanta are the successes. And so already, so every question that gives me like gives me a good idea for a piece, right? Like what's it like to sort of have your job be something that you're gonna fail at almost all of the time you try? Yeah. Maybe it would be great for me if you want to say your name when you ask a question because it's still my first week and I'm still meeting people. So what's your name? Paul, okay. My name's Annalia. Annalia? Annalia. Annalia, I got it, okay. So I wanted to ask a question about your book. What audience do you think shape is aimed at? Oh, so I have a good story for this. So the question was like what, this is the new book shape. And the question was what audience is it aimed at? So I had a great experience with my publisher, Penguin Press, which is published both of my math books. When I first started working with them, when I sold the first book to them and we had our first kind of phone meeting and I was kind of nervous and I was like, okay, we should talk about like, who's the audience for this book? Like who's, who should I be thinking about? Who do I, like, who am I writing for? And what the people at the press told me is, don't think about that, it doesn't matter. Like that's our job. Like we're the people who sell books, that's our job. So we'll figure out how to sell it to you. Your job is just to write the book you said you were gonna write and then we'll figure out who's gonna read it. I found that extremely reassuring. And I've lived by that. So I mean, again, it's this tension between the empathy piece and the marketing piece that like, on the one hand, yeah, you wanna sort of like write something that's readable but the press, I mean, the Penguin Press has basically said, you should write what feels right to you to write and then we'll sell it. And that's whatever the author wants to hear, right? That's the greatest. So yeah, an answer to your question. I don't really write with a particular audience in mind except like humans, I don't know. Okay, I see one in the back that might be hard to see. Let me do the one in the back first. I'm pointing at you, yeah, yeah. You, who just went like this? Yeah, that's you. Hansen, okay. Oh, so this is a great question. So Hansen asks, as you've gotten more into popular writing about math, has that affected your academic writing? And if so, how and like, do I have like broad dicta about like how academic writing needs to change? It's funny, I said good math writing is good writing but I think by that I meant good popular math writing and I actually think sort of the way math papers should be written is in many ways pretty different. So I would say, I would say my popular writing has affected my teaching a lot. I feel like sort of somehow training and storytelling which is what you're doing when you write a book has given me more material for teaching and also just sort of like affected like how I see the project because I see those two things as very tightly related to each other. But yeah, the, I mean, I am not of the party that thinks that academic writing is super bad, to be honest. I think it has a very specific goals and I actually think like, I actually think it's mostly okay. At least in math. Okay, now there was somebody else in front who I, yeah. My name's Samir and I was just wondering actually how your short form writing and these ideas you're contributing to your short form writing, how that's affected your pedagogy and teaching and if you think that, I think that the empathy piece is really important and if there are ways to communicate as a teacher or a tutor that displays more of that empathy and great draws people in. That's a great question. So Samir asks like, how do I feel like writing these short form pieces and trying to sort of do this empathy piece has affected? I'm assuming it's hard for people to hear the questions. Is that right or did everybody hear it and I'm just wasting your time? Okay, good. Also if I paraphrase, then I can re-ask the question as I want to answer it instead of as it was asked. So that's useful for me. And then like, just how has it affected my, oh, what are ways to sort of convey that empathy more in class? I think what I'd say for me, I don't think it's so much the empathy piece. I think that writing a lot of these popular pieces has kind of, it's kind of dug out a good channel for me of like, as I'm thinking about a math concept, thinking about what it relates to like in the world. Or in for instance, I'll give you an example. One thing when you write for like news magazines like Slate or like newspapers or whatever, there's this concept called the news peg. Like if you want to actually sell a piece, like the best way is if there's something that everybody's talking about in this moment that you're like, oh, here's the math angle on that thing that you already know you care about. Right, that's a thousand times easier than selling a piece that's like about math just because I feel like talking about it. So I guess that habit I've gotten into from like selling a lot of stuff of like reading the newspaper and being like, what's the math angle? I mean, you guys probably know this, like an old book by John Allen Palos, the sort of like parent of us all in this realm of who I'm called a mathematician reads the newspaper. So very much that I think having gotten used to doing that makes it easier like in class, if I'm teaching whatever math concept I'm teaching to sort of notice the connections between things that students already know they care about and connecting that to things that I'm encouraging them to learn that they care about. Yes, and there, I agree. Adrienne, okay. Never say never, I don't write fiction anymore now. For the number of years, I always thought, tell me if you guys agree that people who are like old enough to know what I'm talking about. I thought a novel about a thinly disguised Sarah's Lang could be like really amazing. Does anybody agree with me? Like with like the files and the conspiracies, but also the, I mean, someday, maybe. And then you all know it's Lang, right? They'll like review it in the New York Times and it'll be like this fascinating character, but you all know I'm just ripping off Sarah's Lang. Okay. But no, I mean, yeah, I'll be honest with you. I think fiction is harder than this stuff. I mean, it's like really, really, really hard, at least for me. Wait, I forgot the second part. The first part of the question was, are you still writing fiction? And the answer was like, basically no. But there was a second part, Adrienne. Oh, yeah, in the, I mean, a little bit. Like in the novel, I learned a thing. Actually, this is probably appropriate for popular in a way. I mean, I learned a thing from one of my teachers. He's deceased now, but the novelist Robert Stone. Because I had this whole section and there was like four characters and Stone was like, you gotta understand that there's never a scene with four characters. Every scene only has two characters in it. That's the way scenes work. And so you gotta figure out of the four people who are physically in the room in the narrative moment, like which two are actually in the scene at any given time. And then I found that very inspiring and then so basically I like, I organized the entire thing around like four choose two. Then I sort of very explicitly was like, okay, let me go through all of four choose two and have like a section that's this, a section that's that. I don't have anybody noticed, but that was like my organizing principle of the section because I was like taking my teacher's words very seriously as you all should. Yes. Your name's what? Daniel, okay. Oh, so Daniel asked pop culture references yes or no. I would say yes, although it depends a little bit for the newspaper, definitely yes. They tend to date badly. So like in a book, I would do it less because ideally I try to, if I'm writing a book, I try to write a book that you can still read in two years and not be like, who's Kim Kardashian? Like what is that? Like what does that mean? You know what I mean? Like I don't want that to happen. Oh, did we already forget who Kim Kardashian was? Is that already an outdated reference? Okay. Who should I have said young people? Who's PewDiePie? I don't know. What my kids say. Okay, so I would say pop culture references like yes in a newspaper or magazine where people are gonna read it like that day and maybe not if you're writing a little bit for the long term. Yeah. Convey utility, is that what you said? Oh. Yeah, so Hendrick asked the question like how important is it when writing about math to write about like how it's gonna be used? Like how it's applied? And that's a great question because I think my habit as a writer has been to kind of lean towards that, to lean towards like something that's happening out in the physical world and like what is the mathematical side of it? So sort of an application first approach if you like. On the other hand, as I said, I have often found that when I write something about pure math with no application at all like this bouncy gap between prime things, people really like it and I'm always like well maybe I should do that more, I don't know. So I mean it's, I will say this, this gives me an opportunity to sort of say a slogan because there's really no right answer to that question. Like, does it make more sense to kind of just try to convey the beauty of pure math or whether to sort of talk about like real-world applications or sort of connections with a news story? You guys know Stephen Krantz, the professor at Washington University, he wrote a book about teaching that really influenced me a lot where he said like there's no one right way to teach math. You gotta sort of figure out what kind of teacher you are and then be the best version of that. And I think that's super writing too. So I think I figured out what kind of stuff I like to write and I try to be the best at that but none of what I say should be like, I mean, you should write the kind of things I write. You can only write the things you're moved to write. So I would say if what you actually care about, remember what I said about the marketing, if what you actually care about is the beauty of pure math then that's what you should write about and do as good a job of it as you can. Yeah. Wow, that's a great question. So how have my thoughts about writing changed from the first book to the second book? Something funny happened which is that I really intended, I was like, I don't want the second book to feel just like the first book. In the end, I think it feels more like the first book than I thought which is maybe relevant to my answer to Hendrick than maybe like in some sense we are the people we are and if we're writing authentically to ourselves since I'm not that different a person in 2021 than I was in 2014, like it sort of makes sense that the book comes out, it's sort of the same voice. I suppose the main difference is just that after like the first book came out and people bought it, I guess I had sort of somewhat more confidence that anyone would read it. So maybe I wrote with a little more of a feeling that there might actually be readers and whether that makes the book sound different, I don't actually know. Questions, okay. And I'm not looking at my clock so somebody should tell me if we're running into the time. So just someone should tell me why I should stop taking questions. Five minutes, okay, so yeah. Yeah, so this is a good, the question from Johan is about like, what does the actual process look like of like selling an article to a publication? Like how do you start when you haven't done it before? So part of the problem is that this is exactly the thing where my experience is most outdated because in the old days, young people, every city had a kind of newspaper called an alternative weekly that came out once a week and was free and like freaking anybody could write there. Because they didn't pay. But so I used to, does anybody remember the Boston Phoenix? Any Boston people? Okay, I used to write, there we go, okay. Good, a woo for the Boston Phoenix. I used to write a lot there when I was in grad school I would write book reviews and stuff like that. So there used to be more print publications where the bar was a lot lower or even if you didn't really have experience because you weren't getting paid, you could just like, and they needed stuff. And like these magazines, oh, you guys, young people, it was a paradise and they were so thick and they were like full of classified ads. And anyway, but I think in some sense like social media and blogging have taken the place of that. So I would say that probably what I would say is, you're right, you say you put it up on your own website and nobody reads it, but not nobody, right? People sort of do like slowly grow followings and there's like lots of mathematicians out there who have like Twitter followings like in the thousands, right? So somebody's reading it. And I think that that's what in publishing they call a platform. They're like, what's your platform? And if you could say like sort of somebody exists who follows me, you know what I mean? There's some set of people and then you could do a random sample and figure out how many are bots, right? Whatever. But the other thing I would say is, but in terms of the logistics, usually for magazines, you would pitch first. So you would sort of just write an editor and be like, hey, would you be interested in the thousand words about blank? Here's my special angle. Like there's this new story that your readers are already reading about and there's a math angle that's not being covered. And like, would that be of interest? So I usually, I wouldn't write the thing first and then submit. I would like pitch with like a paragraph and then see if they, and then see if they bite. That's the typical way. And obviously everything gets easier once you've done it. Like, because once you've done it, then you have an editor you've worked to then you know who to write and there's like a lower barrier. But with a new place, I think that's what, that's what you usually do. And then, but then that paragraph has got a show that you know what it means to write for the general public. You know what I mean? That's where the work is. So that's where you're really like, is every work doing its work. It's like really important in that pitch paragraph. Yeah, way in the back. Hi Greg. Oh, great questions. So Greg asks, like, what are some principles for portraying mathematicians, whether fictionally or just like, you know, writing. And this is the thing I do in the books a lot. I do a lot of, I mean, I do a lot of history in the books because maybe this is worth saying. Because it's one way to inhabit the minds of people who don't understand something you understand is like literally go back in time and read things that were being written by the people who were figuring those things out. Because they definitely didn't understand the things we now understand because they hadn't been understood. So I find that a very useful, I go back to that well again and again if I'm trying to explain some concept to go back to when it was being developed because there's always a human story there. Like one thing I say in the book, you know, math is made of people and every single math problem is like a human problem. Something human beings were like trying to do. They had a reason to develop this concept and you go back and find it. So that's one thing I would say one way to depict mathematicians is like, not as like floating brains, but as like human beings who were like trying to accomplish something, not just like make up a concept to torment an 11th grader with, right? That's I mean, that's not why we do it. I would also say a couple of things. I mean one is the, I can't emphasize it strongly enough. The mere fact that there are mathematicians who are alive is news to most people. So even just like, this is what I mean, this is what I always say about the TV. I've not been on TV a lot, but like if you're there on TV and like you're there for like 20 seconds, I'm like, if all I did was they see somebody on TV and they're like, oh, there's like a mathematician who's alive. That's great. Like I feel like I accomplished something because you could go to school for a long time and not know that, right? And the other thing I would say is Justin I sort of mentioned at the beginning of that first slide is I certainly, you know, try in the books to sort of get something of like the range of humanity, you know, being mathematicians, like whether that's like, you know, both men and women or like people who are like not from Europe or like what have you, I think it's worth being like somewhat conscious about that when you write, when you're, because we are sort of here trying to like, describe this like barely universal human activity and, you know, given the historical training we get, it can be easy to sort of slot that into a rather narrow slice of humanity if you're not like at least a little bit conscious about it. Yeah, also in the back. Yeah, it's a great question. So as, wait, what's your name? As Jeremy points out, there's kind of an integral over time that I didn't do which is that like at any given moment there might be 30 kids in your classroom but in a life in teaching, those numbers do add up and the reach amounts to a lot over a long period of time. First, and then the question Jeremy asked was like, well, from what I've learned and what I was talking about today, like what would I say about like stuff the teachers should do in classrooms? And one thing is, how to put it? I'm extremely hesitant and I don't know what age groups you teach. I feel like K-12 teaching is kind of its own thing and I'm like very hesitant to say anything about it because I don't do it and I want to tell the K-12 teachers in the room, it's a challenge because people ask me stuff like that all the time and I'm like, I really don't know. I'm not in that room with the eighth graders and there's like a lot of stuff that's like different about teaching eighth graders than teaching college freshmen, which is what I mostly do. I guess with that proviso, what I would say is I've sort of tried to emphasize in today's talk, the continuity between like what I learned writing fiction and what I've learned about how to write popular math in that they both are narrative and both involve storytelling. And I guess I've come to see that as part of my teaching as well, that like when the students come into my classroom, it's in some sense gonna be a story. It's not always clear how to make that work, to be honest. I mean, it's easier to say as a slogan, but I think that there's something, I don't know, even as I say it, I'm questioning it, but you could tell me if you agree. I do think there is something universally humanly compelling about a story and a narrative. And if we can work that into our teaching, I think it's to the good. Do you guys buy that? I'm experimenting with saying it to see how it sounds coming out of my mouth. I think I believe that. Okay, well, I think we're at the end of the hour now. So let's thank Jordan for a wonderful discussion. Awesome. Thanks for coming, everybody.