[00:00:00] Bonni Stachowiak: Today on episode number 429 of the Teaching in Higher Ed Podcast, Who’s Counting? with John Allen Paulos.

[00:00:12] Production Credit: Produced by Innovate Learning: Maximizing Human Potential.

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[00:00:21] Bonni: Welcome to this episode of Teaching in Higher Ed, I’m Bonni Stachowiak, and this is the space where we explore the art and science of being more effective at facilitating learning. We also share ways to improve our productivity approaches, so we can have more peace in our lives, and be even more present for our students.

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Bestselling author, mathematician, public speaker, and formerly a monthly columnist for ABCnews.com, and occasionally for The Guardian and Scientific American. John Allen Paulos grew up in Chicago and Milwaukee, and received his PhD in mathematics from the University of Wisconsin. Now Professor of Mathematics at Temple University in Philadelphia, he’s married and the father of two, and grandfather of three, actually, at this exact date, three and a bit.

In addition to being author of a number of scholarly papers on mathematical logic, specifically model theory, probability, and the philosophy of science, Dr. Paulos has written Mathematics and Humor, I Think Therefore I Laugh, Innumeracy: Mathematical Illiteracy and its Consequences, Beyond Numeracy: Ruminations of a Numbers Man, A Mathematician Reads the Newspaper, Once Upon a Number, A Mathematician Plays the Stock Market, Irreligion, and A Numerate Life. John Allen Paulos, welcome to Teaching in Higher Ed.

[00:02:05] John Allen Paulos: It’s pleasure to be here.

[00:02:06] Bonni: It’s a real honor to talk with you, and I know you even have written an entire book about humor and math, so I thought we might start with a little bit of humor, would you give us the audio version of a joke about, if math went away?

[00:02:23] John: There was a genie out of the bottle, and talks to some little genie like figure, says, “I give you three wishes, what would they be?” The figure says, “I wish math would disappear.” Then, the genie says, “Okay, you’re all done, no more wishes.” [chuckles]

[00:02:42] Bonni: I both loved that joke, because, of course, so many of us can be intimidated by math, but I also particularly found it amusing that when I went into the comments on that particular tweet, the number of people who were like, “Wait, it’s zero, that’s still math.” I thought, “Oh, gosh,” [laughs] I just started shaking my head, “I think you missed the point.” There was another little bit of humor I thought you could start us with. I’ve been enjoying seeing a lot of puns on Twitter, would you give us a pun related to coffins?

[00:03:09] John: Okay. … says, “I wonder if glass coffins will ever catch on.” The response is, “Remains to be seen.” [chuckles]

[00:03:20] Bonni: [chuckles] Thank you very much. Thank you for humoring me. Well, we are not here to talk about that book today, although we certainly already have, but we’re here to talk about your newest book, would you tell us a little bit about the origins of it, and why you decided to write this particular one?

[00:03:36] John: Well, yes, the title is Who’s Counting, the subtitle is Uniting Narratives and Numbers with Stories from Popular Culture, Politics and Whatever. Since … math is a imperialist discipline, almost any other discipline can be invaded by mathematical ideas and techniques. The book covers all kinds of topics, and parts of it comes from my ABC news.com columns that I wrote.

I take some of the most resonant and relevant ones, as well as a lot of more topical columns on recent events, abortion, Trump, COVID, and so on. It’s a collection of columns, and the older ones are put into perspective, and often, unfortunately, things, [chuckles] they don’t change that much. A lot of the things that I complained about in a way in Innumeracy, an earlier book of mine that are still problematic, and is still the case.

So many of our problems that have a mathematical component are not due to, but are relevant to policy issues and so on. In that sense, almost every issue and many minor ones in this country are made more difficult by believing that, or at least acting as if the numbers, probabilities, and relative magnitudes relative to them, don’t really matter. They say, “Societal innumeracy remains a vastly underrated driver of that policy, bad politics, and so on.”

[00:05:19] Bonni: Yes, thank you so much for that. In addition to the lack of the literacy there, but then sometimes the lack of the ability to turn it into that narrative, and make it compelling, so that that tension is certainly important. Well, let’s start with something very important for many college students in some places and cultures, and that is pizza. Often thought of as a typical college food for, again, some students. Why is a medium size, such a bad move for college students, or really anyone?

[00:05:48] John: Well, it shows that people aren’t sensitive to scaling. If you have a pizza, which is one piece of which is 5 inches in diameter, another one, which is 10 inches in diameter, the one that’s 10 inches in diameter is actually four times the area of the one that’s 5 inches in diameter, because the area of pieces scales with the square of the diameter, not with the length of the diameter.

Area scales up with a square of whatever some linear measure is. Again, a 10-inch pizza is four times as large as a 5-inch pizza. With regard to drinks, it even worse, that scales up with the cubic. In fact, if you have one drink, the same shape cup that’s 5 inches tall, and another one is 10 inches tall, and say it’s a Coke or whatever, then the big one is 1,000 cubic, it’s not a cube, but a 1,000 to 125, is almost 10 times as much volume.

If one is conscious of scaling factors, one is aware that you should probably take the larger, even if you don’t finish it, you can get much more pizza, and give it to the person sitting next to you, [chuckles] if you like. Scaling plays a role in all kinds of things, not just in pizza, but model cars, and even a politician. Some policy works really fine in a small town, but it doesn’t scale up to a big town, or much less to a city, because you multiply it by some power of whatever, actual measure is in a small town, and that’s why so many brilliant ideas, which seemed to work in small towns or small states, don’t in large towns or big cities or nationally.

[00:07:45] Bonni: In addition to writing books, you also have been teaching for a while, and before we continue to explore, and we’re going to go into probability next, but before we do that, I’d love to have you share a little story about a time that you realized that students were behaving as if they understood what you were saying, and you realize you had to dig a little bit deeper.

[00:08:07] John: Well, I’ll tell a story. I was teaching for a summer course in Bangkok, Thailand, a friend of mine was there, and was at the university there. I was teaching a class of nursing students, and they were very pleasant and whatever. I was doing some mathematics, I think it was probability or whatever, and they would always say, [foreign language] which is, “Yes, I agree,” and nodding their heads in Thai. [foreign language]

Then, I realized I had misspoken, and I wrote something on the board that was wrong, and I still got the, [foreign language] so I made some other mistakes and the same thing happened. Then, I realized that has nothing to do with an agreement, they were just trying to– The students were delightful, but nevertheless, were just trying to be agreeable. I started saying complete nonsense, [chuckles] I mean, probably a marginally cruel pedagogical practice, I don’t know.

Again, they said. [foreign language] Finally, I said, “No of it, this is nonsense. I appreciate you’re trying to be kind to me, but, no, you should think. You should shake your head, no, and not say, “Yes’.”

[00:09:22] Bonni: To me, your sense of pedagogy came out in the book as well, because you don’t leave it as a only one-way conversation, because you ask a series of puzzles or riddles in the book, and then get us engaged to try to answer them before you give us the answers to how we might think about some of the problems that you pose in there.

Because I do think that some of that can be a cultural thing for sure, but I think there’s is that universal desire to not appear foolish, as if you don’t know, and there is still a lot of respect for teachers that comes in, and some of that can be helpful, but some of it can be really not helpful, in terms of us actually seeing what is actually occurring in someone’s learning process versus– As if we could ever truly see it, but to get better glimpses into how that learning may be occurring, versus the head nods which aren’t going to happen in a lot of different contexts, for sure.

[00:10:11] John: [chuckles] No. In fact, the math in humor, when people think it’s a odd juxtaposition, or a strange combination of topics, but both of them deal with logic, patterns, rules, structure, with a different emphasis, of course. In humor, the logic is often inverted, patterns are distorted, rules are misunderstood, but the transformations are not random. Math and humor are also economical and explicit.

A short proof is better than a long proof, and a long, shaggy dog story, after a while, it’s not funny. The shorter punch lines are. What is the case is that, there’s a continuum between math and humor, and somewhere in the middle are puzzles. Maybe the math is a little bit more significant than the punch lines of jokes, but there’s something similar to it than the– I think puzzles and jokes are a good way to impart mathematical ideas.

In fact, Ludwig Wittgenstein, a philosopher, once commented that he could write a whole book in philosophy that consisted entirely of jokes, jokes broadly conceived. Not just the one-liners, but little anecdotes and humorous little stories. If you got the relevant joke, you get the relevant philosophical point. For these and other reasons, personally, since I included a lot of puzzles in the book, Who’s Counting?

Because, as I say, I think they bring out ideas without scaring people with the equations and formulas. Often, the idea that can be understood by somebody who’s completely innocent of the relevant formal mathematics. There are a lot of puzzles, but not just mathematical ones, ones that have a social aspect as well. I ordered a lot of thought experiments, so to speak, I included as well. I try to get points across via little stories and anecdotes and vignettes, rather than more formal mathematics.

[00:12:22] Bonni: Speaking of philosophers, would you talk about the time you were able to go on The David Letterman Show? [chuckles]

[00:12:30] John: [chuckles] That was a long time ago when he was on NBC. I was impressed with him. He was really engaged and was quite positive. The first thing he says, “I’m going to take you out of that suit, if you bring up numbers again,” [chuckles] or whatever. I think we had a discussion of luck, because why I had Innumeracy, an early book of mine. It was a bestseller.

How did it get to be on the bestseller list? I said, “Well, a big element of it, as in life in general, for people in general, it was luck.” There was a cancellation of this other book on the Today Show, that the author got sick, and I was called in. Then, all kinds of other things developed from that. There was no internet in those days, but there was still a kind of hollow media mirrors, and one thing led to another. The term wasn’t in use in … This was the late ’80s or early ’90s. It wasn’t in use in those days, but because of this … the book went viral. It became a bestseller for five months on New York Times bestseller list.

[00:13:38] Bonni: You make me wonder what it would look like if we were to actually give more attributions to the power of luck in any of our lives to what happens to us, because it’s something that we don’t– I certainly don’t hear that happening very often in public discourse.

[00:13:54] John: Yes, that’s true. People don’t want to admit that if there’s something more than luck, you need skill, you need whatever, but you also need luck often. Many people don’t want to acknowledge that.

[00:14:06] Bonni: Speaking of luck, which would be one type of way of looking at probability, we’re really bad at probability, correct, [chuckles] as a society?

[00:14:15] John: In general, but people don’t realize that formal probability and statistics grew out of everyday language. Think of the terms for central tendency, mean, median, mode, and so on. Those are technical notions, but they grew out of everyday words like usual, customary, difficult, same, standard, stereotypical, and so on. Same thing for the precursors of the formal notions of statistical variation, standard variation, variance, and so on.

Those terms scare people, but they just grew out of, and are refinements of words like unusual, peculiar, strange, extreme, deviant, far out. The probability itself, it’s pre-staged in words like chance, fate, likelihood, odds, and we change our minds if we get new information. Everybody does that, but base them on probability, it tells you how to do that more precisely.

Bottom line is that, probability and statistics are not really alien ideas, even though people are generally quite bad at them. They’re in dire need of distillation, clarification, but too many of us are like the Monsieur character who was surprised that he’d been speaking in prose his whole life. [chuckles] We do speak probability, all of us, throughout our whole lives. Most of us speak it badly, but we do speak it.

[00:15:39] Bonni: You talk a lot in the book about the ways in which our inability to sometimes think in probabilistic terms. with regard to a story, an example you use of COVID hospitalizations.

[00:15:53] John: Right. People sometimes point to states that have a very high vaccination rate. This was earlier in the epidemic. Let’s say a state is such that 90% of its residents have been vaccinated. You look at the hospitals, and half of the people in the hospitals have been vaccinated, and a half have not, so people say, “Oh, see, it doesn’t make any difference. Half are vaccinated, half are not,” but the half that are not, come from 10% of the population that are unvaccinated.

The 10% contribute fully half of the cases, and the 90% also contribute half the cases, but at obviously a much slower rate, because 90%– Let’s say there’s 200 people in the hospital, 100 of them came from among the 10% who are unvaccinated, and the other 100 came from people who were vaccinated. Well, like a lot of things, people don’t critically interpret the numbers they’re given.

Even, for example, the five-year survival rate is often taken as evidence of the total … but that’s not always a good metric, the five-year survival rate. Imagine there are two states, and one of them mandates vaccines, and the other one doesn’t have vaccines at all. There’s a disease which generally arises, let’s say in peoples and one is 50s, and is almost always fatal by the time one is 65.

If you mandate screening tests for people at 50, they’re of course going to have 100% five-year survival rate, and the other state which doesn’t mandate anything, has no screening test, the symptoms show up in the early 60s, and so they a have a 0. Once the symptoms turn out, those people die when they’re 65, so they have a 0% survival rate. The state that has a screening tests mandate has a 100% screening rate, but the same number of people die in both states.

This is a special case, the numbers are just made up to show that, if a condition or a disease … of a good treatment, then the screening tests aren’t going to do anything. In general, people don’t really realize that, don’t critically analyze various metrics that are used. Sometimes it’s because of the innocence of probability. Sometimes it’s because willful– The desire not to face up to the thing, to whatever the situation is.

[00:18:31] Bonni: John, before we get to the recommendations segment, I wanted to ask you one question I’m not sure is fair, [laughs] in the sense of, it seems too broad and too small altogether, but I’m going to give it a shot anyway. We’re really bad at this stuff sometimes, and so what can we do to get better?

[00:18:49] John: I am tempted to give a … answer. Oh, you want my book?

[00:18:53] Bonni: [laughs] I just thought, I figured it’s one of those softball things, there you go, John.

[00:18:57] John: [laughs] Well, just paying attention to– There are reasons that people aren’t good at probability. One is, people don’t naturally think in those terms, but the more you are exposed to them, the more you will think natural in this, but people often– Their vocabulary regarding probability is quite limited. It’s either, sure thing, or 50-50, or impossible, and that’s hardly a discerning set of terms.

The second reason is, people often subscribe to the belief that everything happens for a reason, every great blade of grass, and, certainly, the scourge is trying to understand things probabilistically. Many people, for religious or other reasons, think that everything has a reason. Another obstacle is that much of the reasoning and many of the conclusions in probability are tricky and counterintuitive ones.

Standard issue is a birthday problem. You have 367 people in the room, you will be certain at least two have the same birthday because there are only 365, 366 days in a year. The probably at least two will have the same birthday is 100%, but if you have just 23 people in the room, the probability that 2 people will have the same birthday among those 23, it’s 50%. That seems counterintuitive.

Paying attention to numbers and putting them in context, numbers, probability, logic are still among our most reliable guides to reality. If we don’t critically examine them, then we’re reduced to relying on power and wealth in determining what is true and that, unfortunately, is becoming ever more obvious. That’s a bad thing for society.

[00:20:55] Bonni: Well John, that’s the perfect transition to the recommendation segment, because I actually would like to recommend your book Who’s Counting? I fit in the type of person who probably wouldn’t have maybe picked up this book, if it weren’t for the fact that I was excited about the opportunity to interview you. It was definitely a, “I will do this thing if I get this opportunity,” you know what I mean? [laughs]

I didn’t sign myself up for it, let’s just say that. I think it’s too often, we’re just able to leave ourselves where we are, and think, “I can’t do this. It’s too hard.” It sounds so simple when I say it that way, but I think it really can be a confidence builder. I remembered some of the things I had learned in statistics class and thought, “You remember more of this than you would have thought coming into this,” but yet a lot of it really did stretch me, but you never leave us behind.

I might not have understood [chuckles] every single puzzle or things how they worked out, but you kept me going, you kept my attention, and you cover so much ground in the book. I do really think that you are being both authentic and true and seeing a book like this. Your book can really help people to build confidence and to build competence, because I think confidence without competence is [laughs] not what we need more of.

I really would like to recommend that people pick it up, or some of your earlier books as well. I understand from hearing from other of your people who have been big fans of yours for many years now, also, really great reads. I’ve really enjoyed again, getting to read the book, getting to talk to you today, and I want to recommend that people pick up Who’s Counting? and give it a read, and let me know what you think. If you also get to build your confidence and competence as well.

[00:22:36] John: Well, thank you very much for those kind words.

[00:22:38] Bonni: Oh, of course. Yes.

[00:22:39] John: I really appreciate them.

[00:22:41] Bonni: What would you like to recommend today?

[00:22:43] John: I’d like to recommend a book it’s Thinking, Fast and Slow by Daniel Kahneman. It’s a summary of his work and Amos Tversky’s work in this examining various cognitive foibles to which we’re all subject. The availability area, the anchoring, and mistake confirmation bias. Actually, even there’s another bias that people aren’t aware of, it’s called the conjunction fallacy. If we have time-

[00:23:17] Bonni: Sure.

[00:23:18] John: -I’ll give a brief example. There’s this, Senator Jones, let’s say, and he’s a very upright senator. He’s middle of the road. He’s moderate and everything, he lives modestly, has been married for 40 years, he lives happily with his wife and sick daughter. That’s the background on him. Given that background, which of the following two options is more likely? One, he took a bribe from the lobbyist, and two, he took a bribe from the lobbyist, and used it to pay for his sickly daughter’s operation.

Given the background, many people would say, “Well, the second one is more probable,” but it’s not because it’s always, let’s say, the more conditions you have, the less likely they all are. The probability of one condition, mainly, taking money from the lobbyist, the probability of one condition being true is greater than the probability of two of them. In fact, there’s a trade-off between probability and plausibility. If you want your story to be more plausible, add more details.

Given the internet, there’s all kinds of factoids and non-factoids you can put together and concoct a plausible story, but the more you add, the less probable it is. It is this tension between the two. Again, it’s called a conjunction fallacy. Conjunctions are less likely than simple sentences, and long conjunctions even less likely. Anyway, the book Thinking, Fast and Slow, again, examines all kinds of cognitive foibles. Many of them with a mathematical flavor that can affect all our thinking. None of us are immune from some of these.

[00:25:08] Bonni: I’ve read it and I forgot, though that’s called a conjunction fallacy, but there seems to be such this tension between, like you said, the more details that get concocted, the more some people get swept away in it, and then the razor thing [laughs] where you go like, really, all of these things came together for all of these different actors.

I’ve been doing a little bit of work with the students I get to teach. I certainly don’t want to seem to be poking fun because, thank goodness, they’re in college, and they’re pursuing their education, but sometimes we’re really not getting our news from the greatest of sources out there. I came across a couple recently that really blew my mind, some of the things that they believe in.

You think about, how many people would have had to all be orchestrating together in order to have this happen? [chuckles] Again, I have happy news to report that I even think the one small little bit that we did on information literacy made a small difference. I recently did some reading, where you hear things like, small is all though. It’s those little tiny, small steps that we make that you don’t really know the kinds of changes they might have. We shouldn’t just give up, I guess, is the point I’m making, but, yes, the whole thing between the trade-off between probability and possibility is really, really.

[00:26:30] John: Actually, we started with pizza. Maybe we can end on pizza gate. The reason a lot of people believe that ridiculous stories about that pizza parlor and Democrats are torturing kids there or trafficking kids, it’s because there are lots of details which were absurd, but the more they added on, yes, there’s this and there’s that. There’s a quote I like from Mark Twain basically says, “It’s much easier to con people than it is to convince them, they can’t.”

[00:27:00] Bonni: I like that. I love that you’ve wrapped us up on pizza again, that was very well taught.

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[00:27:07] John: I got to tell it …

[00:27:08] Bonni: That couldn’t have been better, John. Thank you so much for being the guest today, and thanks for the wonderful book. I really appreciate your work and your time today.

[00:27:15] John: Oh, thanks for having me. I appreciate it.

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[00:27:19] Bonni: Thanks once again to John Allen Paulos for joining me on today’s episode. Today’s episode was produced by me, Bonni Stachowiak, and was edited by the ever-talented Andrew Kroger. Podcast production support was provided by the wonderful Sierra Smith. Thanks to each of you for listening.

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