Eugenia Cheng: 0:02

There are many situations where one plus one can equal something else. one plus one can equal zero. For example, if you turn a piece of paper over, and you turn it over again, that's like having turned it over zero times some people say, oh, that doesn't count. you can always say something doesn't count. But that's not what math does. Math goes, this is interesting, let's study it. The ultimate purpose of this is to stop closing off math. Things are not as black and white And for some people that horrifies them because that's been their framework, that's been their safety rails. But for other people, it sparks the curiosity that has previously been squashed out and that's why I want to bring it back.

Vanessa Vakharia: 0:42

Hey guys, it's me, Vanessa, and we are back for another episode of Math Therapy. I have a question for you guys. Have you ever felt like it's just impossible to win an argument or like you just can't connect with someone you totally disagree with, or, I don't know, maybe you have that friend, family member or coworker who gets so stuck in their own way of thinking that it feels like you can never find common ground? If that sounds familiar, you're gonna love today's episode. Look, I am a Gemini, which means I can always see both sides of almost everything. And honestly, it's a blessing, but it's also a curse because on the one hand, it makes me empathetic and open-minded, but on the other it leaves me torn, like, what's the right answer if both sides always make sense. And that's why I'm actually so excited for today's guest. Mathematician Eugenia Cheng is going to totally change the way you think about right answers. Her new book just came out last week, it's called Unequal, and in it she argues that one plus one doesn't always equal two. And I know that sounds wild, but I promise it's going to blow your mind. Eugenia is gonna help you see math and the world in a whole new way. She's going to give you strategies for embracing nuance, for finding bridges in places you thought only walls existed. And for teaching kids and ourselves that math isn't about memorizing right answers, it's actually about thinking deeply and seeing differently. So if you've ever wished you could connect more with someone you disagree with, or if you wanna get better at finding common ground in tough conversations, or if you wanna learn how to bring more context into math class, this episode is for you. Let's get into it. Eugenia. Hello. Welcome to the podcast.

Eugenia Cheng: 2:25

Thank you so much for having me. I'm real excited.

Vanessa Vakharia: 2:27

I'm really, really excited. So I've been a fan of yours for years for a variety of reasons. I will actually say the earliest one, which has nothing to do with what we're gonna talk about, but I remember there was this TikTok that went viral because this girl asked what math was and all these people went on to troll her. And then you sent the most beautiful response saying that actually asking what math is is a very intelligent question and not something we should be making fun of. That was very cool.

Eugenia Cheng: 2:55

I got so many emotional responses to that from people all around the world, and that led to my previous book actually, which was Is Math Real? How Simple Questions Lead us to Mathematics' Deepest Truths.

Vanessa Vakharia: 3:09

What did you tell this girl? That she was like the inspo?

Eugenia Cheng: 3:13

I think that by the time it happened, she was so over it. I think she was just done with the attention that this had got her, and somebody tried to interview her and ask her if she knew about me and there was some kind of weird, she knew about me because someone had made a video making fun of me and that was that. So I thought, you know what, maybe I'll just leave it.

Vanessa Vakharia: 3:34

Oh my God, that's so, but imagine, I love how I was like, I don't wanna talk about this at all, but now I'm like, wait a second. Imagine this beautiful situation where this girl gets trolled for being quote unquote dumb because she asked what was considered a dumb math question, but a mathematician is like, this is one of the most intelligent questions, and now I'm gonna write an entire book on it. Like

Eugenia Cheng: 3:56

It was a whole series of questions that people consider to be well, that people have been told are stupid questions. But actually it's a really deep question. Things like, why does one plus one equal two? Which is a really deep question. And so it unifies people who've been put off math with research mathematicians, because those are the two groups of people who worry about that question. And the people who don't worry about that question are the ones who are like, oh, math is easy. I'm a math person.

Vanessa Vakharia: 4:30

Okay, well this actually kind of leads in perfectly'cause we're gonna talk about your new book, where I believe one plus one equaling two is, is a hot topic. Maybe.

Eugenia Cheng: 4:39

It's always a hot topic, and it's always surprisingly divisive.

Vanessa Vakharia: 4:45

It's, it is very divisive and I can't wait to discuss, but so that we all know what we're talking about. Your new book is called Unequal. I mean, my first question is why now? Why was this the time to write a book about equality and fairness and math?

Eugenia Cheng: 5:00

Well, it's just always a pressing question, isn't it? And I, unfortunately, this book was kind of catalyzed by an absolutely terrible argument I had with someone that just went worse and worse and worse. And I'm not sure if I should really say this because I'm always saying, oh, math helps us have better arguments. And so I thought, I went into this argument thinking, I'm not going to, I'm not going to try and, tell this person they're wrong. I'm going to find a point of commonality using math. Uh, completely failed because this person was so far gone the other way and was refusing to acknowledge that there is anything ambiguous about equality and fairness. And all I was trying to say was, well, we make decisions, don't we, about what counts as equal and what, what fairness really is in society? And he was just like, no, we don't.

Vanessa Vakharia: 5:49

Wait.

Eugenia Cheng: 5:50

don't get to decide what fairness means. I get to decide what fairness means. And I just thought, oh no, that didn't go well at all.

Vanessa Vakharia: 5:58

Okay, hold on. Can we, I need a little, can I get a little more context?

Eugenia Cheng: 6:02

It was a long time ago actually, because books take a long time to come into being, and it was about racism and it was about what he was calling affirmative action. And I think there are different things that we can call it. But he was basically saying that we shouldn't help anyone because that's unfair. Because nobody and his, it got to the point where he was really saying that, that nobody has ever been disadvantaged, so why should we help anyone?

Vanessa Vakharia: 6:34

Ooh, that's a tough one to wade into.

Eugenia Cheng: 6:37

Yes. And so I was, I thought I was going to find some point of agreement where we could agree on at least somebody has been disadvantaged in life and that maybe they deserve some help. But it pushed him to the point, and I think that he didn't really mean it, and that he just couldn't bear to concede anything to me. So I didn't do very well because I, I, and I was really, I really thought we could just agree on one person in the world who had been disadvantaged by something.

Vanessa Vakharia: 7:02

Terry Fox. Terry Fox,

Eugenia Cheng: 7:05

or, you know, someone who's, I don't know whose parents served in a war and were killed and the baby was born, we couldn't find anyone.

Vanessa Vakharia: 7:13

But what about being born like without limbs?

Eugenia Cheng: 7:16

Without limbs, without being able to anything, he, he refused to concede that anyone had any disadvantages. And so we couldn't get anywhere. And so then I thought. I'll write a book about this.

Vanessa Vakharia: 7:31

it's funny you bring this up because just today I was talking about with my friend who's a therapist and we were talking about this idea. I feel like this is probably what this guy was doing even though I was not there, like, I'm sure maybe you even know what this phenomenon is called, but it's like when you believe something, in order to preserve your identity and value system, you have to like double down on that belief. Otherwise it like threatens your entire identity.

Eugenia Cheng: 7:51

Yes. That, it's cognitive dissonance. When the thing that you believe is being contradicted by some other information and you can't deal with that dissonance, and so you have to, in order to resolve the dissonance, you either have to change what you think, which is very hard if you're attached to it or you just have to squash out whatever the other

Vanessa Vakharia: 8:10

The other thing is, and this is yeah, I, that flat earthing documentary, have you ever seen that Flat Earther documentary? It was so fascinating because the guy kept getting evidence that the earth was round. And by the end, and I thought this was very reflective of him, he actually said, listen, like the scientist was like, we've like show all your experiments are failing. Like, and he goes, I cannot admit the world is round because I would lose all my friends. My entire community's built around this, everything I do. And he was basically like, I can't, I can't entertain it because my whole life would fall apart.

Eugenia Cheng: 8:41

Right, and I thought about this argument I had with this guy for a really long time, way too long. But this is what mathematicians do, right? We think about the same thing for a really long time and, and I decided, my best guess is that he feels that he has been disadvantaged himself, but nobody's ever acknowledged it. And because he's a white male American, he feels aggrieved that he never gets to. And so maybe if someone had just taken him aside and went, I see that you were disadvantaged in these ways, here, have a hug or something, now he probably didn't want to hug, but

Vanessa Vakharia: 9:17

But I think you're onto something.'cause that's actually kind of how I feel about math therapy and when people like have such a staunch view of who can and can't do math. When you're simply like, Hey, listen, like, tell me about your experience. Like, oh man, that must have been hard. Like people soften.

Eugenia Cheng: 9:31

Right, exactly. Exactly. What was it that caused this person to think this, it went, goes back something, so something really deep. Some experiences that they had in very formative years and that they can't give up on it now because they've built their, kind of personal narrative on this. And I understand that about math previous experiences, because I had that about sport. And in fact, a lot of what I understand about other people's math trauma comes from my sport trauma, which I feel is quite parallel and quite comparable.

Vanessa Vakharia: 10:01

Tell us.

Eugenia Cheng: 10:02

Well, when I was young, I did not play sport at home with my parents because they were not sporty. They were not into that. They liked music and they liked reading and they liked math. And so I did math at home with my parents in the same way that other people go into the park and they, you know, hit a ball around. And so I couldn't throw, I catch, I never ran, I couldn't jump over things. And so in all sports lessons through my entire childhood, I was so bad at it, so incredibly bad at it,'cause I'd never done it before and everyone else was ahead. And then, because I was good at academic subjects, people liked to make fun of me for when I was bad at something. And, and then the sports teachers liked kind of humiliating, not exactly humiliating, but they, but they were so incredulous that anyone could be this bad that they often said things like, well you are obviously not trying because there's no way you could be trying and still be this bad. And so, of course, what did I do then? I stopped trying. Because I was trying, I tried to get better at everything all the time. And I wanted to get better. I didn't want to be terrible, and the last person chosen on a team every time and everyone makes fun of me. And the kind of thing, especially in team sports, you let the whole team down every single time. And so I tried to get better, but once they said that, that I didn't try anymore because if they were going to accuse me of not trying, then I might as well not try. But then I did what people do with math as well, which is I had to back up my own not trying, by denigrating the entire concept of sport.

Vanessa Vakharia: 11:37

So true. This is such a, I'm, I'm, this is, yeah. Wow.

Eugenia Cheng: 11:41

And it took me years after leaving school to realize that that was maybe not necessary. And I think that some of it, some of it I still believe in because I still don't like the part of sport that involves trying to beat people. And

Vanessa Vakharia: 11:57

Another parallel in my opinion.

Eugenia Cheng: 11:58

Right. right. And so I don't like the part of math that involves com competitions in trying to beat people, and where people get excited because they have beat beaten somebody else. I really, really, that makes me very uncomfortable. But I now am quite obsessed with playing tennis, and so I have got over it, and I only got over it in the last few years. But honestly, when I, we found some free tennis courts in the park near our house, and the first time my partner persuaded me to go and play, I had a full on panic attack, just like people do with math, because I thought, I'm gonna look like an idiot. Someone's gonna humiliate me, I'm gonna miss the ball. We weren't even going to score or try and play a game, because every time we try to, every time in my past when it's, there's been a game with scores, I lose just always. So what's the point? I know I'm gonna lose. There's no suspense. There's no excitement. I just lose. And so I had this whole panic attack and I had to give myself this giant pep talk saying, you know, physical exercise is good for us, especially as we get older. It's really important. And anyone who makes fun of me, that's their problem. At least I'm trying, at least I'm trying to get better. I'm making an effort. I'm trying to be fit, you know, no one, if someone laughs at me, then that shows that they're a horrible person. I, and then once I've got over that and I've got, and I realized that no one was going to tell me I was bad, no one was gonna humiliate me, that it is in fact good for me. And I now appreciate that having some physical exercise is good for me in the same way that I've always appreciated the intellectual exercise is not only good for me, but it's fun. And so much so that, a year later, someone did make fun of me, but I was okay because I'd got over it. Someone said the funniest thing to me, he came up to me and he said he, he'd been watching me play, and he said, I was wondering if you'd like to play with my daughter sometime. she's 10 and she plays competitively. She plays competitively and she plays with me, but, but I, I think it'd be really good for her to play with someone who's average or below

Vanessa Vakharia: 13:53

Oh my God. Did you say yes? I mean, I guess that could be a cool opportunity. Maybe she'd be an Olympian one day.

Eugenia Cheng: 14:02

I don't, I didn't know. I didn't know what to say and because I'm still conditioned to be polite to people, I just kind of went.

Vanessa Vakharia: 14:11

Okay. But this story, this is like a love story. Like this is really giving rom-com because it's like you have the like, oh my God, like it sucked. I was pushed out, dah, dah, dah. And then you like come around, you finally try, you have the breakdown, but then you get over it and you start seeing the value in it. And then again, that bully, the, the childhood bully comes back in the form of a man asking you to play tennis with your daughter. And now you are healed. It doesn't affect you. But everything you've said is so parallel to math. Even the whole idea of like, oh, if I was gonna score, if the scoring happened, what was the point? Right? It's like, well, if grading's gonna happen, what's the point? And almost being like being able to be like, well, I'm not doing it to win the tennis match. I'm doing it because physical activity is good and trying something is good. It's like

Eugenia Cheng: 14:56

Right, and it doesn't matter. I don't need to compare myself with other people. I just want to feel like I'm getting better. As long as I'm on a trajectory that's getting better, it doesn't matter what everyone else is doing because I'm not trying to win Wimbledon or something.

Vanessa Vakharia: 15:11

Totally, but like, I guess like we're talking about school I like, I'm like, yes, yes. And then I'm like, well, it just kind of sucks because school is kind of just designed that way as like to,

Eugenia Cheng: 15:19

Yes.

Vanessa Vakharia: 15:20

you know, to sort of sort people. So it's kind of hard to be

Eugenia Cheng: 15:23

Right. And that was what my, uh, previous book of mine was about X plus Y a Mathematicians Manifesto for Rethinking Gender. All about how one of the huge problems with how we present math and who we keep in and who we push out is because of how competitive it is all the way through. And so it appeals to people who have a competitive nature, but it also puts off people who don't like the fact that they got graded lower than someone else, or that they're humiliated because, so they found it harder than someone else. Not to mention the fact that often the people who find it harder. Aren't worse at it. They're just seeking a deeper understanding. And that's what I always try to say to, to people. If you, if it takes you longer, if you feel like you don't understand something, maybe it's because you are seeking a deeper understanding. And the people who said they've understood it, haven't understood it more, they just weren't seeking as, as much understanding. Oh, an analogy just popped into my head about, about about eating. So if you are full, it doesn't mean that you ate more than someone else. Maybe you just had a smaller appetite.

Vanessa Vakharia: 16:24

That's good. That's a good one. Whereas like the person seeking the deeper understanding, they're not full yet. They haven't gotten what they needed out of it. Like that's it. Okay, hold on. So I wanna, I wanna say one more thing before I get more into your book. But I actually have a question about what you were just talking about, the competitive nature. Like you saying that math appeals to those with a competitive nature. And in your book,

Eugenia Cheng: 16:48

Well, math, math class. Typical math education. Yeah.

Vanessa Vakharia: 16:52

But here's my question. Why is it that math education appeals more to those with a competitive nature as opposed to English education, let's say.

Eugenia Cheng: 17:04

I think it's because it is presented it, it is graded so quantitatively you can really compare people with other people. And it doesn't have to be like that. It could be presented in a way that is more like English, where you, it's not just this person has more points than somebody else. So if English essays were all just, this is how right you were, how right were you out of a hundred then? Then it would be much more obvious that there was a scale to compare people against. But English, everyone kind of knows that English, writing English things isn't about how right you are. And there are so many different things going on. And so if we presented math. In a way that was less about being right and wrong and more about how much you have expressed this and taking a situation in life and seeing what math we can find in it and what we can, how we can use that to illuminate our ways of thinking. Then it would be so much less a thing where we could compare everyone so directly and then it, there wouldn't be so much winning, losing, and humiliation involved. I think.

Vanessa Vakharia: 18:05

Well, this is really making me think too, that, you know, students like know everything. They're like dogs. Sorry, I'm not comparing students to dogs, just so everyone's very clear on that. Like they're very different. But you know how people say like, dogs are amazing, but obviously I'm like, you know how dogs can like sense things,

Eugenia Cheng: 18:23

Mm-hmm.

Vanessa Vakharia: 18:23

can like sense energies in people before, like even people can, I feel students are kind of like that. They're like savvier than we ever give them credit for. So even if we're not publicly, you know, like we used to back in the day, put their grades up. Like my grades used to be up on a door. Even if we're not doing that, I think you're right in the sense that they know that they are being compared to other people in this exact same way that you're talking about. Things are being marked right and wrong. Some person next to'em got more things right than they got wrong, whatever, and that must be, it is for sure, such a key part of math trauma formation is that idea of like shame or knowing you're being compared to, it's like that causes so much anxiety,

Eugenia Cheng: 19:02

Mm-hmm. And, and it, but it, it's backed up. It's kind of, played into by the idea that math is, has right and wrong answers. And so, the whole thing I'm trying to say in my next book, Unequal, it's the math of when things do and don't add up. And it's, it's kind of subtle that, because it's not that they do or they don't, it's that they do and they don't at the same time. And so things are equal and unequal at the same time So even if we say, one plus one equals two. There's something different going on. One of them is a one thing and a one thing, and the other one is two. It looks different. The left and right of the equation are visibly different. They're different. And so there is always something that's the same and different. And we have decided what we're going to take into account. And we make these decisions all the time in math, but we usually suppress them in math education because we're trying to teach people how to do it. And so we say to them, this is equal. We don't say, okay, here's how they're not the same. And so there was this incredibly frustrating memory I have when I was an undergraduate, when I was in a logic class and I didn't understand, we, we were proving that something or other was the same as something or other else. And um, and I didn't understand how they were different. And the tutor just kept saying to me, well, that's the whole point. We are proving they're the same. And I was like, yeah, but there must be something different about them. Otherwise we wouldn't have to prove they're the same. He's like, oh, but they're the same. Like so enraged.

Vanessa Vakharia: 20:33

Well that is, but hold on, you're kind of blowing my mind.'cause what you're saying is so obvious and so not obvious all at once. Like, I've never thought about this. Like, you're right. Like why are you trying to prove they're equal if they're so obviously equal? There has to be some dispute. So what's the dispute then?

Eugenia Cheng: 20:51

Right. And that we hold those things in our brain at the same time, which is an amazing thing to do, but we should realize we are doing it. And so there are all these different ways of thinking where you can change your point of view on what you are going to accept as a difference and what you're going to be really picky about. And this is a really great training for our brain, if we can bring it into our consciousness, because this is what we should do in life with people as well. Like, no, no, people aren't the same as anyone else. No one is the same. And so if we're going to talk about equality in life, which I think we all should think about it, but, we shouldn't think about equality because no one's the same. What we are really thinking about is what differences we are going to accept, and then we have to take responsibility for what we decide.

Vanessa Vakharia: 21:34

Okay, this is, this is wild because I get it and then I don't get it, and then I get it again, and then I don't get it. It's like at each point along, I'm assuming, tell me if I got this wrong. Along the decision making process or at each, basically at each specific juncture, you're in that moment deciding what differences actually matter and which don't. Right? Like, regardless of what we're talking about, what point in time, like, oh, I feel like I've got an analogy brewing. Hold on, hold on. Okay. I feel like this is a, like Coke and Pepsi are totally different, but Right. They're different. Okay. They're different things. There's different tastes, whatever. But like if I am at a bar that only serves a Pepsi product or a Cola product, it's like the same thing. I'm just like, oh, I just want like a cola. Like in that

Eugenia Cheng: 22:17

Yeah. Yeah, exactly. Great analogy. Yeah. Right. But some people would be really picky and be like, no, I absolutely do not want Pepsi in my drink.

Vanessa Vakharia: 22:27

Yeah. I, I would, I would only drink a Coke

Eugenia Cheng: 22:29

Mm.

Vanessa Vakharia: 22:30

Anyways. That being said, unless it was a diet of Pepsi, it doesn't matter. Okay. But, Okay. I

Eugenia Cheng: 22:34

And there is no right and wrong answer

Vanessa Vakharia: 22:35

Well, that was my question. because this is almost going back to like that guy almost, I don't wanna be that guy, but I'm like, well, who, when do we, who gets to, and I don't, I, I, I don't think it's a who, but like, how do we know when a difference is important or not?

Eugenia Cheng: 22:51

That's an amazing question. And that's what I think is really at the heart of math. It's not about this is the right answer, this is the wrong answer. It's not about this person's better, that person's better. It's not about what's the answer to this thing. It's about how are we going to decide what matters? And math is not usually presented as being a system of deciding how we think about

Vanessa Vakharia: 23:16

Like, it's not democratic.

Eugenia Cheng: 23:18

Well. That's a whole different thing. But math has a framework. So all subjects have a framework for how they're going to decide things, and math has a framework for deciding things, but each, each field of math comes up with a different notion of what's going to count as the same for those purposes. And there are reasons behind it, but in the end there aren't that many, there aren't that many reasons something would be a bad choice. So there are some reasons I might go into them in the book. So there are things like that, they're kind of technical. So if something is gonna count as the same as something else, then there are things like, there's reflexivity, which means everything should count as the same as itself and so then there's symmetry, which is that if a counts as the same as B, then B should count as the same as A. Then there's transitivity, which is about whether we can build up long strings of these arguments. So if A is the same as B and B counts the same as C should A count as the same as C, probably. But there are situations where that doesn't, that's not true either.

Vanessa Vakharia: 24:16

Okay. I have to tell you, this is bringing back a core memory of me at 12 years old where I wrote this poem and the poem was, oh my God, I have not even thought about this in so long, but maybe this is gonna be the moment where I finally understand this. I said, nothing is something, and everything is also something. Hold on. I had I, I, it was something like that. It was like If nothing is something and everything is also something, then is everything nothing. And nothing is everything. Like, do you know what I mean? that's

Eugenia Cheng: 24:50

Mm-hmm. Mm-hmm. Yeah. You were thinking about transitivity.

Vanessa Vakharia: 24:54

So in that case, is that true? Is something nothing because'cause nothing is something

Eugenia Cheng: 24:59

Mm-hmm.

Vanessa Vakharia: 25:00

So is something nothing.

Eugenia Cheng: 25:02

Well, it, I don't think"is", is necessarily symmetric

Vanessa Vakharia: 25:07

Oh,

Eugenia Cheng: 25:09

and because, and this is why in math, we use language very carefully. Whereas in normal life, we use language a little less carefully. And that's one of the things that can make math frustrating because we use the same words, but we use them differently. And so to make that more precise in math language, we might say nothing is an example of something.

Vanessa Vakharia: 25:32

Nothing is an example of something. True. But I guess now we're, it's not really transitivity.

Eugenia Cheng: 25:39

Now it's cl, now it's symmetry and it's, it's clearer. It's clearer that it doesn't work backwards because there are way more things that are something than nothing. So nothing is a thing. There are way more things. So not all something are an example of nothing.

Vanessa Vakharia: 25:54

This is so cool. Like, this is like such a, because we are talking about math, but I've always believed, and I feel like this has always been your vibe because you make math so human and, and this book kind of is, yes, it's math, but it's a way of understanding nuance, right? Especially in a world where like, it almost seems like we're so focused on our differences now. Like. This feels like a way to almost, be able to see that we all have at least one thing in common with someone. Am I like taking it too far?

Eugenia Cheng: 26:27

No, that's, that's right. And I think that it is still important. I'm not saying we shouldn't see our differences at all, because it's also important to see our differences, especially when we think about our past experiences. So we've all had past experiences that are really different, and that often plays a really crucial part in why we are the way we are now. Just as we were talking about with the arguments we've had with people and we think, wait, what in your past experience led take this position? And so I think it's also important to understand those differences, but also to, to not take those beyond where is necessary. So we can think about people's past experiences and think about how that affects how they behave now and where they are now and where they have got to. But then we don't need to go, oh, well this means you can't have this job

Vanessa Vakharia: 27:12

Mm-hmm.

Eugenia Cheng: 27:13

that, or the fact that the fact that you've cut your hair short. Does that matter for this job? Well, it depends whether you are trying to play somebody's body double. So I, I was talking about how for a job, it shouldn't really matter what you look like unless you are playing someone's body double.

Vanessa Vakharia: 27:29

Right. Can you tell me about one plus one equaling two? What's, what's the hot topic? What's the debate like? Is there a world where they don't equally each other? Like where can we

Eugenia Cheng: 27:38

Yeah, there are many worlds where it doesn't equal two, and mathematicians all know this, but then people who think of themselves as math people sometimes get so angry that we've changed the rules or we've, we, that this isn't real math or some kind math or this, yeah, exactly. It's some kind of woke math that lets more people in. Oh no, we've let more people in than we did before. But there are many situations where one plus one can equal something else. So, one plus one can equal zero. For example, if you turn a piece of paper over, and you turn it over again, that's like having turned it over zero times

Vanessa Vakharia: 28:16

What? Pause. Hold on. Let me just really observe this. If you turn.

Eugenia Cheng: 28:21

a piece of paper over, and then turn it over again. It's like having turned it over zero times.

Vanessa Vakharia: 28:28

Okay.

Eugenia Cheng: 28:30

And so some people say, some people say, oh, that doesn't count.

Vanessa Vakharia: 28:33

I was about to say that doesn't count!

Eugenia Cheng: 28:38

And the thing is, that's a valid answer, but it's not the one mathematicians give. Mathematicians go, well, that's an interesting scenario. Let's model that with a world. And so we get the Integers Mod two, where every time you do something twice, it goes back to zero. And so you can always say something doesn't count. But that's not what math does. Math goes, this is interesting, let's study it.

Vanessa Vakharia: 29:02

So why am I, let me see why I'm having the reaction that it doesn't count. Because I think I'm like one plus one. That means we're adding two things together. Is me flipping a piece of paper over once and flipping it over again. It doesn't feel like addition to me. Why is that?

Eugenia Cheng: 29:20

Because addition is usually presented in a very restricted context. Because it's often presented in the context where it has to equal two, because we're not trying to let in the possibility of other things. But then there will be some child in the class who's like, I've turned my piece of paper over again, and now it's the same way up. Now what? And so another of my favorite ones is when you press a button, when you call an elevator, you press the button and then someone else comes along and you know that thing where you've already pressed it and it's illuminated and someone else comes along and presses it again.

Vanessa Vakharia: 29:53

Me, I'm the second person.

Eugenia Cheng: 29:54

You are the second person, right. So, well, maybe we're having a theme here, so that does not call two elevators.

Vanessa Vakharia: 30:00

Hold on. You press the button. Yeah, I got, I, I, yeah, but I'm not, okay. Yes, you're right. You're right. But I'm not seeing it as a, why am I not seeing it as addition?

Eugenia Cheng: 30:10

I think because, you have been maybe trained to think of only the natural numbers. But mathematicians go, hmm, something's going on in this situation, let's model that situation. And so then there are places where you can model one plus one is two, and see what happens.

Vanessa Vakharia: 30:27

So here's my question. The juicy question here is, so what do we do now? What you're like, you know, we're in class the, the teacher's teaching math, and there's that kid that's like, but I turned my paper over twice, so one plus one does not equal two. But now we're in this classroom where the goal is they're supposed to be getting two, you know, in order to pass math class or whatever, but they're not wrong. Now what happens?

Eugenia Cheng: 30:51

Well, so then you go that's a really great point, that's a whole different mathematical world. Unfortunately, we don't have time to study that world right now because someone's decided you all have to pass this test. But then maybe if we have some extra time at the end of the week, then we'll investigate that world.

Vanessa Vakharia: 31:12

It is really interesting because it's like we we're in this kind of like very interesting place where, you know, there's a lot of backlash against this idea of like, woke math or like being like, we're letting too many people in by saying, all of a sudden there's no right answers, everyone can do whatever the fuck they want. And it's like, that's not what you're saying here, but I almost am curious what you would say to those who are like, well, hold on a second, you know, if there's no right answers in math anymore and everything is the same and it's different, like my whole world is rocked. Like how are they gonna pass this test? How are I supposed, like what, what do you say to

Eugenia Cheng: 31:42

So I am not saying there are no right answers.

Vanessa Vakharia: 31:45

Yeah. Okay. Let's clarify

Eugenia Cheng: 31:46

I'm saying that, I'm saying that math is a framework for deciding what is good information and that there are many different possible right answers in different contexts. That's the thing. And so it's not that there's no right answers, it's that we should be careful. And so we might say, oh, that's too much for children. And the thing is that if we limit in advance what we think children can handle, then we are limiting them. And I don't want to limit them. And children are usually capable of more than adults think they're capable of.

Vanessa Vakharia: 32:15

Well, you know what else, this is making me think. I love how you just said that. You said, the right answer depends on the context, basically, right? Like whether one plus one equals two depends on the context. And that kind of adds to the whole idea of like, we should be giving kids more context when we teach the math, right? Like we often are just like, just do this thing. And I'm not, I saying we as in like traditionally, historically, that's how math has been taught with very little context. And that's a complaint we hear a lot, right? Is like kids feel disengaged from it, disenchanted because they don't like, there's no context to it. So perhaps, you're kind of adding more like oomph to the reason why context is so important. It's not only will they be more interested, but actually then they have a deeper understanding of what they're learning. And it leads to this whole idea of nuance, which is one of the most applicable skills in the real world. The ability to see that context matters.

Eugenia Cheng: 33:04

Yeah, exactly. And this is the thing that if you, if you teach some rigid thinking, some kids. And they do really well at it, and they call themselves math people and they will then defend it to the teeth, to the death because they got all their self-esteem from being good at it. But then, like you say, it's way more applicable and transferable to learn how to think well than to learn some specific math techniques. And that if you learn some specific math techniques that help you pass a test, then all you learn is how to pass that test. And I get it, and I'm not blaming teachers for this at all, because teachers have their hands really tied by all the constraints that are put on them and how they are judged by what their students do in these tests. But the whole system is the problem. And that that if we teach everyone how to think, well, then hopefully they can pass tests and do more things as well. And so it's that much wider applicability. And too often applicability for math comes out to be things like, oh, we shouldn't just do one plus one equals two. We should go, if you go into a store and you buy eight watermelons and then you, that's not applicable at all. That's just contriving some kind of thing that's gonna count as a real world situation. And like you said, students and children are very perceptive. They can sniff that out from miles away.

Vanessa Vakharia: 34:19

I always say like, I'm like, what's relevant to kids doesn't mean like what is on the planet we call Earth. Like, oh, because you're adding an apple plus an apple. Now it's relevant. Like what's relevant to students is like, what's relevant to them right now in their lives? What's important to them right now? And what's important right now is exactly what you're talking about, the idea of being curious about, Hey, I'm having all these conversations with people, I'm hearing my parents talk about things in the news. Like this makes me wonder, like, is this really the same? Is this really different? Do these two differences between these people matter? Like that's relevant right now?

Eugenia Cheng: 34:54

But also whatever pops into their head is, is what's relevant to them. Because if you try and squash that out, and the thing is, I still have this problem when I'm in a math talk or any kind of talk, if something pops into my head and isn't resolved, I can't concentrate anymore because I'm trying to resolve this thing in my head. And so I find it really hard to concentrate in math talks because if they're interesting, then something will pop into my head and I try and resolve it. And if they're not interesting, then I'm not interested. So, and so if something pops into, if something pops into someone's, a child's head, and they get told no, you're not allowed to ask that question, and no, we can't do that now, then they'll just lose interest because this thing is their pressing question. And so whether it is, why does one plus one equal two, why can't we divide by zero? Or is infinity a number? Or what happens if I cut this triangle in half? Is it two triangles or is it half a triangle and half a triangle? Now, whatever those questions are is what is relevant to them now.

Vanessa Vakharia: 35:52

It's so funny, like just listening to you talk. I'm like, it's so wild that the idea of opening this up to like, Hey, maybe one plus one doesn't always equal two, and maybe we should be looking at all these different angles to this problem. Instead of focusing on the right answer, that gets labeled as like"dumbing math down". Whereas what we're doing is making it like just, even just hearing you talk about it, it's like you're actually making it so much richer and actually in a sense, way more complex. It's like the opposite of dumbing down, right? Like I'm just thinking about me sort of being like, but addition is just compiling things, right? And you're being so like nice and gentle and being like, oh yes, perhaps you were taught that way, but, but I'm like, I'm like, oh man, like I, I'm a trained high school math teacher and I am having trouble being like, wait, how is flipping a, I'm gonna really be thinking about how flipping a paper over and, and flipping it back is actually addition. Like, I'm all of a sudden in this one hour together, my view of addition is becoming so much broader and I'm really gonna interrogate that. Like, wow.

Eugenia Cheng: 36:56

Right, and again, there's no right answer. The question is, does this way of thinking help us?

Vanessa Vakharia: 37:01

Hmm.

Eugenia Cheng: 37:02

Does this open up some new avenue of thinking? And that's what I always think. It doesn't really, it's not really a case of whether it's right or wrong. So some things are wrong. There is definitely wrong answers in math. If you contradict logic, then you are wrong. But in terms of can we count this as addition or not? There are plenty of people who will say, no, we shouldn't. I'm like, well then you've, all you've done then is you've restricted your thinking, and I prefer to expand my thinking.

Vanessa Vakharia: 37:27

Are there situations in which it does not help to think this way?

Eugenia Cheng: 37:32

I, I'm going to guess yes, because there surely is some example that we can come up with. I mean, I don't like making sweeping statements, so if I said this always helps us, then I would feel very uncomfortable. So yes. So there are situations where we could take it too far. For example, if we think that$1 and$1 doesn't make$2, then we are gonna be in trouble with our bank account. And so we can't just make things up. The other thing I like about this approach is it gives students some, uh, of their own contribution into the conversation. So it's not telling them what to do, it's going, okay, can you think of some other situations? What do you think about this situation? What if you mix one color of paint with another color of paint? This was example that an art student of mine came up with actually, and I loved it. You don't get two. If you mix

Vanessa Vakharia: 38:21

you don't get

Eugenia Cheng: 38:22

Yeah, you don't get two, you just get another one.

Vanessa Vakharia: 38:24

You just get another one.

Eugenia Cheng: 38:26

The ultimate purpose of this is to stop closing off math to people. And, and what I would say, if anyone is wondering about specifics is that, if no one questions why one plus one is two, you don't have to go into it. But if someone does question it, and that's why it came up, because it's one of the things that people say, they say, why does one plus one equal two? And it's really hard. And I think it's amazing. We can think about it without really thinking about it. And sometimes there's a lot of math that you can do without really thinking about it. And then mathematicians sat down and went, wait, what even is one, what is two? We whole new field of math to do this. And then set theory was born and then they had to do a whole load of new definitions to go, what is a number? We didn't know? Help. Ah! And, and so that is where, that is why it is in fact very deep. And so if someone, the point for me is that if someone raised this outta curiosity, it would be to acknowledge that there is a really good point in there and that there are situations where it doesn't equal two. And my art students,'cause I teach art students at the school of the Art Institute now, almost all of them got put off math at some point in the past. And one of the things that put them off was the fact that it had rigid black and white answers and they weren't allowed to be creative. And one of the first things I do is get them to just dream up situations in which one plus one can be something else. And it's one of the first times anyone has ever invited them to dream in math class to dream up things to create. And that's what they want to do. And sometimes they say, oh, well I'm a creative person so I don't like math. And I go, wait. That's just how math was taught before. And so I want to offer possibilities for how you can be creative in math class and tie it into things that really are math. Because all of those situations, it's not just semantics or woke math or the girl math or dumb math or dumbing it down. They're all things, and I could say, you know, it's the integers mod two or the integers mod three, or it's a, projection in, in a monoid, or you know, they, they're all genuine things that we study and we then we study what the behavior is and what the consequences are. And so it's just all to say that, things are not as black and white as they seem in math. And for some people that horrifies them because they're clinging to those, that's their, that's been their framework, that's been their safety rails. But for other people, it sparks the curiosity that has previously been squashed out and that's why I want to bring it back.

Vanessa Vakharia: 40:47

It's, you know what is really making me think, the thing that makes people the most uncomfortable is uncertainty, right? And ambiguity. Because it's uncertain. We want certainty. So what's interesting is, what you're sort of talking about is uncertainty adjacent, let's just say, I've been thinking about this a lot lately of just so many of the solutions to our like quote unquote ailments, like whether health related, where, whether it's mental health, whether it's like needing a new hair color, like whatever it is, are targeted, like advertising is targeted as like, just get this thing and you're gonna be sure that blah, blah, blah, blah. Like it wants give you the sure thing. And I think this really all ties in because. That's part of our nature is we want security, we want certainty. We think certainty is gonna lead to security, et cetera, cetera. So yes, this would make people uncomfortable. But I think the other side of that is, I'm on a bit of a, a tear here, but it's to circle back is the point is that certainty isn't always better. Right? like,

Eugenia Cheng: 41:45

Right. If it's wrong,

Vanessa Vakharia: 41:46

if it's wrong exactly. Then not

Eugenia Cheng: 41:48

That is just wrong. Right.

Vanessa Vakharia: 41:50

Or it's like, you know, again, like with the food thing, I'm just thinking'cause like we're thinking of so many food metaphors here. I don't know why, like, it doesn't mean that what I'm eating isn't good, but it doesn't mean there isn't something I might like more, you know what mean? Like, and, and maybe I'm going to be a bit uncomfortable and try a few things that I don't really like and then I'm gonna find some food that I'm obsessed with, like whatever it is. But that is probably one of the like spiritually and philosophically and emotionally and like cerebrally, like, in order to really grow as people, what could be more beneficial than learning to become comfortable with uncertainty at an earlier age. So you can try new things and be open-minded.

Eugenia Cheng: 42:27

Right. And it's not just uncertainty, it's also the context thing so some things will be good in this context and some things will be good in this other context. And it's like the fact when people say, oh, this is good for you. Well, usually they did a study that said it found it was good for a certain percentage of people on average overall.

Vanessa Vakharia: 42:45

Or if it's good, if you don't drink 10 glasses of it.

Eugenia Cheng: 42:48

well, exactly. And so one example for me was the idea that, oh, physical exercise to circle back to where we sort of started, physical exercise is good for your mental health. Well, physical exercise wasn't good for my mental health when I was just humiliated every time I did it or I felt bad about myself, or I had to go into a gym where there were all these muscle bros grunting and dropping their huge weights on the floor and I didn't like that either. And so I had to find something. But I liked. And so it was much more helpful for me to go, okay, what physical activity exists that I could enjoy? Rather oh, I should keep doing this physical activity'cause it's gonna be good for my mental health. And so realizing that there are different contexts and then there's also, at one point I realized that if I was feeling slightly bad, physical exercise would make me feel worse. If I was feeling slightly good, it would make me feel better. And so it's these kind of subtleties that are much more helpful for our personal lives, our own lives, as well as for, it's not just about passing tests, it's about how are we going to live as humans in a community? How are we going to try to have better lives for ourselves and the people that we care about?

Vanessa Vakharia: 43:56

Mm-hmm. I love it. I love this so much and yeah, I'm even like, now I'm like, are ambiguity and uncertainty the same thing? I don't think so. They're not, they're not quite the same. So like I'm kind of went on a big uncertainty tear, but I think I'm more talking about being ambiguous, where there's no one right answer, it is context dependent, which can feel ambiguous when context isn't part of the convo.

Eugenia Cheng: 44:23

Yes, exactly. So sometimes it's not actually ambiguous, it's just that we didn't specify our context clearly enough, and that's what made it ambiguous. And math is a really amazing discipline for clearing all those things up so that we can be much more precise about what we mean, and hopefully we can be much more precise about what other people mean, even if they don't realize it themselves, and then we can all reach a better understanding of each other.

Vanessa Vakharia: 44:48

I cannot wait to read this book. Okay, so we, we have to wrap up but like I could talk to you for 10 years. Um, this is just so fascinating. And I mean, by the time this comes out, your book will have just come out, but I want everyone to know that the release date is September 2nd. And Eugenia just found out that that is Keanu Reeves' birthday. And how do you feel about that? He is an angel among us.

Eugenia Cheng: 45:15

Yes.

Vanessa Vakharia: 45:16

Like, right, like

Eugenia Cheng: 45:17

Yes. I mean, I've heard that he is just such a nice person. And Yeah. And it wouldn't it be great if everyone was a nice person?

Vanessa Vakharia: 45:25

Well, I mean, it would be so great I think that he would love this book, quite frankly. Um, so it's coming out on September 2nd, which is the anniversary of my book that came out last year on September 2nd. So we both share Keanu Reeve's birthday book release in common. So everyone go grab it. It's gonna be out by the time this comes out. Um, if there was, if there was a teacher listening to this, and they were like, oh, I love this so much, what's one way I could like bring this whole context, like equal, unequal, same but different thing into class tomorrow. Is there like a little thing someone could do?

Eugenia Cheng: 45:55

I think just pointing out what is unequal about something. If you are presenting an equation, whatever it is pointing out, focusing on the fact that it's different and the same at the same time, and congratulating everyone if they can hold both of those things in their brain together.

Vanessa Vakharia: 46:11

I love that. I really love that idea of being able to hold both of those things in our brains. I feel we're really, um, part of the reason there's a whole like war between creativity and math is because we like, live on binaries. Like it has to be one thing or the other. Like, do you know what I mean? Like, you're either a cheerleader or you're good at math. Never seen a movie with both. Like it's one or the other. You know what I mean? So being able to hold both things in our mind, first of all, allows us all to bring our full selves everywhere, right? To not feel like we have to fit into a mold.

Eugenia Cheng: 46:39

Yes.

Vanessa Vakharia: 46:41

I just love it. Okay. Are you ready for the final two questions? Question number one, if you were gonna pick one thing to change about the way math class is taught, what would it be?

Eugenia Cheng: 46:51

Uh, no grades

Vanessa Vakharia: 46:54

There it is. No grades at all, ever.

Eugenia Cheng: 46:57

Ever.

Vanessa Vakharia: 46:58

We're gonna have to do a whole other episode on this. Okay? No grades ever. And question number two is if somebody listened to all of this and they were like, no, I get what you're saying. Like, awesome. The thing is, I'm just not a math person. How would you respond?

Eugenia Cheng: 47:15

I would quote the dedication of my previous book, which is if you've been made to feel that you are bad at math, you didn't fail math, math failed you.

Vanessa Vakharia: 47:24

Oh my God, I love that. I love it. You're so cool.

Eugenia Cheng: 47:31

Oh, thanks. This was so fun. I do feel like we could just keep talking for years.

Vanessa Vakharia: 47:35

We, I mean, the metaphors we could come up with. I can't even imagine. We'd get, I think things would get weird, but I'd be down. Um, okay. Thank you so much. I can't wait to read this book. I can't wait for everyone to read it. There's been so much amazing stuff in here. It's really got me thinking, and I'm gonna go leave right now and start thinking of all the ways that one plus one does not equal two. I'm like looking around me. I'm like, I don't even, it's here. It's among, it's among us.

Eugenia Cheng: 47:59

Yeah, right. It walks among us.

Vanessa Vakharia: 48:00

It walks among us. Thank you so much,

Eugenia Cheng: 48:02

Thank you so much. That was great.

Vanessa Vakharia: 48:05

Bye.

Eugenia Cheng: 48:06

Bye.

Vanessa Vakharia: 48:09

Okay, I don't know about you guys, but my mind is spinning. One plus one not always equaling two is going to live in my brain forever, and I, I still need to think about it. But more importantly, Eugenia reminded us that context and nuance change everything, in math, in conversations and in how we see each other. Okay, so now it's time for your Math Therapy challenge this week. I want you to think of one relationship or conversation where you and someone else feel miles apart. Instead of asking who's right, ask what's the context and where's our common ground? And if you're a teacher, I want you to take this into the classroom. Think of one lesson this week where you can invite your students to notice nuance, where the answer isn't just about being right or wrong, but about the thinking and the context behind it. And don't keep this episode to yourself. I want you to send it to that friend, that family member, or that colleague who often gets stuck in black and white thinking. Honestly, it might change the way they think, and the way you two talk. I always wanna hear from you. So if something in this episode inspired you, you can now text the podcast, it's a new feature, the link to text us is right in the show description of your podcast app. You can also DM me on Instagram@themathguru, or email me at vanessa at themathguru dot ca. And remember, Math Therapy is now a weekly podcast, so make sure you're subscribed so you don't miss an episode. A new one is coming your way next week and the week after that, and the week after that, and you get the picture. Math Therapy is hosted by me, Vanessa Vakharia. It's produced and edited by David Kochberg, and our theme music is by our band Goodnight Sunrise. And guys, if you know someone who needs Math Therapy, please share this podcast rate or review it on whatever podcast app you use, because those things make a huge difference. I'm so determined to change the culture surrounding math, and I know you are too, and we all need to help each other out. So spread the word and until next time, peace, love, and pi.