Edward Doolittle

Edward Doolittle 0:03

I want myself and all students in mathematics to approach mathematics as whole people, the mental but also the physical, the emotional, and the spiritual. We often have this deficit thinking around indigenous culture. How do we help those indigenous people get better at math or do better at standardized tests. But we ought to look at also what indigenous people can offer the world. And we have a different point of view that can actually help the whole world and help us all learn mathematics better.

Vanessa Vakharia 0:35

Hello. Hi, it's me, Vanessa, and welcome to your weekly hit of Math Therapy. Okay. To kick things off today, I have a fun little multiple choice question for you. What is my favorite topic to rant about here on Math Therapy? Is it A cults, B astrology, C spirituality in math, or D, all of the above. If you answer D, then you are going to love this episode because it truly was a conversation written in the stars. So last month I was keynoting a conference in Saskatoon, and so was today's guest, Dr. Edward Doolittle, who I happen to have wanted to interview forever. During his keynote, he casually drops this absolute bomb that Pythagoras, like the Pythagoras, may never have even existed by that in entire cult of Pythagoreans did. So I literally went back to my hotel room and emailed him and was like, can I please interview you tomorrow, I need to talk about this immediately, and thank God he said yes, because it was one of the most fun and heartfelt convos I've had this year. Also, because this was a totally impulsive decision, I didn't have a mic with me, so I recorded the entire thing with my phone. Why am I telling you this? Because David forced me to tell you this because he was like, the audio quality is not exactly a hundred percent up to snuff. Please let them know. Okay. Anyways, Edward Doolittle is a Mohawk mathematician and scholar, and he shared a perspective that genuinely stopped me in my tracks, which is this: math is not something we discover or create. It's something we come to know. Seriously, wait till you hear 'em explain, it's gonna blow your mind. We talk about indigenous mathematical thinking, how math shows up in ceremony and games in the body, in the breath, and what it might look like to actually bring our whole selves for real into learning math. We also talk about historical figures like Kepler, who was both an astrologer and a mathematician, and suddenly, aren't you like wait, when did math get separated from all of that? So if you've ever felt like math was cold or rigid or just not for you, this episode might actually change your mind. And before we get into it, I want you to think of that one person who would never expect math to be talked about this way, or even better, that one person who's always hated math because it's so like, meh. I want you to send them this episode. And don't forget to text the podcast using the link in your app and tell me what you think, because I just know you're gonna have big feelings about this one. All right, let's get into it.

Pythagoras started a cult?!

Vanessa Vakharia 2:55

Edward, welcome to the podcast.

Edward Doolittle 2:59

Alright. Thank you. Glad to be here.

Vanessa Vakharia 3:01

I don't even know how to start really. We met, less than 24 hours ago. I saw you do a riveting talk at this conference, and I'm gonna be honest, I have a bunch of questions for you, but the first one I do want to just come in with is one that I've been thinking about ever since I saw you speak, because in your talk you told us that Pythagoras does not exist.

Edward Doolittle 3:22

Well, it's unknown whether he exists or not. I, I, this is certainly a mythology constructed around Pythagoras and I tend to believe that constructed mythology is generally maybe rooted in truth, but not true. And so, the Pythagorean cult certainly existed, whether Pythagoras was their master or not, oh, whether there's a person named Pythagoras, I think no one is really sure.

Vanessa Vakharia 3:45

Okay. But let's back up 'cause you just so casually said, Pythagorean cult.

Edward Doolittle 3:48

Yeah.

Vanessa Vakharia 3:49

Can you

Edward Doolittle 3:49

Well, they,

Vanessa Vakharia 3:50

I mean,

Edward Doolittle 3:51

they were,

Vanessa Vakharia 3:51

just so you know, you don't know this about me, but I'm obsessed with cults.

Edward Doolittle 3:55

Okay. Well then you're talk, you're talking about the right thing here. This was a cult. That worshiped number, and worship is not really the right word, but they, uh, a group of people who believed that, the universe was constructed out of number. And so everything was built from number and that you could influence the properties of something by making say a house out of, uh, 284 bricks would have have certain properties, My car has a license plate that has number 496 on it, and so that's a perfect number. The Pythagoreans would've recognized that as a perfect number,

Vanessa Vakharia 4:29

But hold on a second. Hold on a second. You're saying it's a group of people who believe the world revolved around numbers? Yeah. Isn't that just a mathematician or someone who believes that math is real?

Edward Doolittle 4:41

So I think, I think what we understand the difference between mathematics and a mathematical cult is that the cults can't, can't prove what they, they, they don't prove to be true what they believe. And so a lot of it was pronouncements of their leadership would say, this is the way we are going to be. So they had a number of obsessions. One was number, another was cleanliness, and their motto was Hygieia, clean.

Vanessa Vakharia 5:08

I'm sorry, what? Okay. So a cult that is obsessed with numbers and cleanliness.

Edward Doolittle 5:14

Yes.

Vanessa Vakharia 5:14

This honestly doesn't sound like a bad idea.

Edward Doolittle 5:16

Yeah. No, I, I might have joined if I were back in the day.

Vanessa Vakharia 5:19

Would you have?

Edward Doolittle 5:20

I might have I, we'll see what the membership fees are and so on,

Vanessa Vakharia 5:24

Because cults, I mean, they start off as a well-meaning thing, but they usually go awry.

Edward Doolittle 5:28

Yeah. And, and in this case, uh, they made a number of, of very interesting discoveries. And so our musical system, we were just talking about music. Our musical system is based on their discoveries.

Vanessa Vakharia 5:41

Like what?

Edward Doolittle 5:41

Uh, so they apparently, the, the myth is that one of their members was walking past a blacksmith who was hammering, and the hammering was making harmonious musical sounds. And so he investigated. Why, why? What are you doing? Well, here's, here's what I'm hammering is what blacksmith said. And the pythagoria noted that, well, these are two rods that are of, uh, rational ratio that they, that they have two to three or or four to five or some, you know ratio like that. And so they, they come, came to understand that, these ratios, certain ratios, certain rational numbers as we say now are associated with harmony and music and it's true. And so we built the entire musical system, 12 foot scale out of that.

Vanessa Vakharia 6:29

I am sorry. This cult sounds like it's actually been very productive and helpful.

Edward Doolittle 6:33

They were, yeah.. And so, um, they discovered, well, they didn't discover, but they may have proved, it's not really clear. Uh, the Pythagorean Theorem. Pythagorean Theorem was known for millennia, probably before that. Right. But, uh, they certainly gave a proof of it. Probably their best mathematician was a person named Hippasus, and Hippasus probably proved the Pythagorean theorem. A myth is that they sacrificed 160 oxen. When they discovered the Pythagorean theorem,

Vanessa Vakharia 7:01

Why? Just to celebrate?

Edward Doolittle 7:02

Uh, yeah.

Vanessa Vakharia 7:03

What about like a confetti cannon? Why did they do that?

Edward Doolittle 7:06

Well, what would please the gods? It's not a question of what would please you. You, confetti cannon. 160 oxen, gods.

Vanessa Vakharia 7:16

Got it.

Edward Doolittle 7:17

Okay. So, uh, yeah. And, and, but, but on the other hand, they're also known to have been vegetarians and so it doesn't make sense that they would sacrifice animals. To please the gods.

Vanessa Vakharia 7:28

That's actually, sorry, this has nothing to do with math now, but now I'm actually wondering if the two are similar. Like I guess if you're a vegetarian, maybe you care somewhat about animals. You're not gonna just sacrifice them.

Edward Doolittle 7:39

Yeah, I guess, I mean, it kind of makes sense so it raises questions anyway.

Vanessa Vakharia 7:43

It raises a lot of questions.

Edward Doolittle 7:43

The mythology is not, you know, self-consistent for one thing. they're also known, uh, to have hated beans, so they, they wouldn't eat beans.

Vanessa Vakharia 7:52

How can a vegetarian subsist without beans, what's their protein source,

Edward Doolittle 7:55

I don't know. So, again, doesn't make sense. so they had the thing against beans. Um, they,

Vanessa Vakharia 8:05

okay. This sounds like a really weird, kind of like picky, a bit of a diva of a cult.

Teaching through storytelling

Vanessa Vakharia 8:10

So here's the thing though, right? Is like you telling us this, I don't know if you noticed this, but when you started talking about this yesterday, people's eyes lit up, right? Everyone was like, what? Tell me more. And I've actually always thought, you know, we talk so much about like making math engaging and relevant, and we're trying to compare it to things in kids' lives. But this stuff, it's fascinating. Like if we, if we told more of. The history and the lore of the mathematicians that created the math we use, like, don't you think people would be like more into it?

Edward Doolittle 8:40

Yeah. I think it's, I think it's really interesting and it's part of what interested me in mathematics as a young person. Uh, I think we ought to integrate this into, into our education system. I don't know why it's not done that.

Vanessa Vakharia 8:52

Well this kind of leads to what you're all about, right? I mean, I actually heard most, the thing I heard most about you and saw that really intrigued me was that you tied, um, for lack of a better term, just using kind of the title of your talk yesterday, you tie spirituality and math together in a way, right? Like you're, talk you in a, in a sense, appreciate the spiritual nature of mathematics, would you say?

Edward Doolittle 9:14

I, uh, I'm, I guess I do, I'm very interested in it. Yeah. I mean, that's the idea of, of basic research, just curious about it and trying to learn about it. I believe it plays a prominent role, but, I don't think we understand that. We tend to shy away from that as a culture. I believe. You know, kind of western or, you know, non-indigenous culture shies away from spirituality. I believe that's probably due to a history of, of, uh, religious fanaticism and warfare in European history.

Vanessa Vakharia 9:48

Wait, tell me more about that. So you're, you, tell me more about how you feel that has caused us to sort of, shy away from anything mystical or spiritual when it comes to math.

Edward Doolittle 10:01

Uh, let me give you an example. I told you yesterday, one of my, my mathematical ancestors is Johannes Kepler. So he supervised someone who supervised someone and so on and so on. Who supervised my supervisor, who supervised me. So,

Vanessa Vakharia 10:13

so guys, this is like Six Degrees of Separation with Kevin Bacon, but for math, right? Right.

Edward Doolittle 10:18

Yeah, right.

Vanessa Vakharia 10:19

Okay.

Edward Doolittle 10:20

and people even more famous than Kevin Bacon. So, uh, Johannes Kepler,

Vanessa Vakharia 10:26

come on.

Edward Doolittle 10:28

But, Kepler, um, so he made his living doing astrology.

Vanessa Vakharia 10:33

I'm sorry, what?

Edward Doolittle 10:35

Yes. Yeah, Kepler was. Oh, kind of.

Vanessa Vakharia 10:39

I'm freaking out

Edward Doolittle 10:39

as foundation an astrologer.

Vanessa Vakharia 10:41

Okay.

Edward Doolittle 10:42

And so that's why he became interested in astronomy is to draw more accurate astrological charts.

Vanessa Vakharia 10:49

Okay. When, when was this? Like, what year are we talking?

Edward Doolittle 10:51

Uh, 1530, I think. Okay.

Vanessa Vakharia 10:54

Can I just say why this is. So crazy. Okay. Because I'm not into Duns and Dragons. Yeah. But I'm so into astrology.

Edward Doolittle 10:59

Right.

Vanessa Vakharia 10:59

Okay. Like I'm obsessed. I still dunno your sign, I will guess by the end of the interview. Okay. But what's so crazy is I always talk about how Astra like, 'cause people are always like, people will say to me, how can you be into astrology and all this woowoo stuff and also be into math. Yeah. And I'm always like, you know, back in the day, astrologers were mathematicians. You actually needed to be a mathematician to cast a birth chart. Right. And as a result, by the way, all of the astrologers were men. Right, because women were too fragile or whatever to do math and they weren't allowed to. So we took astrology very seriously, right. Now, things have shifted. There's an app you can use. You don't need to like, actually do any math to cast the birth chart. So it's not taken seriously. It's been divided, but, but astrology was considered one of the math. So I, and I didn't even know about Kepler, so this is actually very cool that you're bringing this up.

Edward Doolittle 11:45

yeah.

Vanessa Vakharia 11:45

Okay. Go ahead.

Edward Doolittle 11:45

Well, Kepler is, is kind of one of my heroes too. I'm, I'm, I was very, uh, gratified to, to learn that Kepler was one of my ancestors, just because he kind of crosses two worlds. That mystical world on the one side and, and the very rational. Uh, mathematical world that, that, and he, you know, found, he was at one of the giants, Isaac Newton said, I, you know, I've seen this further than others because I was stood on the shoulders of giants, he meant Kepler.

Vanessa Vakharia 12:09

What did he do that was like, so cool mathematically?

Edward Doolittle 12:12

Well, he, he took all these astronomical observations and then synthesized them into Kepler's laws that these are the laws of the motion in the solar system. and, uh, he actually made four Kepler's laws, I think. but we only, we only have three because his fourth law was not correct. So he was wrong. But that, that's part of being a scientist or being a mathematician. You're wrong a lot of the time too. And eventually you have to be able to double check everything that you say. And so that's an important part of mathematics and science. Kepler's a very interesting person. Kepler wrote, science fiction. He's credited with writing the first science fiction novel. Yeah.

Vanessa Vakharia 12:50

What this guy is really on it.

Edward Doolittle 12:52

Yeah. Yeah. Okay. Insomnia it was called. In Insomnia, uh, a demon communicates with the protagonist's mother and gives information.

Vanessa Vakharia 13:03

Okay.

Edward Doolittle 13:03

And so, uh, church, uh, Lutheran church found this and, actually seized Kepler's mother and threatened her, threatened to torture her. Oh, yeah. And so they didn't actually torture

Vanessa Vakharia 13:17

Well, because they thought that he was writing it about his own mother

Edward Doolittle 13:19

Yes.

Vanessa Vakharia 13:20

And that she was like demonized.

Edward Doolittle 13:22

Yeah. Yeah. Oh, wow. That he, that he and his mother had had, communicated with demons. And so they brought her in because they really wanted Kepler, but he was traveling. He was away. So they, they, uh, brought in his mother and then laid out the instruments of torture to show her these are the instruments of torture. This is what is going to happen unless your son comes back. So Kepler immediately returned to, uh, his hometown and, and, um. Face charges for, for, you know, this.

Vanessa Vakharia 13:51

For what?

Edward Doolittle 13:52

For, uh, for writing about demons. And, you know, I mean, he had to, he had to show that he wasn't under actual demonic influence, I suppose. I don't know. I don't know the details

Vanessa Vakharia 14:00

of you weren't like alive at the time.

Edward Doolittle 14:01

Uh, well,

Spirituality and math

Vanessa Vakharia 14:03

it feels like there was a time when mathematics and spirituality were not divorced from one another. when you read about like these ancient mathematicians and their discoveries and their lives and like all the stuff you're talking about, all the Lord, it feels like there was a point where mathematics was more mystical.

Edward Doolittle 14:19

Yeah.

Vanessa Vakharia 14:19

Am I right?

Edward Doolittle 14:20

Well, I, I, I believe so. And, and one way I think about this is to run it backwards in time. Mathematics was essentially astronomy if you go back years. Yeah. So they weren't called mathematicians. we were called astronomers. Yeah. And so

Vanessa Vakharia 14:39

this is blowing my mind.

Edward Doolittle 14:41

So, yeah. And so as time has passed. These disciplines have kind of, become their own thing. They're separated. So mathematics is separate from astronomy now, and separate from physics is separate from chemistry and so on. But you start to run it backwards in time and these things become integrated. And so, you know, it kind of makes sense and, and the philosophers and historians of, science and so on would known better than me. But my belief is that knowledge was much more integrated in the past. and possibly there are a time and time in the past where you just didn't have someone who was a mathematician, a medical doctor, whatever. You just had a sage. Knowledgeable person. And so, uh, I believe that indigenous cultures are still in that state. Where we don't have that division of knowledge.

Vanessa Vakharia 15:30

I would love to even just pick up on what you just said. That is such a beautiful thought that there's this idea of a wise one.

Edward Doolittle 15:36

Mm-hmm.

Vanessa Vakharia 15:36

Right. And mathematical knowledge is just integrated as part of their knowing.

Edward Doolittle 15:40

Yeah.

Vanessa Vakharia 15:40

Can you tell me a bit more about how it works and how indigenous mathematics is different in that way?

Edward Doolittle 15:47

Yeah, so I think, I think we need possibly to think about mathematical thinking as opposed to mathematics.

Vanessa Vakharia 15:54

Love

Edward Doolittle 15:54

it. We talk about mathematics. We we're creating that, that crystallizing that as a subject in itself. But when we talk about mathematical thinking, we can think, think about it in relation to all kinds of other thinking and, and it's really something all people can do. It's something that's universally open to us as human beings, mathematical thinking. It's just like, if I do this, what's the consequence? Or if I want this to be the consequence, I'll run that backwards. What I have to do,

Vanessa Vakharia 16:19

I need to stop actually hitting the table. Take the microphone. David will actually kill me. It was not me for once in my life. Oh my God. The bracelets. Oh, he's gonna kill me. Oh no, they're probably jingling. Oh my God. Fuck. Okay. Don't worry. It's all good.

Edward Doolittle 16:36

Okay.

Vanessa Vakharia 16:37

Okay, keep going.

Edward Doolittle 16:37

We're gonna need AI to process this thing. It's like, please take out the bracelet jingling.

Vanessa Vakharia 16:44

Okay, there we go.

Edward Doolittle 16:46

Anyway, uh, yeah, so,

Vanessa Vakharia 16:47

so mathematical thinking versus mathematics.

Edward Doolittle 16:49

Yeah. Mathematical thinking. And so I, I, you know, it's open to all people and we see it in children, we see it. It's part of being human, is being, is having mathematical thinking. and so, uh, we, I am interested in searching for mathematical thinking in my culture and other cultures, indigenous cultures. I'm currently working on a paper on this with my, uh, grad student, and he's done a lot of the work, we just have to kind of form it into a paper and submit it now. So the idea is that there's a ceremony we have and it's a game. So our mid-winter ceremony, my culture is, uh, Rotinonhsyonni, or, uh, Mohawk culture. Iroquois cultures, how it's often said in English. Uh, Rotinonhsyonni, we have made the longhouse. We have midwinter ceremony. It's our, probably our oldest ceremony. And so I participated in midwinter ceremony. And at the end of midwinter ceremony there's a game, Peach Stone Bowl game. And we, um, usually we put up stakes and it's, uh, usually something handmade, something very valuable in the sense that it's taken somebody a long time to make. A, a year of, of, you know, sometimes beautiful jackets that are beading on, put up on stakes. And, at the end of the game, there's a winner and, and the winning side will take all the stakes and the losing side gets nothing. The real interesting question, uh, which is raised to me by people playing the game, is how long is it gonna take to finish the game. Because it's part of ceremony, once you start, you have to keep playing it. And you know, you, you may have other things to do, a job to attend or whatever, but

Vanessa Vakharia 18:26

how long does this go? Like

Edward Doolittle 18:27

It could take about a week.

Vanessa Vakharia 18:29

Well, I Under a week.

Edward Doolittle 18:31

Yeah. So it kind of goes back and forth. You, you know, you, you get a winning outcome. You get some beans from the other side counters.

Vanessa Vakharia 18:39

Oh my god and Pythagoreans would have hated those beans. Am I right?

Edward Doolittle 18:41

That's right. yeah. and then so we use these counters and the game can go back and forth. Okay. And it can go on for days. And, and so the real interesting question is how long it does it go on for? It was designed to go on a certain amount of time, I believe. Consciously designed. It's not just random, but it's designed to take between three to six days or so.

Vanessa Vakharia 19:06

Okay.

Edward Doolittle 19:07

How do people arrive at that?

Vanessa Vakharia 19:08

At that, yeah.

Edward Doolittle 19:10

And

Vanessa Vakharia 19:11

you'd have to, like,

Edward Doolittle 19:11

we used markup theory and we used computer simulation. Those were not accessible to my ancestors.

Vanessa Vakharia 19:18

Hmm.

Edward Doolittle 19:18

How did they answer the question?

Vanessa Vakharia 19:20

Hmm.

Edward Doolittle 19:22

It was trial and error. I'm sure. I mean, I, I, I'm, I'm not absolutely sure, but it seems to be the, the, the obvious answer. That trial and error, was used to develop how many, how many counters to, to use to, to make the game last a certain amount of time. But there has to be some kind of mathematical thinking there.

Mathematics or mathematical thinking?

Vanessa Vakharia 19:40

Okay. I am totally with you here. This idea of mathematical thinking. Let me, tell me, I'm gonna reflect back and tell me if I've like understood this. Yeah. First of all, I love the idea that mathematics as a field is different than mathematical thinking. And tell me if you agree. I feel like mathematics as a field is very constrained. Like we're kind of like, it's this, that, the other thing, it's these things. Whereas mathematical thinking feels so much more open. Yeah. And broad and infinite. Right. There are so many ways to think mathematically about everything. You don't need to be thinking about something in the category of mathematical field.

Edward Doolittle 20:18

Well, yeah. I, I think that that is exactly one reason why we use mathematical thinking is to broaden our, our area

Vanessa Vakharia 20:26

and to show that it can be applied to anything, like, which is the next piece where you're like, mathematical thinking in Indigenous math is integrated into everything, including this game.

Edward Doolittle 20:35

Right. and so then the question becomes, what do we really mean by mathematics? I think mathematical thinking is also a little more well defined perhaps than mathematics. Some people say, well, mathematics is proof. and that's, it is kind of a go-to many my colleagues mathematicians will say that mathematics is, all about proof. And if you're not proving something, it's not mathematics.

Vanessa Vakharia 20:55

But if you're adding two plus two.

Edward Doolittle 20:57

uh, you can prove that two plus two is equal to four. But, uh, but, but

Vanessa Vakharia 21:02

go on. We have to think about this.

Edward Doolittle 21:04

There are, there no, there are lots of problems with that there, there's a conjecture as part of mathematics. Well, there's no proof there. It's just like a kind of an, uh,

Vanessa Vakharia 21:12

a hypothesis.

Edward Doolittle 21:12

A, you know, it could be intuition, it could be experiment that gives you inspiration. It could be just a lucky guess or whatever. These are, these are where conjectures come from. So these are not proof. So there's a whole bunch of stuff. But I think one of the best arguments is that we go back to a time, the first mathematician in the sense of proving things was Thales of Miletus, greek. And, uh, what about before him? Was there any mathematics before him? Well, yes there was. Of course, of course there was. There was. There's a whole, whatever human race, uh, we've had mathematics, we've had counting, we've had, um, you know, ideas of, of consequence. Uh, this happens then that happens in order. Uh, so ideas of ordering. All these things go back well before the existence of proof. So proof is not the fundamental part of mathematics. It becomes then more difficult to define what mathematics is. That's really the problem. And I ask people, and I get different definitions from everyone I talk to.

Vanessa Vakharia 22:09

What's your definition?

Edward Doolittle 22:11

My mine is going to be very strange. My definition

Vanessa Vakharia 22:14

Shocking.

Edward Doolittle 22:17

My definition is that mathematics is a spiritual being is like a being. Yeah.

Vanessa Vakharia 22:24

Say more.

Edward Doolittle 22:26

Yeah. Well, one thing we want to know is, is where did mathematics originate? And as I mentioned, it originated at the time of creation. The world was created, the universe was created.

Vanessa Vakharia 22:36

does it mean something to you that I have Euler's Theorem tattooed to my finger.

Edward Doolittle 22:40

Uh,

Vanessa Vakharia 22:40

in terms of what you're saying,

Edward Doolittle 22:41

I strongly approve. Uh,

Vanessa Vakharia 22:44

but don't you think it proves what you're saying?

Edward Doolittle 22:46

That, I dunno,

Vanessa Vakharia 22:49

because the reason I have it

Edward Doolittle 22:50

Okay, yes.

Vanessa Vakharia 22:51

Is because Euler's theorem, part of the idea of this proof is that it is so perfect and it's complexity that there's no way that humans could have created it. It must have been something that was more akin to what you're saying. It's something we discovered when we came here because it already existed.

Edward Doolittle 23:09

Right. So, so part part of my point of view on this is that we don't discover and we don't, we don't create.

Vanessa Vakharia 23:19

Okay.

Edward Doolittle 23:19

Okay. So you're a human being. You're a being.

Vanessa Vakharia 23:22

Yeah.

Edward Doolittle 23:23

I didn't discover you today. I mean, that's just wholly inappropriate to say, oh yes, I discovered you.

Vanessa Vakharia 23:27

Right,

Edward Doolittle 23:28

right. maybe your producer discovered you, but I didn't discover

Vanessa Vakharia 23:31

No, but that's like an ego-based way of thinking,

Edward Doolittle 23:33

Right, it's ego-based.

Vanessa Vakharia 23:34

Yeah. I see what that

Edward Doolittle 23:35

Exactly, it centers us. As human beings. The same with creation. It's centers us as human beings. But that's not, you know, mathematics exists independently of us and is something much larger than us. And so we come to know, we don't,

Vanessa Vakharia 23:51

we come to know,

Edward Doolittle 23:52

yes, we come to know mathematics. We don't discover it. We don't create it. So the debate in mathematics is, is discovery versus creation. And that's all ego driven. And we take the ego out of it. We humble ourselves, which is the indigenous way, and we now come to know mathematics as a being. And some of us come to know this being better than others. We, we, you know, I, I, I feel I'm on a first name basis with mathematics, but many people are not.

Vanessa Vakharia 24:22

I am getting emotional over this. That was really, really beautiful. And I think I, that I feel like that I just checked myself there to be like, oh wow. Like we are so egocentric in the way that we talk about math. I also think it's so beautiful, this idea of some come to know mathematics more than others. And I think this is beautiful because, you know, at its core my work is about math, trauma, and, and our relationship with math. And we come to know mathematics, at different levels, at different stages of our lives. And just because we, some of us are not on a first name with math, you know, a first name basis with mathematics now doesn't mean that we can't be. And I really wonder if, like, if we looked at math as more inter integrated in this way, you know, the way that we're like, you know, there are, we know things, we are engaged in mathematical thinking in all that we do. If that would change the way so many of our students feel about math. 'cause right now it feels like we are looking at math in the in, in isolation. Like it's this like bucket, like this, this what you were talking about. Like, oh, math is proof, or math is this thing. And if you're not in that bucket, then you are not engaged in mathematical thinking and you can't access it. But the way you're talking about it as, hey, math is here. We come to know it at different stages in our lives, in different ways. It is, it is there, you know, ready to greet us when we're willing and able, if that would change the icky, horrible way that most people feel about math.

Edward Doolittle 25:49

Yeah. I, I, I agree. I think, uh, this is an example of how indigenous culture can teach the whole world something. So we often have this deficit thinking around indigenous culture. How do we help those indigenous people get better at math or do better at standardized tests? But we ought to look at also what indigenous people can offer the world. And we have a different point of view. We have a point of view that can actually help the whole world and, and help us all learn mathematics better.

Vanessa Vakharia 26:23

How can we, if someone's listening to this right now, which I know they are being like, I love that. I love this. I teach math. How do I, what's something I can do right now in my classroom to bring this way of knowing into my room? What can they do?

Edward Doolittle 26:36

Yeah. that's a, that's a harder question. My, my area, of research is, is I guess you might call 'em basic research. I'm interested in ideas and, I'm not good at the implementation. I don't have a degree in education, for example. So I, I try to partner with other people to bring this into the classroom, and it's a slow process because there's so many factors. You know, I think there's this, also this religious. we were, we were talking about Kepler and, and the point of Kepler is that there are a lot of religiou violence, socio religious fanaticism in European history. The Lutherans versus the Catholics and on and on and on and wars fought and so on. And so I think that's why people shy away from that. And, they mistake spirituality for religion. And that's something our elders tell us is very different that we can, we can introduce spirituality into the classroom without interfering with people's religious beliefs. I believe that's true, but it takes agreement and it takes, you know, understanding. So it's gonna take time to build that. But I, I aim for that to integrate, I want myself and all students in mathematics to approach mathematics as whole people, which is the mental,, but also the physical, the emotional, and the spiritual. So we as if we, in that integrated way, address everything that we learn, including mathematics, I think it leads to much healthier outcomes.

The difference between religion and spirituality

Vanessa Vakharia 28:08

Can I ask to you what the difference is between spirituality and religion?

Edward Doolittle 28:13

Yeah, it's one of those things that, uh, I, I know it when I see it.

Vanessa Vakharia 28:17

Same like, like, so I'm only asked because I am the exacts. Like, we're so similar in this one. We're so dissimilar in many ways, but we,

Edward Doolittle 28:27

I don't know,

Vanessa Vakharia 28:29

but we are so similar in this, and I've always felt the exact same way. You know, to me, math has always had a spiritual, uh, role it's played in my life. Like the meditative nature of working through a, a beautiful problem. And, what that does for your mind, the curiosity and wonder and those aha moments, the questions, the curiosity, the, the intuition you get when you feel like you're on the right track with something, when you don't even know what a formula is that you might need. Right. Yeah. There's a lot in there. But you're right. As soon as you start talking about spirituality, a lot of people get their backs up, and I think you're totally right. It's because from spirituality, we get to religion. From religion, we get to divisiveness because that's how it's been used so often and there's kind of this shutdown. But if I never say the word. Right. And you probably, maybe, maybe experience this sometimes if I never use the words like spirituality or, you know, magic or whatever. And I, I, um, engage someone in an experience where they themselves feel it. There's something that happens, right there is this spark, there is something that happens there. So I wonder how like we as educators can create those moments for our students, without wading into the hot button of having to label the moment as spiritual.

Edward Doolittle 29:45

Yeah, I think possibly, yes. one of my models for this is not just indigenous culture, which is my main interest in being an indigenous person, but

Vanessa Vakharia 29:58

do people associate, when, when you're talking about indigenous culture, do you think people automatically start thinking religion?

Edward Doolittle 30:06

when we talk about things like creation story

Vanessa Vakharia 30:09

Yeah, I got

Edward Doolittle 30:09

people, people talking. You know, there's a famous example in, uh, the writing of Vine Deloria Jr. God is Red. Excellent book, I highly recommend. but he writes a story about, a, Christian religious figure, I believe a Catholic priest, I'm not sure though, in conversations with some indigenous people. And, and, they tell him their creation story and then he's listens patiently and then he replies with his creation story. And he said, they asked him, do you believe it? And they say Yes. Do you believe ours? And he says, well, no, we can't believe the two things. And he said, no. We said, we believed your story. Why won't you say you believe ours? That was kind of unresolved. So we have a history in North America, at least as indigenous people, of religious tolerance or spiritual tolerance that goes back to time immemorial that we, uh, respect the spirituality of religion of other people. And that was one of the most kind of confusing and astonishing things about the settlers is their lack of tolerance. So I don't think we've ever really come to, come to grips with that. But, we,

Vanessa Vakharia 31:25

so I guess we're like, we have to, in a way to make math like palatable to everyone, you have to take any sort of spirituality, like I'm using air quotes out of it.

Edward Doolittle 31:35

Well, it's, I don't know. I mean, not not just math, but all of education.

Vanessa Vakharia 31:39

Sure.

Edward Doolittle 31:39

Right. And people recognize that teachers are in a powerful position and can manipulate young minds. And so they want to be careful that the teachers don't contradict their religious beliefs and you know, I, I guess it's a, it's a complicated thing. So, yes, I think it's possible to maybe stealth introduce spirituality, but that again runs counter to some basic principles of informed consent. You want people to, give consent and give informed consent when we engage in any kind of relationship. And so education is exactly one of those things. And I think that's a problem with our education system, it doesn't have conform consent. It's a government saying, this is the curriculum, this is what happens in the classroom. It's not a matter of negotiation, which is really what it should. And so I think, I think that, that there's probably a better way to build a classroom, uh, experience is to gather everyone involved, uh, together and just say, here's what we hope to do, or here's what we hope to accomplish, let's all agree on some basic principles. Let's agree on the curriculum, let's agree on so on. And I think it's possible. Uh, I think this happens all the time. Uh, and for example, diplomatic protocol. You bring people from different countries together and, uh, they have to come to an agreement about how they're going to behave. You know, how hierarchical relationships are represented. Who goes first? Who goes second? Are you gonna shake hands? Are you going to bow? Are you going, whatever. So these things are, it's understood. Human cultures are different, but when we come together, we have to come to an agreement about how we're gonna proceed. And this is not done in education now, and I think it needs to be done. So I think that we need to, not try to sneak this in, is we have to be very clear and open about it and everyone should agree on it. But on the other hand, I think we can introduce spiritual experiences that are maybe universal. One of them is giving thanks. We all give thanks, all cultures give thanks and, you know, we do it maybe in different ways, but we can find a way to give thanks together.

Bringing your whole self to math class

Vanessa Vakharia 33:38

Tell us how to give thanks in the math classroom.

Edward Doolittle 33:42

Uh,

Vanessa Vakharia 33:43

you've got this.

Edward Doolittle 33:44

Yeah. I don't know. I'm sorry. I don't got this. Uh, I,

Vanessa Vakharia 33:49

no, let's, let's talk it through though, because, because let me tell, tell me if I think I'm on the right track. Something that actually a lot of research is now showing can be really useful is a gratitude practice. Yeah. And I actually think this is, well that's similar, right? Exactly. Yeah. And so a lot of our tutors before a student's math session will say, before we start our session, let's give gratitude to three things. Okay. Like, just three things you're feeling really thankful about. Doesn't actually even have to be about math. Anything. I'm thankful that I woke up this morning and it was sunny. I'm thankful for the, you know, my, the amazing breakfast my parents made me. I'm thankful that I, I walked here 'cause I live so close. It actually puts your body, mind, soul in a, just a better mind frame with which to tackle the challenge that lies ahead, which might be mathematics. So what about that?

Edward Doolittle 34:31

Yeah. Yeah.

Vanessa Vakharia 34:32

There we go. We just did it together.

Edward Doolittle 34:33

We did it. Alright.

Vanessa Vakharia 34:34

We did it. Yeah.

Edward Doolittle 34:35

High

Vanessa Vakharia 34:35

five. High five. Okay. But that's, I think that is, and I think, you know what, I do wanna pause and be like, look, we're using the word spirituality, we're talking about religion. It can seem like we're going really deep, but at when we strip it away to its core principles, I really like what you said. You are calling it a spiritual experience that is almost cross-cultural that, you know, giving thanks. A lot of people would be like, what the fuck are you talking about? That's not a spiritual experience. And that's why it's like, forget the word. It's like I say, we don't need to use the word math trauma, okay. We're just talking about did you have an icky root experience with math? Don't bring up a heavily loaded word if you think it's gonna throw people off. But, when we're talking about universal experiences, I think you and I would probably be like, a lot of those are spiritual experiences. Giving gratitude to me is a spiritual experience. But if that word is throwing you off, you don't have to use it. But I think those little moments, and I'm saying they're not little, but I'm saying something that takes five minutes, can actually put somebody into the mind frame of getting to know mathematics, right?

Edward Doolittle 35:36

Yeah. And so another example in indigenous culture that we use very often is kind of a cleansing ritual. Say for a pipe ceremony, we'll gather together, but we begin with smudging. So we'll smudge and it, it is symbolic cleansing. It's like washing, you know, I'm, I'm cleansing. Uh, and, and now I know to be clean. I, I'm going to treat others with respect in this space and so on. So that's another thing that can be done.

Vanessa Vakharia 36:01

I love this. Sorry,

Edward Doolittle 36:03

we're going back to the Pythagoreans.

Vanessa Vakharia 36:04

I was just about to say the obsession with cleanliness. Yeah. Um, but the idea of cleansing is to, tell me the purpose of it, because I, I also like cleanse my space. I do it. I did it before my talk yesterday. Like, I always like to, I kind of feel like I'm resetting myself and the energy.

Edward Doolittle 36:20

Uh, I am no expert. I'm no ceremonialist and so on. I participate in ceremony and I think that's an important part of, of becoming a complete human being. So, I long ago left a Christian Church. But I

Vanessa Vakharia 36:31

You were in the Christian Church,

Edward Doolittle 36:32

uh, was raised as a Christian.

Vanessa Vakharia 36:34

Interesting. Okay.

Edward Doolittle 36:35

Yeah, till 14, I, and it's kind of a long story, but I disavowed that.

Vanessa Vakharia 36:40

Got it.

Edward Doolittle 36:40

At that point. But I still feel the need for spirituality in my life. And so, uh, I have found an indigenous spirituality that, fulfilled that need. But, I've had to learn all of that because I wasn't raised that way. And so I, I attended many sessions all over the place. But when I was a student at the University of Toronto, they brought in ceremonials and spirituals and so on. Joe Couture was the name of one man, and he I think had a degree in spirituality and religion. So he talked about it in a rational way as well as just doing it or experiencing it. He said, this is kind of a reset. I, I'm not sure what the exact words he used. It's like a reset or a centering thing that you do is this cleansing ritual. So it prepares us for, to open ourselves up to the next experience that comes.

Vanessa Vakharia 37:29

Well, and I would say to that there is so much work around, um, the nervous system and polyvagal theory. So like, let's put spirituality aside for a second and talk about like cold hard science the idea of, we are evolved to co-regulate, right? So we walk into a room and everyone's energy around us, we absorb that energy. We want to kind of match people's energy. So the idea of a reset is very in, and the idea of grounding and centering, how much work have we done lately on the idea of putting our feet on the floor, you know, all of the breathing exercises, taking a walk around the room, use of fidget tools for kids who just need to center. So I would say that those two are basically the same thing. Again, we're using lang, different language. We're using the term cleansing and reset, or we're using the term centering, grounding, regulating, right? So I would say these things happen in math classrooms and can happen on a greater level. And I love the idea of really making it a part of, and again, I'm gonna use the word ritual, which you might be like, oh my God. But it's like, what is ritual? It's the same thing as a routine. That's all it is. Right? A routine that gets kids coming in. A nice cleansing might be something like, let's all take three cleansing breaths together, center ourselves, those breaths. Yes, they emotionally regulate us, but they actually put us into parasympathetic. They actually regulate our nervous systems. So we're now relaxed and ready to absorb the material. So I think in these ways, like these things you're talking about can really be brought into education. And you're kind of saying like you've seen through your research that indigenous mathematics creates this more whole picture of a person who's able to bring their whole self to mathematics because mathematical thinking is not separate from who they are as a person.

Edward Doolittle 39:10

Mm. That's the one of the major messages of indigenous education in general. So we we have to address the whole person. The whole person includes these spiritual things. And, and it's not like they're just made up. They're actually connected with the human, the human nature that these, I prefer the term ceremony to ritual. I don't know why. Okay. But that's been my education in this, is that we, we tend not to talk about rituals and why in, uh, in, I don't know, uh, in, in indigenous culture, we just don't use the word ritual.

Vanessa Vakharia 39:39

Interesting. And some people may hate both of these words.

Edward Doolittle 39:42

Well, too bad for them. I'm gonna say ceremony.

Vanessa Vakharia 39:48

Yeah. It's your interview. Say whatever you want.

Edward Doolittle 39:51

Okay. So, yeah, so, so. We, uh, we talk about ceremony all the time, and, and it's, I mean, there, it's so complicated and it's something that people study, you know, spend their lifetime studying. So I can't, I can't contribute very much to that. It's not my field. I can just say, people have thought hard about it that, that there's part of a universal human experience and that we all, you know, we have needs that need to be addressed, and ceremony helps with that. and, and, you know, anyone I think can benefit from engaging in ceremony.

Vanessa Vakharia 40:23

Ugh.

Edward Doolittle 40:24

And they find the ceremony that suits them. Yeah. It's, you know, I,

Vanessa Vakharia 40:28

I feel like going to the gym is ceremony for some people, like going to a spin class.

Edward Doolittle 40:32

Sure. I, I would not say no. I mean, people will find that they will find their own level, they will find their own way. But

Vanessa Vakharia 40:41

Did I misuse that term? I'm sorry. I don't wanna like

Edward Doolittle 40:43

No, no, no, no. It's not, uh, it's not a misuse, but it's just, what's may be lacking and, you know, may or may not be lacking in that context is guidance. And I think that's one thing that we need is guidance from, uh, more knowledgeable, more sage, more, uh, wise people in spirituality.

Vanessa Vakharia 41:01

Yeah. What an amazing talk. We have to wrap up.

Edward Doolittle 41:04

Okay.

Vanessa Vakharia 41:05

I mean, I can't believe it.

Edward Doolittle 41:06

Yeah.

Vanessa Vakharia 41:06

It's been so fun. Did you have a good time?

Edward Doolittle 41:08

I did. I did. I,

Vanessa Vakharia 41:10

I feel

Edward Doolittle 41:10

like I'm just getting started here.

Vanessa Vakharia 41:12

I know, right? We could talk forever. Yeah. This has been so amazing. There's, I mean, you're such a wealth of knowledge. We are just getting started. There's so much more I wanna ask and I just, I think it's really cool because I think sometimes we neglect like, like you in a way, like are, you have all of this wisdom, you don't have an ED degree, so it's not in the classroom, but it is wisdom that would be so useful in the classroom. So I really hope this bridged a bit of the gap and I'm really excited for teachers to hear this. If you're listening to this right now, I'm sure you have a million questions. Not only about like the Pythagorean cult, but about a lot of what, Edward has said here about how we can create a more holistic picture of mathematics. So text the podcast. You can hit the link in your show notes and text us.

Q1

Vanessa Vakharia 41:53

And I have to ask you the two questions I always ask everyone. Okay. Okay. You have 30 seconds to answer each one. Okay. Ready? The first one is, what is one thing, if you could pick one thing to change about the way math is taught in schools, what would it be?

Edward Doolittle 42:07

I think we should use manipulatives at all levels.

Vanessa Vakharia 42:09

Oh, whoa. Throwing in manipulatives out of nowhere.

Edward Doolittle 42:12

Well, it's, it's part of the whole person. Mental, physical, social, spiritual, or, sorry, emotional and then spiritual, uh, manipulatives are like the easiest way for people to get started in that journey around that circle. We're all mental now, and we need to break that and introduce more of the physical. So,

Vanessa Vakharia 42:29

oh my God. We could do a whole interview just on that. Okay.

Q2

Vanessa Vakharia 42:33

And now imagine someone's listening to this and they're like, oh, I love what you said about getting to know math and you know, all of this, but honestly Edward, I'm just, it's never gonna happen for me 'cause I'm not a math person. What would you say?

Edward Doolittle 42:46

Well, it's, it's part of the universe around us as part of ourselves. We're all math people. I'm sorry to say.

Vanessa Vakharia 42:52

Okay. And

Can Vanessa guess Edward's star sign?

Vanessa Vakharia 42:53

finally. I need to guess your sign, and I'm honestly, this is a tough one for me.

Edward Doolittle 42:57

Really.

Vanessa Vakharia 42:58

It, I,

Edward Doolittle 42:58

I think it's obvious, but

Vanessa Vakharia 43:00

do you I do. Okay. That

Edward Doolittle 43:01

I, I, on the other hand, I don't know as much about astrology as I should. I do draw charts for friends and family.

Vanessa Vakharia 43:08

You do,

Edward Doolittle 43:09

I Do

Vanessa Vakharia 43:09

You draw birth charts?

Edward Doolittle 43:11

Uh, yeah. I can draw various charts. just using, existing software and so on.

Vanessa Vakharia 43:16

Oh my God.

Edward Doolittle 43:16

Okay, so give me your date of birth. Your time of birth, and your location.

Vanessa Vakharia 43:20

You sound like me. This is what, this is now what I'm supposed to say now. Okay, so let me just do a mental, I'm trying to see if I'm getting an intuition. For some reason, I'm getting an intuition that I really don't think there's something about it that is not tracking for me, but I'm going to put that in the pocket of one thing and then I'm gonna go with my rational mind and think. I just, it, it's, honestly, this is a really tough one for me. Really?

Edward Doolittle 43:43

Okay.

Vanessa Vakharia 43:43

Are you an earth sign? Okay, so my first guess is actually gonna be Aries.

Edward Doolittle 43:48

No.

Vanessa Vakharia 43:48

Okay. And I didn't think that was right. Okay, so fine. My next guess is going to be,

Edward Doolittle 43:55

there's only 12 guesses.

Vanessa Vakharia 43:56

I know there's only 12 guesses. I know there's only 12 guesses. I get that

Edward Doolittle 44:00

Right.

Vanessa Vakharia 44:01

Like are you,

Edward Doolittle 44:01

Which, which again is. Uh, in my culture there's 13. What? There's 13. We divide the year into 13 phases now. 12. So, so, uh, you know, this is, uh

Vanessa Vakharia 44:13

oh. Okay. Wow. This just rocked my world,

Edward Doolittle 44:16

right?

Vanessa Vakharia 44:16

Are you a Taurus?

Edward Doolittle 44:17

No, my sister is.

Vanessa Vakharia 44:19

Are you a cancer?

Edward Doolittle 44:20

No.

Vanessa Vakharia 44:20

Oh my God, guys, I just, I can't after three.

Edward Doolittle 44:23

I'll give, I'll give after three. That's it.

Vanessa Vakharia 44:25

It's over. Fuck.

Edward Doolittle 44:27

Isn't it obvious?

Vanessa Vakharia 44:29

I was, I'll tell you what honestly happened. Okay. I was between Taurus and Virgo. Okay. I was like, okay, it's gonna be one of the earth signs. But you don't actually seem particular enough to be a Virgo.

Edward Doolittle 44:39

Oh, you don't know me either.

Vanessa Vakharia 44:40

It's true and I don't. Right. That's the thing. That's his.

Edward Doolittle 44:42

I'm so particular

Vanessa Vakharia 44:44

You are. You seem so chill.

Edward Doolittle 44:48

I try very hard to be chill. Oh, internally. It stresses me all the time, and I'm not particular.

Vanessa Vakharia 44:55

Okay. And so, so

Edward Doolittle 44:56

I'm very, very, I'm, you know, it's, it's something I'm trying to overcome.

Vanessa Vakharia 44:59

You're working on it. Do you know what your rising sign is?

Edward Doolittle 45:02

no.

Vanessa Vakharia 45:02

Do you know when, oh, you do know when is your birthday?

Edward Doolittle 45:05

My birthday?

Vanessa Vakharia 45:06

Yeah.

Edward Doolittle 45:06

Uh, September 6th.

Vanessa Vakharia 45:07

Keanu Reeves is September 2nd.

Edward Doolittle 45:09

Really?

Vanessa Vakharia 45:09

Yeah.

Edward Doolittle 45:10

Okay.

Vanessa Vakharia 45:10

I know, right. And he seems really chill. Maybe he's really particular.

Edward Doolittle 45:13

Okay.

Vanessa Vakharia 45:14

Thank you so much.

Edward Doolittle 45:14

I think that's how we can meet Keanu Reeves is to say, I know someone who has, um, born just a few days apart from you.

Vanessa Vakharia 45:20

I really appreciate you trying to help me out here. Okay. We're ending this, the end. Oh, I have to say bye. But it's weird to see by your next. Okay. Bye. Okay, well that was easy.

Outtake - Dungeon Master?!

Vanessa Vakharia 47:15

Tell us what a dodecahedron is. Is that 10?

Edward Doolittle 47:17

It's 12. Dodeca is 12 in Greek. Yeah.

Vanessa Vakharia 47:19

So 12 sides. 12 angles.

Edward Doolittle 47:20

12. 12 sides you've probably never played Dungeons and Dragons, but if you do, how

Vanessa Vakharia 47:25

do you know that? Why did you make that assumption?

Edward Doolittle 47:28

I just, I don't know.

Vanessa Vakharia 47:30

I have not.

Edward Doolittle 47:31

Okay.

Vanessa Vakharia 47:32

I mean, I am interested in Dungeons and Dragons, like I would like to play a game, actually, to be honest. I don't understand how it works, but yes, you're right. The so back,

Edward Doolittle 47:40

it's still great fun. Yeah. I'll hook you up with a dungeon master. You can, anyway.

Vanessa Vakharia 47:46

Told you this is gonna be salacious interview.