Ideas - Why spirituality is central to Indigenous mathematics

Episode Date: December 19, 2025

Indigenous math isn't just about numbers and equations, it involves culture, spirituality and more. Math professor Edward Doolittle, a Mohawk from Six Nations in Ontario, sees math as something embedded in Creation itself. In his Hagey Lecture at the University of Waterloo, he describes Indigenous mathematics as being grounded in cognition, emotion, the physical world and community. Indigenizing math, Doolittle hopes, will make it more approachable and meaningful to Indigenous students — show them how entwined it is with everyday life and something much bigger than ourselves.

Okay, so there's this new play about the Rogers family and their battle for control over the gigantic telecom empire, and I cannot stop thinking about it. I'm Alameen Abdul-Mahmoud. I host a pop culture show called Commotion. This week, we're talking about Rogers v. Rogers, and on the show, we'll get into what this corporate story actually tells us about our national mythology and why Canadian theater audiences are craving more and more homegrown stories. Find and follow Commotion on YouTube or wherever you get your podcasts. This is a CBC podcast. Welcome to Ideas. I'm Nala Ayy. That's the University of Waterloo student drum group, performing the Mohawk Friendship Song, in honor of mathematician Edward Doolittle, who delivered the 2025 Hegey lecture at the university.

Yeah, I like this one. Yeah. All right. Thank you all for the invitation and for the Thanksgiving address. Now, when most of us think of math, it might be numbers, logic, and equations that come to mind. Maybe something forbiddingly abstract and difficult. Perhaps the laws that govern everything in the universe. That's a pretty big picture. But Doolittle's trailblazing work in indigenous mathematics considers a compelling even bigger picture, one with cognitive, emotional, physical, communal, cultural, and spiritual dimensions. He is Mohawk from six nations of the Grand River in southern Ontario. He's also an associate professor of mathematics and associate dean of research and graduate studies at First Nations University of Canada in Regina.

And in his lecture, he describes the project of grounding mathematics holistically in indigenous worldviews, a project he hopes will transform how we learn and think about math. I'm here to talk to you about this, indigenous mathematics foundations. And I think it's really been about 40 years that I've been working on this since I was an undergraduate at the University of Toronto, I was thinking about indigionizing and indigenous mathematics. And I had a lot of bad ideas. So it's taken 40 years really for me to kind of sort this out. It's complicated, but it's quite rewarding. And I hope to share with you some of the rewards of this journey today. I just want to draw your attention to some of the words here.

rainy, that was the best that I could do. It's kind of an acknowledgement that we're not perfect, that we still must try to do our best. So in that spirit, I'm going to do the best that I can tonight. And we'll certainly see that these ideas that I have can use improvement. I want to talk about myself just a little bit. So going to go hageny, from Flint Nation, Mohawk. Osweigal native Waganah from Six Nations, Underwillow, and Wagoniata Turtle Clan. So that's indigenous side.

On the other side, I've got PhD in Pure Mathematics from the University of Toronto. This talks about my identity, which is like a current snapshot, but we also have these concepts of relationality. How am I related? Many mathematicians have this. That's my academic genealogy. And anyone who's had a Ph.D. in mathematics, who has a mathematician as a supervisor, now has this genealogy. That goes back a thousand years. One of the earliest ancestors I have is Avicenna, a great Persian scholar. My thesis advisor, Peter Greiner, and his thesis advisor was Felix Browder and so on.

And I'm proud to say that the great Johannes Kepler is one of my ancestors. I want to talk in this introduction about some tools that I use to try to understand indigenous mathematics and so this is one of the most useful. It's very basic and very simple, but it is powerful, this four directions medicine wheel,

this idea that we want to develop the whole person and not just the mental faculties of us. And so we have the mental as an important component but we also have the physical, the emotional, and the spiritual. And we need to consider all of these and develop all these in possibly an equal measure to be a complete human being. So the mental is fairly well understood in the context of mathematics. That's what most people will say it's all about.

The physical is something we don't often see at the higher levels of mathematics at high school, university mathematics. But I think our colleagues in elementary education have a good idea when they talk about using manipulatives, using physical objects to teach mathematics. And I did some of that today with a session with the faculty of mathematics and some of my colleagues. I showed them how I use the physical manipulatives to teach some topics in mathematics at First Nations University. The emotional too is very important. I often ask my students how they feel about what we're doing. And usually I get, I hate mathematics.

I wish I didn't have to take mathematics and so on from my students. And that is a problem. But mostly what I want to talk about today is spiritual, which has been the biggest challenge for me. and for many of us to indigenize mathematics. Here's another tool that I learned from. This is the creation story. And so this is actually, I think, critical.

If we're going to indigenous mathematics, we need to connect it to creation. What is the creation story about? Here on Wendat scholar, Nicholas Renaud, I think, has put his finger right. on it, he says it holds instructions about how to make the world. And so any new endeavor that we engage in is creation. This endeavor of developing indigenous mathematics, this endeavor of First Nations University, we are moving into a new space, indigenous people moving into the space of creating our own university. That's another kind of a creation story. And even in the personal changes, the personal movements that we make,

I talked to some students today, and they asked about how can we frame this, this work that we have to do, this space that we're engaging in that's different. Our families haven't done this before. You know, I was the first in my family to get a PhD, depending on what size you mean by family, of course. But that was my answer, that we already have paradigms for this. We have this paradigm of moving into a new space. So, you know, it's critical that we look at what has already been done in our culture to help us move into new spaces.

And that's encapsulated and embodied by our creation story. So here's some questions. What is inclusion? What is cooperation? circular relationship with everything that lives. What does it mean to respect women at the center of the human world for their power to give life? What gratitude do we owe to animals because they sustain us? Those are all questions asked by Nicholas Renaud that are answered in the creation story.

So those are some of the things that I consult to build, to move into this space, Indigenous mathematics. So let me just say a few words about the Menthol. So one tool that I have found helpful is this tool called back translation. It involves indigenous languages. So let me give you an example. One of my colleagues in Quebec, early in days of working in indigenous mathematics, didn't know what to do. So thought, hmm, let's translate the. Quebec math curriculum into Cree.

Okay, let's try to take the word triangle. Translate that into Cree. How do you say triangle in Cree? And so Cree speaker gave her a word. What does that mean if you translate it back into English? That's back translation. And he said it's something with three sides. Fine.

Okay, so she goes to the community down the road and said, so this is going to be our word for triangle. They said, no, that's not right. Well, how do you say triangle? They gave a different word. Let's backtranslate that. It's something with three points. So right away, we're starting to encounter cultural difficulties, misunderstandings.

What's right? Well, they're both right. This is an important point that we'll circle back to. Anyway, it's a valuable tool. I highly recommend it. We not just stop at what's the Mohawk word for this or what's the Cree word for that. But we ask that next step. Let's translate that back into English and see, maybe learn something about how that's constructed. Something else I'd like to talk briefly about is logic.

And this is something I had trouble with. As a student, I found a difficult to let go of logic that was. the foundation of the mathematics that I learned. But there is a movement since the 1960s to study alternative logical systems, paraconsistent logics, which embody contradictions. So what's the word for triangle? Well, it's this word, and is that word. There's, you know, a potential for contradiction, but we're not forced to say one is right and the other is wrong. And so paraconsistent logics, try to navigate that. There's a nice survey article by Rebecca Sinclair. American Indian logics are paraconsistent logics. That is, they support the possibility of true

contradictions. For many American Indian communities, true contradictions are a crucial manifestation of the belief in a non-discreet, non-binary world. In short, indigenous logics are characterized by the acceptance of a principle that classical logic fundamentally rejects, making it difficult for Native Americans to make legible, important ecological, and political realities that classical logic fails to see, quite illuminating both the relationship between indigenous thought and paraconsistent logics. Something else about the mental is this way we structure our mathematics education. And so we have the curriculum.

And I've been in conversation. with a colleague Brent Davis at the University of Calgary. He said a curriculum is a rut. It's literally a rut, and that's a Latin word. It's a track that a cart would run in. And it's kind of essentially a one-dimensional thing. That's not going to suit everybody. So I think we have to think hard about alternatives to the curriculum.

The way I learned mathematics was more like exploring a field. I taught myself mathematics in a public library. And I would just go pick out a book that appealed to me. I'd pick out a different book or I'd pick out the same book. It's more like exploring a field, wide open field, than it is traveling along a track. I never liked school mathematics. It never did very much for me. All right, just a few words about the physical.

The physical can get into mathematics through the relationship between mathematics and movement, dance, for example, drumming and so on. on mathematics and sports. Some of my colleagues in Australia are working hard on this. They find it very appealing to the indigenous students to make connections with sport. Land-based

education is another physical manifestation of mathematics. There's a research project in which I'm involved right now called STEM is Place. It's essentially about land-based education. Now I've just thrown this little quote from an

adventurous guide to number theory. This is one of the books that I learned mathematics from at Terry Berry Library. I started reading it when I was nine years old, ten years old. And it starts with these lines. Every number has its character. I picture seven is dark and full of liquid, like oil when it oozes from the ground. Three is lumpy and hard, but dark also. And four is soft and doughy and pale. Five is pale, but round like a ball, and six is like four, only richer like cake instead of dough. It's, sounds crazy, but it was quite appealing to 9-year-old me. I think I checked this book out of the library about 40 times because it really goes somewhere. It starts out with this, but then

he starts talking about algebra, about arithmetic, about modular arithmetic, about all the topics of basic number theory culminating in proof of quadratic reciprocity. So, I mean, I learned about proof, the concept of proof. I learned about mathematical induction. from this book. It's a treasurer. I still have a copy. So there's the perceptions that we have mathematics that are part of our, you know, almost the physical manifestation of it for us. And somehow it was very effective in drawing me into mathematics when I was a young person. The emotional, I, again, just a few words about the emotional and mathematics struggling with many of us to understand these things. But emotional is something particularly challenging for me. I don't know.

I just love it. I love mathematics. I always have. People ask me why. I just say there's a mathematical shaped hole inside me. That somehow appeals to me. But it certainly does not appeal to everyone. So here is one of my colleagues, Cori, at the University of Calgary. And she's written this, this is a PhD thesis in mathematics education. Angry, boring, crazy. I feel sad. I I don't feel good. Sad faces. A sad yellow bear, which breaks my heart. And then a self-portrait with the words math is fart.

So we got a problem. That's probably like the lowest insult that a child could make right there. Math is fart. Yeah, we got a problem. So what does positive emotion look like? Here's an example. My colleague, Kamala Young, other indigenous mathematicians that he's gathered around, we build community, we enjoy each other's company. I mean, we need to make these opportunities available to our students as well, all of our students.

And so I try my best to create positive emotional experiences for my students in the mathematics classroom. One of the keys I found is allowing students to work together in groups instead of by themselves. at their desks. The big problem, though, for me, the kind of big gap is the spiritual. And again, I can't claim any particular skill or knowledge or capability in the spiritual domain. I do the best that I can.

I don't know how much got thatily wet grainy. But, you know, I think, I think I have made some progress in this way, and I hope to open it up for others who will continue this work. So let's take a look, another look, at the creation story, and this is a spiritual experience, the creation story. It begins with Skywomen, and Skywomen, in my knowledge of the story, was pregnant in Skyworld, but she craved a certain root. And so she would dig and dig to find this root. And she dug and dug and dug and dug looking for the root and eventually made a hole so deep that she fell through from Skyworld

and started falling into our world. And I think there's an important message there. We need to pay attention to these stories. And that message about digging for the route is about search, about research, you might say. And so it all begins with research. It all begins with the search. It all begins with this.

We're not entirely content. And we are going to try to improve ourselves or our world or something. And that is the driver of the story. So to me, it tells me about the primacy of research. It's one of the most important things that we can do. And so I have taken. on that responsibility at First Nations University of trying to help my colleagues to become better researchers and to develop research at our institution.

Now, you'll see Sky Woman in this image has plants in her hands. She grabbed plants as she was falling through. She grabbed whatever she could grab, but of the plants that were nearby. And so she took those valuable plants with her from Skyworld into our world. And that tells us something about what's important. When we move from one place to another, those plants are of vital importance. And we must remember to bring them with us. And going back to what Nicholas Renaud said, it's about how to build a world.

Well, that's what our ancestors would do. they would move from one place to another. They'd move the town, as we say in Mohawk, and they'd move the town from one location to another. They would farm and they would exhaust the soil and then move to another location. And so every time they're building a new world and the plants they take with them

so they'll have something to plant in the new location. Now, the animals, you see the geese are supporting skywomen. She was falling too fast and she would have injured or killed herself when she landed where the geese broke her fall. This tells us what we owe the animals. The animals saved us, and we owe our lives to the animals. You know, sometimes people say,

well, this mathematics thing, we didn't have it in ancient times. So what's the big deal? Why do we need to learn mathematics now? It's not one of our traditions. And my answer to that is, we've screwed the world up so bad that we need to do whatever we can to repay our debt to the animals. We need to do what we can to help the world and to save the world from the damage that

has been done. And to do so, we need whatever tools are available. And mathematics is one of those tools. Mathematics has helped us to screw the world up so bad. So if we're going to unscrew it, mathematics is one of the tools that's going to work. So that's why I think our young people should learn some mathematics. It's necessary for us to repay our debt to the world, to the animals in particular. Now, I've also mentioned the numbers. The numbers are in the creation story. And so this is really what drew my attention to the creation story in the first place.

So this man, Saguanyunguas, Tom Porter, that has written about many, many things. And Grandma said is a book that he wrote. This is what he says. By the time we finish the creation story, we've counted from one to ten. And this counting from one to ten for the Urquois is our chronological telling of the major events of our creation. So every time we count from one to ten, we are telling the major events of the creation story. And that's wonderful for kids, for people to know. know that never did. A lot of people haven't heard that before. So they're really, you know,

part of the story. The creation story is among other things. It's a teaching tool, especially for young people. So we're teaching the numbers simultaneous to teaching all the other lessons of the creation story. There's another aspect of this that's interesting and that's an oral tradition, it's a challenge sometimes to remember the things that we're supposed to say. I believe the creation story is structured around the numbers from one to ten as that memory aid. So it goes both ways. The story tells us about the numbers. The numbers tell us about the story. It reinforces our understanding. So that is an example of what opening ourselves up to thinking more spiritually about these things can do for us.

But what this says something else to me is that the numbers are embedded in creation. The numbers are part of creation. You know, I said to an elder once, this tells me that mathematics is ancient. And the elder corrected me. She said, it's not ancient. It's intrinsic. It's built into the universe. It's part of the fabric of the universe.

Mathematics is an essential part of our universe. In the beginning was the number. That's Edward Doolittle, a Mohawk mathematician who teaches at First Nations University of Canada in Regina. This is Ideas. I'm Nala Ayyad. Do you suffer from television optionitis? Are you paralyzed by too much choice? You're not alone.

One in three people spend up to 30 minutes choosing what to watch. Try one of CBC's shows, like Dragon's Den, small achievable goals, and Murdoch mysteries. Side effects may include loss of boredom. Find all your favorite shows on CBC TV or stream anytime on CBC Gym. Emotion, culture, community, spirituality. Things we typically think have no bearing on mathematics. The Pythagorean theorem holds true regardless. of your religion, your ethnicity, where you live, or what kind of mood you're in.

But that way of looking at math might be to invoke a mathematical operation, reductive. To Edward Doolittle, indigenizing math invites us to see math through an expansive lens that pulls emotional, physical, cultural, and spiritual aspects into focus, along with the cognitive. As you'll hear in the second half of his lecture, do little conceives of indigenous math as a distinct practice rooted in a different worldview from a more traditional approach to math. And so one day I'm trying to think about

what is indigenous mathematics? What does it mean? And it occurred to me that maybe we ought to try to look at the opposite. What is the opposite of indigenous? Well, that's global. And so Indigenous is local to place, and we have this concept in mathematics, local, global. So it's got me thinking that perhaps global is a better word. Often we use the word Western mathematics, which I don't like at all.

So we talk about Western mathematics, Western science. It's based on a misconception. It's a distinction between the West and the East, the Occident and the Orient. It's not real. It's kind of a fantasy. So I don't like the idea of Western mathematics. There are various attempts to use different terminology here. And you see mathematics, which is nearly universal and conventional mathematics.

There's really no good solution. I'm proposing global mathematics, but some of my scientists' friends object to that. They say, well, global science is something different. Global science is really science about the whole globe, like climate science being an example. So I don't think I've got the right terminology yet, but I think I know what's wrong or what's bad, and that's Western. So I'm going to say global mathematics for now. And so I want to look at examples of spirituality in a situation that may be more familiar to many of you. So here is one of my favorite paintings of all time.

Pythagorean's Great the Dawn by Bronikov. Pythagorean's were a cult. And they worshipped, it's probably not the right word, number. They believed that everything was constructed from number. The whole universe was built from number. And so they actually did some interesting work in music to show how music, the pleasant harmonious sounds of some kinds of music were related to numbers. And we still use their kind of analysis now in our music system.

You know, numerous other discoveries they're known for the Pythagorean theorem, which probably they didn't discover, but anyway, there is definitely a spiritual element to the Pythagorean cult in Pythagorean mathematics. And so there are antecedents for the spiritual in mathematics. One of my personal heroes, Ramanogen, brilliant, brilliant, brilliant man, died at the age of 33 from malnutrition, Essentially, he was living in the UK during the First World War. They were under blockade. He was a vegetarian, and so he couldn't get the vegetables that he needed and essentially weakened himself and eventually succumbed to some illness.

But he would, his family's goddess, Namakal. Namakal would come to him in dreams and bring him mathematical theorems. he'd go to sleep at night, he'd say, Namakal visited me and brought me this theorem. And that was part of his spiritual practice. Strangely enough, often Namakal brought him theorems that were incorrect. So God's toy with us, don't they? So anyway, you know, the man who knew infinity, you can watch a movie about him. You read the book by Robert Cannigal.

J.H. Hardy writes a little bit about him and the mathematician's apology, too. Here's another example of the connection between mathematics and spirituality, the number zero. So one thing I do with my students is I show them how to figure out the day of the week given the date, which is remarkably challenging, but it's not beyond our capability. And the students in my first year, Mathematics 101 class eventually learned to do that. I tell them, I can show you guys how you can do it in your head. And they're like, oh, no, do we have to? So anyway,

In the process of analyzing this problem, I say, we're going to go backwards to year zero. But, big but, there is no year zero. We actually go from one AD to one B.C without a zero in the middle. So I say, we're going to make up a year zero, and it's going to work for us. So we're just going to pretend that it existed. And we're going to take the Gregorian calendar, and we're going to run that backwards. And there's actually a word for that. It's a proleptic Gregorian calendar.

We go into the past with the Gregorian calendar, even though it was around 50, 90, the Gregorian calendar was first introduced. So it didn't actually exist in that time period, but we can imagine it. So we run that backwards until we hit you another imaginary thing, year zero, and that actually gives us a way forward to calculate the day of the week easily from the date. Why is there no year zero? There's zero is missing from a lot of things. We start looking for zeros in places we should find them and we don't.

But this year is one thing. I believe that there's a spiritual something at play here. That it's a Christian thing, that God is everywhere. And if there's something where there's nothing, then there's no God. And that's a very troubling thing. So we can't have a zero. We can't have a year's zero. It would be a very bad year. So I just wanted to draw your attention to there being spirituality in mathematics already. These seem to be rare moments. For example, G.H. Hardy said his work with Ramanogen was the only romantic episode in his whole life.

So I gave a talk at the Canadian Mathematical Society conference about two years ago. And many of them came to me afterwards and said, yes, I feel something spiritual when I do mathematics, something many mathematicians feel. There's just a kind of a purity about it, and otherworldliness. There is something spiritual in the way mathematicians engage with mathematics. And that's something that we don't bring to our students. And so that's something I'm asking us all to work on and to consider how we can bring spirituality into mathematics. So this is one of my attempts to do so, the mathematics bundle. And I just want to give you another example of how spirituality has actually done something for me,

actually constructive, you know. I think many people think of it, oh, it's just about feelings or it's just about, I don't know, it's very personal thing. And I think that that way of thinking about spirituality as being personal is something that is part of colonization. that spirituality is not something that's just personal. Spirituality is something that we experience together, we experience collectively, we experience this, you know, as a nation, as a community. I think we ought to open it up more.

And I've seen examples, the Ontario Math Curriculum, for example, has a medicine wheel in it, but they put spirituality in the center of it. I mean, it's central, but on the other hand, it's not connected to the outside. It's not part of the interface. And so that, to me, is a flaw in their design. I guess really I'm looking for a more systematic way to build spirituality into mathematics. And so this is a project called the Mathematics Bundle, which I've been working on since about 2013. It originated at the Banff International Research Station, which mathematics research facility in Banff, Alberta.

There's been a long history of indigenous mathematics workshops at BIRS, BIRS, and I've been involved since the early days in those projects. myself and Florence Glanfield, who's been a tireless indigenous person working on indigenousizing mathematics education for longer than I have been in it. So Florence and I organized a session, and we asked Elder Betty McKenna from First Nations University to come along. And, you know, it's always a struggle, how are we going to indigenous mathematics? And Elder Betty said, here's something that it worked when she worked, with a prison system.

She introduced the concept of a bundle to the prison system. A bundle is a container for important objects and for memories and thoughts and ideas and it's a spiritual thing.

It's got a life of its own. So Elder Betty suggested why don't we create one for mathematics? So we did in 2013. We didn't do much with it until this year. Florence and Elder Betty and Corey Chui and I organized another workshop at the Banff International Research Station. So we invited some Australian colleagues, United States colleagues, and we've got others from all around the world.

We hope to involve in the mathematics bundle over the years. This includes three of the four indigenous PhDs in mathematics in Canada. I think there's only four. We depend on self-identification, so we're not absolutely sure how many indigenous PhDs are in mathematics. And so we've been working hard to develop indigenous mathematics talent. So we put together an all-star team, I think, when we assembled this group. The Banff Center is in the foot of Tunnel Mountain. This picture was taken from Sulphur Mountain.

I took a gondola trip up there. The foot of sulfur mountain is where the hot springs are. in Bav. It's a healing site. It's a sacred site. It's been sacred for time in memorial. And so there's something very special about this place. What is it about what happened in Bavv? Well, I came with an open mind and willingness to experience spiritual phenomena that I haven't really had until this point in my life. And, you know, long travel. eight, nine hours of driving, a very tiring process of organizing our event and so on. And I finally got to my room that night and I closed my eyes and this is what I saw.

Some of you I know in the audience will recognize what this is, but I'll describe it to you. It's called Gaswanta, and often translated in English as Turo Wampum. I've sought the translation of Gaswanta. The best that I've been able to find is that Gaswanta is a description. There's many words that Mohawk are. So we have statives and actives, statives describe the state of something, and an active describes an action or a movement. And that's mostly what Mohawk is built from.

There are no nouns in Mohawk. That's a bit of an exaggeration. It's all about the verb is what my language teachers told me. And so Gaswanta is, I believe, a stative. And what it describes is something that is flowing but structured. That's Gaswanta. And so, for example, Gaswanta would be an apt description for the spine because it's flowing and it moves, but it is structured.

And so we see that this is structured, a beadwork, and it's very orderly structure. And sometimes I use examples of this kind of beadwork for my students to talk about composite numbers and prime numbers and factoring. It also flows. If you pick it up, it flexes. It bends. And so that's a general property of these Wampum belts. This one is very special because it represents to Iroquoan people, to Rodinusone people, the grandfather of the treaties, it's sometimes called. This is the framework from which our treaties and our agreements and our relationships derive.

I saw this and I thought, why am I seeing this? This is a vision. It's a true vision experience. I'm not a visionary. I don't experience visions on a regular basis, but I did this time, and I have maybe four other times in my entire life experience very strong visions like this. They're often associated with very important moments in my life and decisions that I have to make.

So I thought, why am I seeing this? Then I heard a word, a phrase, actually. The phrase that I heard was closing the gap. Closing the gap is something we talk about in mathematics a lot. The Canadian Mathematical Society has a program called Closing the Gap. In Australian, they've made it a part of their, education efforts for indigenous people closing the gap. Do you see us closing the gap? You do not. We are not closing the gap. All right. So the two purple represent two pathways. De Yoja Hage,

two pathways, two roads. And one of them is the settler people and the other is the indigenous people. And we have separate pathways and we are maintaining separate pathways. That's what this says. We are going to maintain separate pathways. It's more elaborate than that. But you'll notice they don't merge. They don't come any closer, these two pathways. This is intentional. This is by design. Closing the gap is a mistake. So that's my message. Closing the gap is a mistake. That we are going to waste our efforts, waste our energy on that. We have to find another thing to do. isn't closing the gap. And as a bonus, I had another auditory hallucination, you might say. I heard another word, and that's braiding. And again, do you see braiding in this? You do not.

You do not see these two purple paths braided together. You know, what we often hear in science where we have to braid indigenous knowledge and Western scientific knowledge is the phrase I've heard countless times. But again, this Deswanted does not agree. We're not braiding here. Our knowledge system stands on its own is what this says. All of our systems stand on their own. We don't need to braid them together. We don't need to close the gap. This, I think,

is a message of spiritual origin to me. I'm taking it very seriously. So that is kind of a concrete result of being open to spiritual experience that occurred in BAMF for me. Now, what do we have as an alternative if we're not closing the gap? Well, notice that there's not just purple in this swan, there's white as well. And the white means something too. The white can mean peace, friendship, and respect, for example, is the white. So, you know, this is not a recipe for apartheid. This is not a recipe for separation. These spaces can be crossed. We can cross over. And that's what I did when I did my PhD. I crossed over. And I joined the global system.

Then I crossed back when I went to work for First Nations University. So we can cross over. We can cross back as individuals. Just as collectives, as societies, we're not joining. We're not crossing. We're not braiding. But as individuals, we can do this. You know, this also gave me another thought. I've been pondering this, and I think it's going to provide for us a way forward as we navigate this complex future. And this comes from work by Willie Irmine. Willie Irmine is a retired professor at First Nations University. I know him personally and I know him personally and well. And he writes about the ethical space of engagement. It says the ethical space is formed when two societies with disparate worldviews are poised.

to engage each other, it is the thought about diverse societies and the space in between them that contributes to the development of a framework for dialogue between human communities. So this, I think, helps to indicate what the way forward is. Imagine two people playing chess. Imagine during the Cold War, Eastern Block player, Western block player playing chess. So they're not arguing about their political systems. They're not talking about religion. They're not talking about anything other than the chess game. They're playing the chess game, which is the ethical space between them. In that chess board, there are rules of engagement. We have well-established rules that both

sides have agreed on, and that allows us to accomplish something together, in that case, to play a game of chess. But I think this way of thinking can help us to do many things together. And I think about our education system, how so much confusion, what's the right way to indigenousize, and to help to welcome indigenous students and so on, and also even students from other parts of the world who are coming to visit Canada or to immigrate to Canada. We have a lot of complexity in our classrooms. So what I'm suggesting is that we look to develop them as a kind of ethical space, is the space between, is one of those white patches on the two-row wampum. And in order for us to do so,

we must explicitly negotiate the rules of engagement. Another example I think about is diplomatic protocol. When diplomats get together, they have rules diplomatic protocol for how they're going to engage with one or another. They don't just willy-nilly slap each other on the back or shake hands or whatever, but they have agreed upon ways of greeting one another. They have ways of expressing hierarchical relationships that are acceptable to all. So that's diplomatic protocol. Why can we not have the same thing in our classrooms? Something that we can all agree on, that this is acceptable. For example, students may occasionally need to be disciplined, young students. How are we going to do that? Well, we know the wrong way to do that. Now, let's talk about,

what's an acceptable way to both of us. For example, what are we going to learn? These are the topics. These are acceptable to both of us, to both parties, all parties involved. This can be a negotiation. This can be an agreement. And that's not the way it's done now, in my experience, that we're simply told by governments what's going to happen in the classroom, which is hardly any better than the residential school system that we've soundly rejected. So that, to me, gives me an idea. And that, I think, is a part of the outcome for me. And it's really, you know, got its spiritual origin, spiritual roots for me.

I'm going to close with one more example. This one's a personal example. And this one's hard to explain. It's mathematics. It's very deeply mathematical. And so it goes back to my PhD when I study partial differential equations. I studied something called Hamilton-Yocobi theory. So the great Irish mathematician, William Rowan Hamilton,

a brilliant, brilliant man, studied optics because his interest was astronomy, but developed methods for solving partial differential equations, which were still exploiting to this day. And so I have some knowledge of optics that came from my learning the background to partial differential equations. All right. So the situation, we held mostly ceremony at the mathematics bundle. We opened with a pipe ceremony.

We open each day with a smudge. We had talking circles, which are essentially ceremony. And then on our last evening together, we had a full moon ceremony, which was conducted by Elder Betty McKenna. During that last day at Beers, it was raining all day. We wanted to have a fire outside, and we were worried we wouldn't be able to if it continued raining. Well, it stopped raining. So we walked down to the site where the fire was to take place.

Somebody said, it rained all day, well, where's the rainbow? I turned around. There's your rainbow. Okay, so this is a rainbow like I have never seen in my life. Maybe, you know, probably some of you are luckier than me. Have you seen nice rainbows? I was astonished by this rainbow. I want to tell you a little bit about how astonishing this rainbow is.

You know, here is a common model for understanding rainbow. And this is something called geometrical optics. So we trace the rays of. of light as they go through the droplets of water. And this lower one gives us the primary rainbow, the bright one on the bottom. Notice that there's one internal reflection. There's two refractions and so on.

So it acts like a prism, and that's why we see the colors and blah, blah, blah. Okay, so the model is called geometrical optics, and it's how Hamilton solved as differential equations. He split them into two pieces. The iconal equation, we call it, is the geometrical optics equation. And so we can essentially understand it by ray tracing.

Now, how do we get the second rainbow, the double rainbow? Well, there it is. There's two internal reflections. Lights coming in the same direction from the sun, but now it's bouncing around in a different way, which causes the second rainbow to flip upside down, for one thing, but it's also a bit dimmer because some of the light escapes every time there's an internal reflection.

All right, so all this stuff, went through my head in the moment that I saw the rainbow. I said, I know this. This is single rainbow, double rainbow. This is internal reflection. Look at what else I saw. But you see underneath the first rainbow, there is more rainbow. This is something called a supernumerary rainbow. It's a rare phenomenon, but not unheard of. And I realized geometrical optics cannot describe the supernumerary rainbow. All in a moment, because I had been prepared by

my learning, by my knowledge of partial differential equations and geometrical optics and all this other stuff, I realized at a moment that that theory does not describe the supernumerary rainbow. What you're seeing here is evidence of the wave nature of light. And when I went to school, we studied Young's double-slit experiment, which we were told is the first evidence of the way of nature of light, but it is not. This is the first evidence of the wave-nature of light right here. And, you know, just that moment, I realized all kinds of things. I realized that's wrong. Young's double-slit experiment is not the first, you know, because these rainbows are not that unusual. And so it was a moving spiritual experience to see this.

The realignment that occurred in my mind at that moment is something I can talk about, but I can't really fully explain. I can't really transmit to the feelings that this caused. But it was an extraordinary personal moment for me. You know, there's power in the mountains. and many places, other places. We just need to begin to open ourselves up to this power. And this is what spirituality can do for us.

Edward Doolittle, delivering the 2025 Hegey Lecture at the University of Waterloo. Following the lecture, he took a question from the audience about what's at risk if math curricula focused too much on closing the gap between indigenous and non-indigenous knowledge. So one of the problems with closing the gap that I didn't go into is that it is rooted in deficit thinking, that we are lesser than others. We have to improve our scores and improve our standardized test results or whatever. And that ignores all the strengths that we bring. So I think that's the biggest problem. It's that kind of framing of it. It's just not right. And if we start with something that's not right, we're going to keep going

wrong from there on. The other part of it is, it assumes that we all want the same thing. But who asked us what we want? And I've done this. I've gone to indigenous communities, and I say, what do you want? What do you want from mathematics? And I get all kinds of different answers. One community in Northern Ontario, they said, we want our young people to be able to survive in the bush, want them to be able to go to the bush and to live and to survive and to, you know, hunt and gather and build shelter and whatever, that's what our goal of education is. I'm like, well, you know, the Ontario math curriculum is not going to help you with that, right? So who are we to say that's not a worthwhile pursuit? That's not our business. That is their business, right? So we have to ask people, and part of this is informed. consent. We need consent and we need informed consent. So we need to say to the people, this is what

we can do with the mathematics that we have and is this what you want? Maybe yes, maybe no. These are parts that are missing from closing the gap, the rhetoric. I visited Six Nations maybe it was a decade ago, maybe longer, and they just done some standardized testing. and I think one of the schools from Six Nations came out 997 out of 1,0002 in the province. What good is this? It's no good at all. And then they said, here's the English test that they gave. They showed me that it's got like words from Dickens, right?

Like nobody says these words anymore. What value is this? So the tests that were using, the standardized tests, they have to be calibrated somehow. And the calibration, like, who decides? There's just so much wrong. You know, we really need to get more involved in our education and to be more assertive about what it is that we really want out of it. And that's what I mean by we need to negotiate this space.

What are we going to teach? How are we going to teach it? We need to have those explicit conversations instead of accepting what's offered, which is, in my view, generally inferior. Naomi Paul, Leith Bullford, and Kara Loft of the University of Waterloo student drum group will take us to the end of this episode with The Traveling Song, which they played before the lecture.

Edward Doolittle is an associate professor of mathematics at First Nations University of Canada in Regina. Special thanks to the University of Waterloo. This episode was produced by Chris Wadskow. Our website is cbc.cais. And you can find us on the CBC News app and wherever you get your podcasts. Technical production, Emily Kiervasio and Sam McNulty.

Our web producer is Lisa Ayuso. Senior producer Nicola Lozschech. Greg Kelly is the executive producer of ideas, and I'm Nala Ayyat.