Listen to the stories of Our Indigenous Mathematicians

In partnership with Native Stories
Episode 1: Dr. Craig Young
Native Stories (00:02):
Welcome to our first interview in collaboration with Indigenous Mathematician, Indigenous Mathematicians was created to bring all Indigenous Mathematicians together for the purpose representation in the field of mathematics for the future generations. Craig young is from Rough Rock, Arizona and was nominated recently for their presidential awards for excellence in mathematics and science teaching. The nation’s highest honor for STEM and computer science teachers. Welcome Craig. If you could tell us a little bit about where you’re born and your genealogy,
Craig Young (00:41):
Uh, young, when I got so translation, I was, uh, speaking in Navajo introducing myself. So my name is Craig young. I’m de uh, I hail from the great Navajo nation from, uh, rough rock Arizona. Uh, currently I reside in tuba city, Arizona on the Western side of the Navajo nation. Uh, just a little background in Navajo nation is roughly the size of West Virginia. Um, and we have about about 400,000 members in a, in the, in a tribe in, uh, the Navajo nation is about roughly 27,673 square miles. Uh, it covers, uh, states, uh, with, uh, New Mexico, Arizona, Utah, and, and borders, Colorado. Um, so that’s where we’re, I’m from. Um, so a little bit about my background. Um, so I, I, uh, I’m a, I’m a mathematician, uh, right now I’m teaching, uh, in, in the elementary setting. I’ve, I’ve taught in high school and, uh, also in, in college, uh, for adult learners and, uh, this is who I am and then, you know, welcome and thank you guys for having me here and, you know, let going to get started. <laugh>
Native Stories (02:05):
Yeah, I’m gonna add a little bit more. So he is a stem educator at no mathematician after school program staffer at tuba city boarding school. Um, he maintains a family is working on doctorate in leadership at fielding graduate university. He also run TCBs Thunderbird robotics team is part of the Alliance of indigenous math circles runs tuba city’s Arrowhead math circles runs a few more camps across the nation, um, and runs warrior of hope in character development for boys who get in, um, who need a little bit more help at school. Uh, so that is a lot, uh, I, I have some background in math cuz uh, my electrical engineering, uh, degree, but I forget everything. So I’m like, okay, algebra is the far further this I could go in this conversation. So welcome Craig, uh, to NA stories. And we’re gonna jump into the questions right now. Okay. Um, so who are your ancestors that you are grateful for or wanna talk about?
Craig Young (03:16):
Well, um, I wanna start off with my, my, uh, maternal grandparents. Uh, uh, we, I would call my maternal grandmother, uh, Shamus, which is, uh, my, my maternal grandma. Uh, her name was Ida. She Harvey. Uh, she was from rough rock Arizona. Um, she was the matriarch of, uh, our family. So we’re we, we come from or the Navajo culture, uh, you know, we are the females, they’re the leaders in our family and um, our art clans, my first clans Kane, uh, one who walks around clan and uh, you know, that’s, that’s who I am. That was my grandma’s clan. That’s my, my, my mother’s clan as well. Um, and then also my cha my maternal grandfather, his name was, uh, Carl C. Harvey. He was a, a council delegate here on the Navajo nation, uh, for, for rock point Arizona around that region, uh, for a number of years back in this seventies and, and eighties.
Craig Young (04:18):
And, uh, that’s where the leadership, uh, comes from from my family. Um, on my father’s side, uh, you know, I have, uh, Julius, uh, young who, who was also well he’s, he was also, uh, a person that I looked up to a lot. And, uh, you know, I just wanted to say, thank you for him, uh, for everybody, uh, who, who has been there, but also going back further, um, you know, we, we mentioned ancestors and I wanna acknowledge the, the indigenous, um, our indigenous warriors way back when, uh, who were a part of their resistance to, to maintain, uh, culture and maintain language and, uh, you know, fight for, for, for identity, um, and our way of life. And, uh, you know, I, I know that, um, uh, I guess, uh, colonization, um, really hit a lot of indigenous communities. We very hard throughout history. And, you know, when we, when we look at mathematics in, in a way it’s, it’s, it’s other, uh, it’s westernized concept that, that type of thinking that a big disconnect. And so my, my, uh, my job here, uhhu mathematics, um, and reconnect that with, uh, our Navajo culture, you know, to, to have our kids see value in themselves and have our kids see value yeah. In, in their identity. Um, yeah,
Native Stories (05:57):
It’s so important to see ourselves and our ancestors, you know, even if you look at just one generation, maybe that’s not the best generation to look at, but if you look throughout history, the many that have come before us have fought, and you’ll always find someone to, you know, um, model after and wanna do better for your peers and for future generations. So yeah, just great points on, on what you’re doing and how it’s connected to that, but how you see yourself, you know, who you spoke about, right. Mm-hmm, <affirmative> your grandmother, your, and then just, you know, the people that you see, you, you hold responsibility within yourself because of the people that you feel connected to. So good points. Yeah. Um, so how did you get into math and what does that journey look like?
Craig Young (06:59):
Well, that to get there, that’s a good question. Um, I, I actually had a, a different experience, uh, with mathematics. So I always tease my, oh, I don’t tease, but, um, I always tell the students in, in a teasing way and older than the internet and, you know, like kinda like, what do you mean you’re older than the internet? How old are you? And you know, that that’s also a realization of how far and how fast technology has come and, and, uh, the what little surface we scratch and how much more we, we have, uh, to explore the technology. Um, so way back before the internet even existed. Uh
Native Stories (07:41):
<laugh> I feel like I’m just as old as you, so yeah, I preach <laugh>. Yeah. I was definitely a, a senior in high school when we had dial up and AOL dialup. Yeah.
Craig Young (07:57):
I think middle school, um, maybe <inaudible> or something, but, you know, I had great teachers, they were great. Um, but one thing that I’ve noticed, my teachers also struggled with was teaching mathematics. Um, so in a way, you know, they did their best, but I felt like they, well, most of them, you know, they, they struggle with, uh, explaining mathematics, uh, to us and I kind of fell behind. Um, I didn’t have that great experience with math. I, I, I actually didn’t like math. Uh, I disliked it. I thought it was too hard. Um, but I got discouraged and, um, you know, way, way back then there was no internet, no, nothing like that. It’s really hard to learn. Uh, if you weren’t paying attention in class or if you were absent or you just didn’t understand, uh, maybe the teacher was mean, I didn’t want to explain it.
Craig Young (08:57):
Uh, right. Um, so there’s a lot of variables involved. Um, but I didn’t, I didn’t get it. I struggled through school. I struggled all the way through high school, barely, um, you know, pass, uh, mathematics with high school. And, and then I went to college when I went to college, I actually started with remedial education, uh, with mathematics, like everything I should have learned from, uh, elementary school, middle school, high school, everything I should have learned, I should have been at that college algebra level. And it was a struggle. Um, so actually dropped outta high. Oh, I’m sorry. I dropped outta college my first year. Uh, I, you know, I just, I, I scraped by, you know, with like a C average and, and I, I thought, you know, math, math, wasn’t, I’m not a math person. I thought that, uh, college wasn’t for me.
Craig Young (09:50):
So I left after a year, I came home, uh, back to the, to, uh, KTA, uh, um, KTA, Arizona was where I grew up. Um, and, uh, I came back home and one summer I went over to, we have this place called, uh, uh, workforce development office. So I went there and they had this opportunity where they sent, uh, Navajo, uh, uh, anybody who was really interested that they, they sent us to Chicago, Illinois. Um, so we went to this iron working school, uh, I forget what it was called, national iron working school for native Americans, I believe. And that was in Broadview, Broadview, Illinois. And I stayed there for a bit, um, that that was a, a trade school basically. So, um, I learned how to become an ironworker. I learned about structural working or ornamental, iron working, reinforcing iron working. And I, I graduated, I got my certificate and I, there was a, a big, uh, reality check for me because I saw that, um, in, in the trade field, uh, I guess, no matter what you do in the trade field, if you wanna be electrician, if you wanna be a plumber worker, um, uh, alignment, you know, uh, just to name a few, everything is mathematical and blueprint blueprints.
Craig Young (11:18):
You know, you have to know what you’re doing on that job. Yeah. And safety is also a big thing. And so I stayed in Chicago for a, a number of years with local union one, and then I came home to Arizona. So I transferred unions and I, I joined local five here in Phoenix, um, in, in, uh, you know, all the older guys, they were telling me, go back to school, go back to school. And they were showing me like, you know, their, their, their hands and their hands, they were losing cartilage in it. And they were having a number of health problems, things that they breathe in when we were working up there and the sky with the skeleton of, uh, skyscrapers. And, and it was like, you know what, you’re, you’re probably right. I probably need to go back to college, but, you know, I was still intimidated, um, with, with mathematics.
Craig Young (12:08):
And so I got the hang of math, you know, just learning to trade, but, um, and then I ended up joining the Marine Corps. Um, so I really didn’t listen, you know, I just kind of like, you know, what, I wanna do something that wanted, that I want to challenge me. I wanted a, a challenge where, you know, something that I can prove to myself that I can do it. So I joined the Marine Corps and even in the military, everything is mathematical like the precision that you need to be, uh, so accurate with, with what you’re doing. Um, uh, when you’re, when you’re out there, when you’re, I guess, with all the, the pounds and weight carrying, um, uh, you know, just to name a few things, uh, everything was mathematical. And so a part of that, that journey, you know, I was thinking like, man, everything really revolves around this foundational skill, uh, problem solving, analytical thinking.
Craig Young (13:05):
And so I went back home after I got out of the military is like, what do I do next? You know, there’s life after the military. So for those of you guys who thinking, you know, don’t let the military use you, but use the military for, for, you know, all those great benefits, no matter what job you, you do, you’re gonna get the same benefits. Um, but I thought to myself, what do I do next? And one really unique thing, um, that I’ve I’ve realized was I could have went back to that same college that I originally started at, or I could have went to a big university, probably wouldn’t have been successful. Um, but one thing that stuck out to me was this college, our, our local tribal college university called the net college. And what was really interesting was that their core curriculum and values and everything revolved around the Navajo culture with the, uh, philosophy of, um, and this is, uh, that philosophy is grounded in Navajo culture and traditions.
Craig Young (14:08):
Uh, it speaks to organizational change while living in harmony with a natural world. And, and so that’s what it really drew me to that college. I learned more about who I was and my identity as a, not only as an indigenous person, but as a Navajo young man. And so I never thought I would become a, a teacher or an educator at all. That was the last thing on my mind. Um, but I’d always say that, uh, teaching found me and there was this, uh, individual there named, uh, Henry Fowler. He, he, he was, he’s a, uh, a Navajo mathematician and a professor at the net college that inspired me to, to, to become an educator. But that was where my foundation was set, uh, learning about who I am and reconnecting to my culture. And so that was part of my journey that launched me into getting, uh, my master’s degree and now working on my doctorate degree.
Native Stories (15:11):
Wow. So you like math now? <laugh> um, <laugh> yeah,
Craig Young (15:17):
Actually, um, I took a lot of hard work, especially with, uh, starting where I was at, um, learning about adding fractions, multiplying fractions, and just like taking that time during, uh, my personal time, uh, outside of class hours to, to catch up. And it took a lot of work and, but, you know, once it, things started to click in, in, you know, like things that things started to click and started to make sense. And that’s when I, I started to like math more and more and more and more. And now I’m, mm-hmm <affirmative> so it, it, it can be learned. It takes a lot of hard work, no matter what you do, there’s no practice hard work in practice.
Native Stories (16:04):
<laugh> lots of practice mm-hmm <affirmative> and I think confidence is the first thing that you have to overcome, right? Oh yeah. In order to, you’re gonna get it wrong. Yep. But the good thing about math is it’s black and white. It’s like it’s right or wrong. Mm-hmm <affirmative> whereas I feel like my, I had the same, I was always good at math. Um, I think my family just does numbers just for fun in, you know, con regular conversations, but, um, I hated history and now I love history <laugh> this is why I had this podcast. Uh, and native stories is cuz I was got so interested in so kind of opposite story, but, um, but you, I remember just having like taking a whole day or, or three days to do like one theory, math problem, you know, you gotta, you gotta keep trying, <laugh> get it right. But once you get it, it’s like, yay. Um, it’s worth it.
Craig Young (17:05):
Bring up a good there’re we’re gonna struggle a lot process.
Native Stories (17:16):
Yeah.
Craig Young (17:17):
Making mistakes, anything. And uh, you, once you develop that confidence, you stick with it, you persevere
Native Stories (17:24):
Persevere.
Craig Young (17:26):
Exactly. And you know, you’ll, you’ll get it eventually.
Native Stories (17:29):
So, and anything you do in life, you learn a lot through from math that you can get through it. Definitely. Um, so let’s skip to the next, uh, question, which is regarding teaching. Um, and, and I’m not sure. Can you first, uh, explain what ethno mathematics is? And then can you provide an example of what that is?
Craig Young (17:57):
Sure. Uh, well, ethno mathematics, that’s where we, we there, I guess you could say that, um, you could think of it like a, a paradigm shift of integrating, um, Navajo culture or even any indigenous culture and traditions into the Western math education and curriculum. And so we’re blending those two together. And what we wanna do is we wanna engage, uh, young indigenous students to, to build a strong cultural self identity. Um, but also connect them with our elders who are, who are the, uh, language holders, who are the, who are the ones that hold all the knowledge that, that knowledge we could say, knowledge holders. And there seems to be a lack of community, our elders and their native. We can that gap. I think that would be, uh, very, very good for, for people. Um, but ultimately mathematics, um, I guess culture tradition with mathematics.
Native Stories (19:11):
Can you provide an example of how you would do that for your students and how you do it now?
Craig Young (19:17):
Um, so, uh, you know, with, with the current curriculum that we have, it’s like a, a one size fits all approach with, with everybody, um, Caucasian students, African American students, Asian students, um, native American, all the way across the curriculum and all the way across, uh, I guess you could say the world. And so you could look at it as a one size fits all approach, but here’s the thing. Uh, everything is driven by data. Everything is driven statistics. Um, however, we’re at a disadvantage because we don’t really connect to what we’re reading in those, in those books. There’s not really a lot of, um, books out there that, um, focus on our, our, our culture, um, or indigenous cultures. And so there’s a disconnection there too, uh, where our kids don’t see themselves in mathematics, they don’t see themselves, uh, being valued as I mentioned before.
Craig Young (20:22):
And so, um, I honestly, you could say I kind of go rogue. I don’t really follow the curriculum, but I’ll, I’ll look at the example problems, but I’ll give it my own twist. Um, so I’ll, uh, this is where I’ll explain, um, for example, fractions, uh, decimals and percents. A lot of times our students see them as separate, uh, entities. They think that are separate from percent and that’s separate from, uh, decimals, but the way I would approach it is I would relate it to like what I mentioned earlier, we call it or, uh, kinship when I introduce my clan, uh, Chi Craig young, like this is who I am, my name’s Craig young. <inaudible>, that’s my mother’s clan. Cheneys Ching that’s my father’s clan. Uh, okay. So, um, Ash Chi, right? That gosh, Chi that’s my paternal grandfather’s clan my third clan and then not, well didn Ney.
Craig Young (21:30):
That’s my, my cha, uh, that’s my, uh, my, my last clan, right. Ash is my, um, Ash. I’m sorry. I’m, I’m kind of like, um, there, but Ash is my Cha’s clan and not to go, KA is my paternal grandpa’s clan, but I have those four clans where everybody in, in, in my, um, my kinship is represented. And so just like fraction, decimals, and percents, they’re all the same. They’re I, I, I, how I would explain it is that they’re siblings, fractions, decimals, and percents are sort of, um, they come from the same parents, you know, multiplication, multiplication begins with the letter M so we could say that’s the mom division <laugh>, um, division is also starts with the letter D that’s the dad. So when you think about it, multiplication and division, their inverse operations, um, they can undo each other. They can, uh, compliment each other.
Craig Young (22:35):
Right? So those, those are the parents of fractions, decimals and percent when we convert between them. That’s cool. And if, yeah, if you think about it, um, if you think about it, we could say that let’s say, for example, let’s say 90%, we could represent that same value, 90% as a decimal, as nine tens. And we can <crosstalk> that same, uh, fraction end. I’m sorry. We can represent that same percent end decimal as a fraction with 90, over a hundred. Okay. So we’re representing that same value in three unique ways. Like those siblings they’re, they’re all related, but maybe they’re different. Maybe one likes country, one likes, uh, maybe gangster rap, the other one, like <laugh>, you know, they, they’re all unique. They have their own personalities, their own likes dislikes, but no matter what, they’re siblings, they’re related, mom and dad, multiplication and division.
Craig Young (23:36):
And so if you think further, you can think of it as, uh, you know, addition, for example, addition or multiplication, just repeated addition. Right. Um, and if you think about division, that’s really just repeated subtraction. So if you integrate those four operators, multiplication, division, addition, subtraction, that would be their, their, their clans. Okay. Mm. Yeah. So relating that to, to my Navajo students, with their friendship system, let’s pretend that, um, percents, fractions of decimals let’s pretend that they were actual people and they would introduce themselves with their, so that would say multiplication. And then, because that’s mom’s plan multiplication mom’s plan, right. Multiplication <inaudible> then they would say, division bus team division would be their dad. And then, um, they would say addition that should cha Chay is actually, uh, the maternal grandfather. And then they would say subtraction dash. So,
Native Stories (24:50):
So you’re doing genealogy.
Craig Young (24:52):
Yeah. And
Native Stories (24:53):
So <laugh>,
Craig Young (24:55):
Once they
Native Stories (24:55):
See that by going up levels.
Craig Young (24:57):
Oh yeah. And they see that they’re like, oh, okay. That makes sense. Because you know, that, that’s how I would introduce myself with Mylans. And so once they see that, uh, they’re all related, they’re not separate things. And then things will just start to click and, you know, they, they use, for example, mom, and dad’s help to help them convert between fractions. They have the, and percent mm-hmm <affirmative>. And the funny thing too, is like, let’s say for example, um, they’re using division to help them convert between those, those three, but guess who helps out, uh, dad who always checks out dad’s work to make sure that it’s right. Mom,
Native Stories (25:35):
Mom, <laugh> parents are learning. Yeah.
Craig Young (25:48):
That would be one way.
Native Stories (25:49):
Huh? That’s super cool. I never thought about it that way, but that’s really, really cool. Um, thank you. Um, so, um, reading off your resume of projects, are there any project you’re working on now that we should be in the loop about? And, um, maybe in the future too, that you wanna shout out if there’s any fundraising opportunities or good opportunities to shout those out too right now?
Craig Young (26:22):
Yeah. Yeah. Actually my I’m working on a, a dissertation research right now. Um, and the title of my dissertation research is called, um, math, uh, math learning experience and traditional Navajo culture using math circles and Navajo culture to improve sixth grade students’ attitudes towards learning math and improve attitudes, uh, toward the value of traditional cultural wisdom. So not only are we, um, coping
Craig Young (26:53):
Change our attitudes towards math and, and, and, um, collapse that I guess you could say, um, I’m Henry Fowler coined this term collapsing, the fear of mathematics, not only are we doing that, but we’re also improving their attitudes toward learning about their language and culture as well. So we’re doing things at time. Um, so we’re doing that. And, uh, along with that, we do have a robotics team that I feel that is also very important with stem education and what we need is a, a, a greater representation of indigenous people’s and careers. There’s not many people in stem careers that are indigenous and, um, you know, guys out there. Um, what we wanna do is, uh, we want to inspire, we wanna inspire the next generation and, uh, to help, uh, you know, your, your, uh, language to help your, your people and come back and beat that change agent for, for, for your, uh, for wherever you come from. And so we wanna preserve our language. We wanna preserve our culture, our heritage, and, uh, hold onto our, our traditions in a way of life and, uh, taking that, uh, traditional holistic approach to, to compliment your life and, and way of thinking. Um, I feel that all, uh, indigenous communities have a history of complex mathematics and sciences throughout culture. Um, and honestly we are mathematics. Our, our language is full of mathematics. Our cultures are so rich with mathematics, um, every culture where
Native Stories (28:41):
Before there was engineering <laugh> we were already sustaining ourselves without polluting the earth. Like, how did that happen? Right. That, that was science mm-hmm <affirmative>, that was data collecting over years and, and implementation into the best fit, into, you know, sustaining our bodies and our minds. Um, yeah. So looking to your history <laugh>
Craig Young (29:08):
Yeah. And the thing with the, you know, the indigenous approach was, uh, we were living in harmony with, with natured and natural universe. Everything was, was, uh, everything was working well together until the influence of colonization. Yeah.
Native Stories (29:26):
We were better off <laugh>, but now we have this challenge, right. And using math and the Western way of, uh, doing things. Uh, but with the background of, for instance, being Navajo, uh, you can make some serious, uh, change in a positive, positive way.
Craig Young (29:58):
Definitely.
Native Stories (29:58):
I see indigenous people as the, the answer to climate change, to the answer, to cuz they’re the ones that care, um, and have the history on and on the engineering part. Um, and the math part. So, you know, it’s one of those, like why fix something that was broken now we have to reinvent something that has been broken, um, but still use our way of living as the foundation.
Craig Young (30:34):
Exactly.
Native Stories (30:36):
So you kids out there, you’re the next generation
Craig Young (30:42):
<laugh> yep. And you know, the last thing I wanna, you know, say to them too, is, uh, your listeners is that, you know, mathematics, it, it is a great teacher, uh, in life, for example, you know, we can’t run from our problems. Math teaches us to, to face those problems, face them, deal with them, head on, find a solution, solve it and then solve the next one and the next and the next, you know, that’s life there. So math, uh, is it’s always a great teacher in that sense where, you know, it teaches you guys, uh, deeper thinking skills to, to overcome any challenges that you have in life. And remember that you are that warrior, you come from a long lineage of warriors, it’s in your blood. And when, if we go back further, you know, when you were, when you were a baby, your parents held you in their arms, they saw you looking back at them, you know, that’s what they saw. They saw hope you are that hope. And I wanna leave that with you guys. And, uh, now do well, you know, if you’re struggling, you know, just stick with it, persevere, overcome, and, uh, preserve your language, your culture, your heritage, and, and we need you guys in those stem careers. We need more of a representation. We need more of your warriors out there to, to, to help, um, to help the world really.
Native Stories (32:10):
So thank, yeah. Mm-hmm <affirmative> thank you, Craig. That was a beautiful closing out <laugh> of this interview. Um, I just wanna say Mahalo for joining us. And, uh, we were, I thought we were gonna talk about math, but <laugh> instead, uh, Craig is such a great example for, uh, you know, future generation, anybody struggling with math. There’s definitely, always hope. So. Um, that being said, anything else, Craig, you would like, like to add?
Craig Young (32:43):
Um, you know, that’s all I really have right now, but um, you know, I do, I do have other projects that we have going on and if you, uh, if you can find us on, uh, Facebook, uh, I do have, uh, a mass circle program that are run called Arrowhead math circle. Um, if you go to, if you Google, um, the Alliance of indigenous mass circles come join us. We, we have mass circles every other Monday from five to 6:00 PM, um, mountain time. And we actually, uh, we have, um, collegiate mathematicians who, who work with our indigenous youth and it’s open to your grandparents. It’s open to your parents, you, your friends, um, really anybody who’s interested, but we’re called the, the Alliance of indigenous math circles. You can,
Native Stories (33:35):
We’ll get those links to, um, our listeners. So yeah, go ahead and said that to me, Craig, thank you so much.
Craig Young (33:43):
And it it’s free, you know, you don’t pay anything. It just, you show up. And the last thing I’ll say is, uh, what books are reading play is with, so just fun play with the numbers.
Native Stories (33:59):
Awesome. <laugh> uh, okay. So mahalo everybody for listening. Check us out on, uh, our website, nativestories.org, Facebook or Instagram, our native stories download the, um, mobile app where we have all the previous podcasts and walking tours. Um, if you have any suggestions for stories, go ahead and, uh, Instagram or our website is the best way to contact us. Everybody take care.

In partnership with Native Stories
Episode 2: Dr. Edward Doolittle
Speaker 1 (00:04):
Welcome to the indigenous mathematician series. Indigenous mathematicians was created to bring on indigenous mathematicians together for the purpose of representation for those indigenous kids. Jake looking in the field of mathematics or stem. Uh, we have today, Dr. Edward Dole. He is Mohawk, uh, one of the six nations in Southern Ontario. He earned his PhD in mathematics, partial differential equations from the university of Toronto in 1997, in 2000 slash 2001. He was a member of, uh, the language group studying the Mohawk language full time in immersion in his home community. In 2001, he joined the faculty of first nations, university of the federated college of the university of Regina, where he is now associate professor of mathematics. Dr. DLI is interested in probability, particularly in relation to partial differential operators. He is also interested indigenous mathematics and related concepts, concepts like indigenizing mathematics, traditional mathematics, and ethno mathematics, and the educational possibility afforded by those different views of mathematics. He is the recipient of a governor General’s academic middle, and an audible mention in the William Lael put mathematical competition. Welcome Edward.
Speaker 2 (01:36):
Well, thank you.
Speaker 1 (01:37):
So we’re gonna get into the questions and, and actually before that, can you, um, give us, uh, you know, in your own language where you’re from and maybe some background on your genealogy?
Speaker 2 (01:52):
Yeah, certainly. Uh, so I’m, uh, Mohawk, which in our language we say go Haga and that really means Flint nation, literally. Uh, so that’s how we’ve referred to ourselves in our language. Uh, and, uh, I am from six nations, which, uh, includes the Mohawk nation in Southern Ontario. It’s, uh, largest reserve in Canada. It’s got, uh, uh, 20,000 people on reserve. Um, and then there are more associated with the reserve. Uh, and, um, the, uh, the language group that I studied Mohawk in is, is qua. And that literally means our language group. And so I studied Mohawk in immersion in year, 2000 after I finished my PhD. This is one of the, the projects that I engaged in is try to become, uh, conversant if not fluent in my language. And, and, uh, it’s something I’m still working on.
Speaker 1 (02:53):
Great. Thank you so much. So getting to the questions, um, who are ancestors you are grateful for?
Speaker 2 (03:02):
Yeah. I’ll tell you a little bit about my genealogy. So I’m, I’m in some sense, half Mohawk I’ve got, uh, my, my father is Mohawk and my mother is of English descent and her family, uh, comes from Lancaster in Yorkshire and England long time ago. Uh, and my father’s side, uh, we’ve been in Canada since the, uh, the, uh, um, revolutionary war where our people left the United States to, uh, to move to Canada. And, uh, I have, uh, genealogy that goes back, um, a long time. Uh, I’d like to remember in particular, my grandfather, Clifford Doolittle, uh, I’m told I resemble him, uh, strongly. Uh, he died when my father was five years old and it really affected my father’s family and my father’s life. Um, but, uh, I was privileged to meet a, uh, uh, a friend of Clifford’s who was an old man when I met him. And he said, you look just like my best friend. So that was a, that was a good thing.
Speaker 1 (04:14):
Hm that’s great. So did he inspire you to get into math <laugh>
Speaker 2 (04:19):
Well, yeah, I mean, my, my father, uh, dropped out of school in grade nine and so did my mother, uh, but my father was always very interested in mechanical things and very handy, uh, and, uh, just very clever. He, he would repair an automobile that was broken down on the side of the road, just with tools that were available to him. He would pull it apart and put it back together again on the spot. So, uh, you know, I, I, I think that, uh, there’s, you know, in our past, probably there was mathematical abilities and, uh, you know, I don’t think my ancestors maybe had all the opportunities to, uh, develop those abilities that I was, I was privileged with.
Speaker 1 (05:08):
So how did you come across math as your path?
Speaker 2 (05:16):
I, I actually encountered mathematics really. I mean, aside from school mathematics, which I never found very interesting, uh, I encountered mathematics in the Hamilton public library, so I grew up, my family raised me in Hamilton, Ontario, and my parents would take me every week to the public library to pick out a few books. And so I sort of gravitated to the science section. I started reading about puzzles and, uh, Martin Gardners focused on mathematics, and then, and then started getting more and more into the, the pure mathematics side of it. This was about when I was about 10 years old.
Speaker 1 (05:55):
Wow. So, and what, how did you kind of pick it as your, you know, where you’re at now and, and traveling along that path?
Speaker 2 (06:07):
Yeah, it’s, uh, it’s when I went to school at the university of Toronto, I started in 1985 and I was interested in artificial intelligence at that time, but, uh, in 1985, there wasn’t the computing power that we have now. And so artificial intelligence was all theoretical and it really poorly developed, and it never did anything practical and it wasn’t sustainable in my opinion. So after a year of that, I decided I I’m going to study something that I’m really good at. And that’s mathematics. Mm-hmm, <affirmative> feel sort of like, I’m a native speaker of mathematics since I started teaching to myself from a young age. And so I was, I, I could be a top student in mathematics and, and that’s why I chose to field. I would get top marks in it.
Speaker 1 (06:58):
So I’m very interested in what is indigenizing mathematics, traditional mathematics, ethno, mathematic mathematics, in your opinion, and how is that used?
Speaker 2 (07:12):
Right. So this is, uh, kind of a current topic in, in research trying to sort all of this out, but I think I have an idea now how to frame these questions. And, um, I think that most of the mathematics that we learn in school, so there has been a, uh, a lot of discussion about what we should call it. You know, this school mathematics, this sometimes it’s called Western mathematics. Sometimes it’s called European mathematics or Eurocentric mathematics or, or whatever. Uh, I think that the term global mathematics is appropriate because it is a worldwide phenomenon. So I could take my knowledge of global mathematics and I could teach mathematics anywhere in the world where English is spoken or French. I could travel anywhere pretty much in the world and teach mathematics. So it really is a global phenomenon. And that situates this question about what it is alongside questions about other global phenomena.
Speaker 2 (08:26):
So global economic phenomenon, global, you know, globalization of all kinds. So this mathematics that, that is now taught in universities is kind of a globalized phenomenon. And so all the, uh, the issues that we may have with globalization, I think carry over into global mathematics. So on the other side, what is, what is the alternative? Well, that is, is indigenous mathematics. An indigenous mathematics is mathematics, which is local localized, it’s localized to a particular geographic area or a particular people. And, um, it may vary greatly from one place to another, from one culture to another. Uh, but there probably are, you know, I can’t say for sure, but I believe that there are underlying phenomena, what makes this mathematical rather than some other kind of, you know, so, so that we look for the mathematical in different cultures, we will find general themes, but the detail always be different because that’s just the reality of, of indigenous cultures is that they vary from one place to another.
Speaker 1 (09:41):
That’s interesting way of thinking about it. So it’s like you can use the Western type to get to a conclusion one way. Um, but there’s these indigenous in mathematics that can take their own personal or not personal, but community type of mathematics to get to the same answer, but just in their own, uh, process, I guess
Speaker 2 (10:09):
That is probably true in many cases, uh, I, I, I will say it is true in many cases. In other cases, the two may not really be entirely, um, consistent or comparable to one another because they address different issues, different, you know, for example, um, in indigenous cultures, very often we work without the aid of technology. And so a lot of mathematics now is really technological in nature. And by technology, I mean, not just high tech, but even things like paper and pencil. So as mathematician to solve a problem without paper and pencil, and they’re often, you know, uh, um, just, uh, hopeless at it. Oh yeah. But that’s not universally true. You take somebody like our committees who was famous because he, he would draw in the dirt with a stick. So all he needed was a stick and right. But’s not true of all mathematicians. I think many mathematicians, these days need some kind of technology. They need their libraries and their books and their pencils and their paper, if not, uh, calculators and computers and slide rules and so on. So, so that is one thing that kind of, I, I think one factor that may distinguish indigenous mathematics from global mathematics and that’s the, the personal nature of indigenous mathematics where we, uh, approach problems just with our own bodies, our own minds, and very little else.
Speaker 1 (11:48):
Can you give an example of maybe a, you know, a, uh, equation or not equation, but some type of, um, yeah. Example of how you’ve used this in, in the classroom setting with your students, if that’s possible.
Speaker 2 (12:09):
Right. Sure. Um, I will try to, to give some examples. I mean, this is, like I said, it’s all, it’s all current research and people are still thinking about these things and uncovering the things that have been hidden. You know, I, I think a lot of our past has been, has been hidden, has been lost, have been forgotten. And, and a lot of that is on purpose. And we see that in Canada with our residential school system, it was really, you know, uh, our, our languages, our culture, but also our mathematics was suppressed. And so we’ve got to try to bring this back and it’s a long, difficult process to do so, but I will try to give some examples of indigenous mathematics. So, uh, one question that I’ve had in the past is, um, where in mathematical knowledge might be located in indigenous cultures.
Speaker 2 (13:07):
So in the global culture, I guess, I don’t know if you can say that, but in global mathematics, uh, mathematical knowledge is generally located inside books and papers. And so it’s written down using the writing technology and it is stored in libraries and stored in journals. And, and so on. Uh, uh, indigenous mathematics, I think is stored, uh, one place where it can be stored is in oral traditions. And so that is one place I’ve looked for. Mathematics, indigenous mathematics is inside oral traditions and, and trying to find the right oral tradition is, is our first question. And so I’ll tell you a little bit of what my culture, we have four major oral traditions. We have the creation story, we have our, uh, Thanksgiving address. Uh, and so the creation story talks in my, in my understanding of it, the creation story is about things that are, uh, unchange.
Speaker 2 (14:10):
So names for the plants and animals, for example, you’re our change. We’re we change slowly. We can’t make up our own names recently. Cause then we couldn’t communicate with each other very well. Our client system, which essentially tells us who we can and cannot marry is, is, uh, also fundamental to our culture. And we find it the immigration story and also games rules for games. The way games are played. Those are in the creation story. Uh, now the Thanksgiving address is more about knowledge, which is, which is, uh, acquired. And so I was talking to a Mohawk midwife once, and she said that, you know, this, all this business about all of our lost knowledge being a tragedy, she says, it’s not a tragedy because we didn’t always have this knowledge. We acquired this knowledge and we can reacquire it if we need to. So, um, uh, uh, you know, a lot of our knowledge has been learned and acquired over time through observation and through experimentation and through, uh, you know, communication with one another and with the, the rest of the living world, all of our, all of our relations, the animals and the plants and so on.
Speaker 2 (15:37):
So, uh, the Thanksgiving address can carry this knowledge, which we acquire and which may change from time to time and which may grow. Uh, and so we talk everything in creation. We give thanks to all the things in creation, beginning with the, the earth, our mother, and, and all the way up to the, the, the heavens. And, uh, uh, so that’s one place where knowledge can be stored. Now we have two other oral traditions, which are more about dealing with human beings. So we have, uh, uh, guy Gowa, which means the great, good way is often translated into English as the great law of peace that tells us how to get, get along with each other within our nations. And so there’s a, a long, long oral tradition, uh, around that. And, uh, there’s also, uh, the great, the, the good news, which is, um, how to get along with the settlers, the newcomers.
Speaker 2 (16:40):
And so we have, you know, these, uh, these things about dealing with nature and also about dealing with people. And so the mathematics, I have searched for it, and I think the appropriate place to locate it is within our creation story. And, and so looking at the creation story, we see some interesting phenomenon. So things like rules as in the rules of games, uh, have a mathematical structure. We can find them in the creation story and the, um, uh, the counting numbers, for example, 1, 2, 3, and so on. Uh, we find those also in the creation story and they play an interesting role in the creation story. The story educates young people about the numbers. It can be used as an educational tool, but, uh, it also the numbers in a flipped way kind of structure the story. So, uh, if you are telling the story, you need to remember the sequence in which things events occur.
Speaker 2 (17:47):
And so the numbers help with that. There’s one of something. And then there’s two of something. And then there’s three of something and there’s four. And so in oral tradition, uh, memory prints are important because they help you remember and not leave anything out. So that’s one place where we can find search for indigenous mathematics is in our creation stories. So I, you know, all nations should be looking for this and they may find that it’s been suppressed. They may find that it’s been transformed and hidden, and, and our responsibility now is to bring it out and to bring it back to life. So that’s one place where we can find it. Um, and, and the games, um, games in particular are, are a great thing, a great source of mathematical structure. So we have, we have a games in, in my nation called the peach stone bowl game, and it’s practiced in ceremony. So it’s not really an appropriate thing for us to, to write about or talk about as, as mathematics, but it is mathematical and it’s highly structured. So it has kind of pointed the way, the direction to me to look for mathematics in, in my culture.
Speaker 1 (19:06):
Wow. That’s interesting. I completely understand where you’re coming from, cuz we have our own creation story and yeah, there’s the counting and I could think we do have games and there’re, you know, obviously gotta put the sticks, can’t put it like five feet away. It has to be like, you know, a hand width away or whatever it is. So I, yeah, that, that makes sense that there’s math in, in these stories. Mm-hmm <affirmative> um, and so what do you, what do you see in the field right now? Like, do you have classes that teach this or what’s what’s out there? What do you see as like projects that people are doing?
Speaker 2 (19:48):
Well, much of this is in research. And so I have, I, I’m a, I’m a, co-applicant on a AHI grant, which is our, one of our major humanities granting institutions in Canada, uh, to, to look at how indigenous people, um, use, uh, the world nature, the land and so on as a teacher and, and in particular to teach mathematics. So this is, is really kind of current research, I guess, but it’s something that I would like to bring into my classroom and, and I find that some mathematics courses are more well suited for this than others. So for example, uh, calculus, I, I struggle to introduce indigenization into calculus. Calculus is very fast and, and there’s just an enormous amount that has to be covered so that the students are ready for the, the next course in calculus. Go on. So if I take time out to kinda talk about indigenization and a calculus course, it’s counterproductive. I think unfortu,
Speaker 1 (20:56):
Yeah. It’s like adding history to <laugh> math class. Yeah.
Speaker 2 (21:00):
Right, exactly. Yeah. And you know, every time I talk about history or tell stories in any of my, uh, in my calculus math class or my statistics, math class, the students just shut down, they kinda story their on theirs.
Speaker 1 (21:18):
Yeah. I don’t have to pay
Speaker 2 (21:26):
Is, is, uh, really suitable for this is, is something we call math one oh one in, in my, in, in my university and it’s, uh, introductory finite mathematics. And it’s for, uh, mostly for teacher candidates. So, uh, who want to become elementary teachers are mostly, uh, mostly who take this course. And so that gives me an opportunity to reach that important demographic of those who are going to become teachers of our indigenous young people. And I hope to transform the way they see mathematics so that they teach it better and raise up new generations of students who are, uh, going to enjoy mathematics more, or they’re going to see themselves in it instead of seeing a free culture. Uh, so I try very hard to indigenize math 1 0 1 and I’ve been moderately successful. So just to give you a quick rundown of some of the things that I do, um, I talk about arithmetic.
Speaker 2 (22:29):
I start by doing arithmetic with the students. So we just learn how to add and subtract and multiply and divide. Now, many of them already know these things, but I do it from a slightly different perspective. I want to eventually show them how to do arithmetic in other bases and do arithmetic and say base four or base 12 or base 20 is the way many people around the world have done arithmetic in the past. And, and so there are indigenous cultures in north America, which use base four, some use base five, the Mayans use base 20 and so on. And, and so I want to, broadener my students perspective and to show them that the base 10 arithmetic that you learn in school as the truth is not the only way to do it, that there’s so much variation and variety around the world. And, and so I, I teach them, uh, arithmetic in other bases.
Speaker 2 (23:23):
And like I recently came across a very interesting video where a woman, a very old woman was speaking CRE the Creek language from Canada. And, uh, there were, there was, uh, well, I, I, I sent it to my CRE speaking friends and they translated it for me, but this woman was saying, okay, in, in olden days, we didn’t use our fingers to count. That’s the English way. We use the web between our fingers and there’re only four of them rather than five mm-hmm <affirmative>. So it’s a very natural way to, to introduce a base four system. When you count using the spaces between your fingers, instead of your fingers. Now you, it’s sort of like if you only speak English, you only know English and you can’t imagine what other languages might be like, well, this is the same with counting. If you’ve only counted using your fingers, you don’t know how else people count, but it’s very natural to use the web between your fingers.
Speaker 2 (24:19):
And she also said, we also sometimes use the knuckles. And so if you call it the knuckles, there’s four, four fingers with three knuckles, each makes 12 mm-hmm <affirmative>. And so there’s one very natural way of introducing a base 12 system. So I talk about this with my students and, and I say that, uh, learning about other bases will help you understand base 10 better. And it will also help you understand that mathematics is, uh, an indigenous phenomenon, not just, uh, not just the global based 10 mathematics that we’ve all learned. So that’s the introduction to my course. And then I go on and talk about other things. So there are some interesting examples of, uh, uh, using base 16, which is computer based. So computers are phase 16, but, um, in, in some modern times we type indigenous languages using something called Unicode and Unicode is based on a base 16 framework.
Speaker 2 (25:22):
And so I show the students, the Unicode tables for the Canadian Aboriginal CELs, which is a way that we write, uh, in indigenous languages, some indigenous languages in Canada and each symbol in the Salic alphabet has a base 16 number associated with it. First shows that base 16, uh, and or that arithmetic and other base, it’s not just an old thing and a long gone thing, but it’s a very modern and current thing for indigenous people. And it’s important, you know, it can be valuable to know this. So that’s one other example that I give, uh, just to give you, uh, flavor of, of a few other things. I, I, uh, talk about modular arithmetic, which is arithmetic on a circle. And so, uh, many most arithmetic that we learn is on the number line. And when I went to school, we all had a number line on our desk and, you know, learned to count backwards and forwards on this number line, but you can do the same thing with the circle, the number circle.
Speaker 2 (26:29):
In fact, that’s how computers do arithmetic. They do arithmetic on, it’s a big circle, uh, with, you know, 32,000 numbers around the circle, but it’s just like, if you overflow, if, if you add, keep adding one to a number you’re gonna get, you know, 32, 7 67, and then you’re gonna go back to zero again. So you can, you eventually computers actually do margin arithmetic. So you know, that again is a modern thing, but it’s an old thing. And we talk about the difference between the circle and the line a lot in Canada. Uh, you know, part part, part of our indigenous tradition is thinking in terms of circles. And so the whole circle arithmetic that we can do. And, and then I use the circle arithmetic to talk about timekeeping. So, uh, this gives me an opportunity to talk about the way different cultures, keep time and, and have their own calendar.
Speaker 2 (27:28):
So my culture, for example, uh, has, uh, 13 months with 28 days each, and that makes 364 days. And it’s almost exactly the length of a year. And then we take a day or two at the end of the year to reset the calendar. And, uh, and then, you know, we start all over again. So it’s so accurate and so perfect. <laugh> yeah. Whereas you know, this, this crazy system that we have now where, you know, 30 days and 31 days and 28 days or 29 days and so on and so on what a crazy system. So haw keeping time was much more logical, I think.
Speaker 1 (28:11):
Yeah. That I think we have a similar, um, different amount of months and the same amount of days per month. Right. Uh, and then catch up at the end, <laugh>
Speaker 2 (28:26):
Make an astronomical observation that the year begins at this point in time where, uh, certain constellation play DS or the seven sisters is directly overhead at, at a certain time of day. So it’s, it’s all, you know, we have, we have it all <laugh> yeah, we have it
Speaker 1 (28:44):
All. Yeah. It’s just a different way of thinking. Come out to the same. Yeah. Basically the same conclusion. <laugh>
Speaker 2 (28:53):
All right. And there’s one other, uh, thing I’d, I’d like to talk about in my indigenization and that’s the use of bead work designs and indigenous other indigenous art designs in, um, uh, teaching number theory. So part of the, the 1 0 1 course is teaching about greatest common advisor underneath common, multiple, and factors, uh, prime numbers and so on. And, and all of these things have analogs in terms of bead work designs. So that’s something we’re working on.
Speaker 1 (29:27):
Thank you. Um, do you have any advice for students aspiring to be mathematicians in indigenous cultures?
Speaker 2 (29:37):
Uh, well, I don’t know. I, I, I, I can advise people that I love mathematics and, you know, I, I think that young indigenous people should, uh, experiment and try out a whole bunch of different things. And that’s what I did at the library. That was one gift that the library gave me is that I could read about anything I could read about science, about engineering, about mathematics, about, you know, on and on and on. And I just explored and sampled everything and, you know, found that I was better at some things than others. And that’s just, you know, luck, I guess. And, uh, what I was, I was fortunate to find the things that I was really good at. So that’s what I would think advise, uh, all young people is that you should really try out a bunch of different things and find out what you’re naturally gravitate towards, what you interested in and natural you’re gonna become naturally good at you hardly have. I feel like I hardly have to put in any effort at all to, to become a mathematician just because I loved it. I loved it. I loved it so much. I would love sitting down. I would fill books full of algebra and it would just be like poetry to me. So, you know, I think that’s, that’s the first thing that I think, uh, and make sure that they’re studying something that they really love. Um,
Speaker 1 (31:05):
Yeah. Great advice. Yeah. Um, and I wanna close out right now. Is there anything you wanna add? And, um,
Speaker 2 (31:14):
Yeah, I think, uh, I think there’s one more thing I’d like to say and that’s that I think we should be aiming, uh, for a world that doesn’t require us to give things up in order to acquire other things. So to become a mathematician, to become a scientist, we should not have to give up our indigenous identity. And, and so this is the problem that we have had in Canada for a hundred years in order to attend school, students had to give up their language, they had to give up their culture and, and, and I want to see an end to that. It hasn’t yet come to a moon. You know, we, we’re talking about truth and reconciliation in Canada, but we, we can’t really reconcile until we acknowledge that this hasn’t yet come to an end and we still have to stop it from happening.
Speaker 2 (32:04):
But a lot of the way that our science and mathematics is taught, asks indigenous people to turn away from their culture. I think so I think we have to do a lot of work to understand this effect and to oppose it and to stop it. And so I think, I think, you know, I, I would say to any young indigenous person out there, if you’re interested in studying mathematics or science or whatever, that you go ahead and do that you do that. If you love it, it’s your right to, to do that. But you should also be aware of when people are going to ask you to stop being yourself, to stop being an indigenous person, and you should resist that. I think, I think those, you know, we haven’t really fully understood this, but I think that that is still happening.
Speaker 1 (33:01):
Wow. Great message. Um, thank you again for spending time with us, uh, so much to unpack there. And, um, I’m actually excited to learn more about my own culture and, and the math. I’m gonna start asking questions. So everybody out there start asking questions of your own culture <laugh> and see, you know, how math is involved. Um, I guess, uh, to close us up, you guys have any questions. You can find us on native stories, uh, dot org, our website, or our Instagram or Facebook at our native stories hall, everybody for listening. Uh, and thank you so much. And Edward for joining us
Speaker 2 (33:53):

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Episode 3: Dr. Kori Czuy

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