Evelyn Lamb: Welcome to My Favorite Theorem, the show where we ask mathematicians what their favorite theorem is. I’m your host Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. I had to wear a sweater yesterday.

EL: Oh my goodness! Yeah, I’ve had to wear a sweater for about a month and a half, so.

KK: Yeah, yeah, yeah.

EL: Maybe not quite that long.

KK: Well, it’ll be hot again tomorrow.

EL: Yeah. So today we’re very glad to have our guest Henry Fowler on. Henry, would you like to tell us a little bit about yourself?

Henry Fowler: I’m a Navajo Indian. I live on the Navajo reservation. I live by the Four Corners in a community, Tsaile, Arizona. It’s a small, rural area. We have a tribal college here on the Navajo Nation, and that’s what I work for, Diné College. I’m a math faculty. I’m also the chair for the math, physics, and technology. And my clan in Navajo is my maternal clan is Bitterwater and my paternal clan is Zuni Edge Water.

EL: Yeah, and we met at the SACNAS conference just a couple weeks ago in Salt Lake City, and you gave a really moving keynote address there. You talked a little bit about how you’re involved with the Navajo Math Circles.

HF: Yes. I’m passionate about promoting math education for my people, the Navajo people.

EL: Can you tell us a little bit about the Navajo Math Circles?

HF: The Navajo Math Circles started seven years ago with a mathematician from San Jose State University, and her name is Tatiana Shubin. She contacted me by email, and she wanted to introduce some projects that she was working on, and one of the projects was math circles, which is a collection of mathematicians that come together, and they integrate their way of mathematical thinking for grades K-12 working with students and teachers. Her and I, we got together, and we discussed one of the projects she was doing, which was math circles. And it was going to be here on the Navajo Nation, so we called it Navajo Math Circles. Through her project and myself here living on the Navajo Nation, we started the Navajo math circles.

KK: How many students are involved?

HF: We started first here at Diné College, we started first with a math summer camp, where we sent out applications, and these were for students who had a desire or engaged themselves to study mathematics, and it was overwhelming. Over 50 students applied for only 30 slots that were open because our grant could only sustain 30 students. So we screened the students and with the help of their regular teachers from junior high or high school, so they had recommendation letters that were also presented to us. So we selected the first 30 students. Following that we expanded our math circle to the Navajo Nation public school system, and there’s also contract schools and grant schools. Now we’re serving, I would say over 1,000 students now.

KK: Wow. That’s great. I assume these students have gone on to do pretty interesting things once they finish high school and the circle.

HF: Yes. We sort of strategized. We wanted to work with lower grades a little bit. We wanted to really promote a different way of thinking about math problems. We started off with the first summer math camp at the junior high or the middle school level, and also the students that were barely moving to high school, their freshman year or their 10th grade year. That cohort, the one that we started off with, they have a good rate of doing very well with their academic work, especially in math, at their high school and junior high school. We have four that have graduated recently from high school, and all four of them are now attending a university.

KK: That’s great.

EL: And some of our listeners may have seen there’s a documentary about Navajo math circles that has played on PBS some, and we’ll include a link to that for people to learn a little bit about that in the show notes for the episode. We invited you here to My Favorite Theorem, of course, because we like to hear about what theorems mathematicians enjoy. So what have you selected as your favorite theorem?

HF: I have quite a few of them, but something that is simple, something that has been an awe for mathematicians, the most famous theorem would be the Pythagorean theorem because it also relates to my cultural practices, to the Navajo.

KK: Really?

HF: The Pythagorean theorem is also how Navajo would construct their traditional home. We would call it a Navajo hogan. The Navajo would use the Pythagorean theorem charting how the sun travels in the sky, so they would open their hogan door, which is always constructed facing east. So once the sun comes out, it projects its energy, the light, into the hogan. The Navajo began to study that phenomenon, how that light travels in space in the hogan. They can predict the solstice, the equinox. They can project how the constellations are moving in the sky, so that’s just a little example.

EL: Oh, yeah. Mathematicians, we call it the Pythagorean theorem, but like many things in math, it’s not named after the first person ever to notice this relationship. The Pythagorean theorem is a2+b2=c2, the relationship between the lengths of the legs of a right triangle and the hypotenuse of a right triangle, but it was known in many civilizations before, well before Pythagoras was born, a long time ago. In China, India, the Middle East, and in North America as well.

HF: Yes, Navajo, we believe in a circle of life. There’s time that we go through our process of life and go back to the end of our circle, and it’s always about to give back, that’s our main cultural teaching, to give back as much as you can, back to the people, back to nature, back to your community, as well as what you want to promote, what you’re passionate about, to give back to the people that way. Our way is always interacting with circles, that phenomenon, and how the Navajo see the relationship to space, the relationship to sunlight, how it travels, how they capture it in their hogan. Also they can related it to defining distance, how they relate the Pythagorean theorem to distance as well as to a circle.

KK: What shape is the hogan? I’m sort of curious now. When the light comes in, what sort of shadows does it cast?

HF: The Navajo hogan is normally a nine-sided polygon, but it Navajo can also capture what a circle means by regular polygons, the more sides they have, drawing to become a circle. They understand that event, they understand that phenomenon. The nine sides is in relationship to when a child is conceived and then delivered, it’s nine months. The Navajo call it nine full moons because they only capture what’s going on within their environment, and they’re really observant to how the sky and constellations are moving. Their monthly calendar is by full moon. And so that’s how, when the light travels in, when they open that hogan door, it’s like a semi-circle. In that space they feel like they are also secure and safe, and that hogan is also a representation that they are the child of Mother Earth, and that they are the child of Father Sky. And so that hogan is structured in relationship to a mother’s womb, when a child is being conceived and that development begins to happen. Navajos say that the hogan is a structure where in relationship there are four seasons, four directions, and then there are four developments that happen until you enter old age. There will be the time of your birth, the time when you become an adult, mid-life, and eventually old age. So using that concept, when that door is open, they harvest that sunlight when it comes in. Now we are moving to the state of winter solstice. That happens, to western thinking, around December 22. To the Navajo, that would be the 13th full moon, so when that light comes in that day, it will be a repeated event. They will know where. When the light comes into the hogan, when the door is opened, it will project on the wall of the hogan. When it projects on that wall, they mark it off. Every time, each full moon, they capture that light to see where it hits on the wall. That’s how they understand the equinox, that’s how they understand the solstice, in relationship to how the light is happening.

KK: Wow, that’s more than my house can do.

HF: Then they also use a wood stove to heat the hogan. There’s an opening at the center of the hogan, they call the chimney. They capture that sunlight, and they do every full moon. Sometimes they do it at the middle of that calendar, they can even divide that calendar into quarters. When they divide it into quarters, they chart that light as it comes in through the chimney. They find out that the sun travels through the sky in a figure eight in one whole year. They understand that phenomenon too.

KK: Ancient mathematics was all about astronomy, right? Every culture has tried to figure this out, and this is a really ingenious solution, with the chimney and the light. That’s really very cool.

HF: The practice is beginning not to be learned by our next generation because now our homes are more standardized. We’re moving away from that traditional hogan. Our students and our young people are beginning not to interact with how that light travels in the hogan space.

KK: Did you live in a hogan growing up?

HF: Yes. People around probably my age, that was how they were raised, was in a traditional hogan. And that was home for us, that construction. Use the land, use nature to construct your home with whatever is nearby. That’s how you create your home. Now everything is standardized in relation to different building codes.

EL: So what have you chosen to pair with your theorem?

HF: I guess I pair my Pythagorean theorem to my identity, who I am, as a Navajo person. I really value my identity, who I am as an indigenous person. I’m very proud of my culture, my land, where I come from, my language, as well as I compare it to what I know, the ancient knowledge of my ancestors is that I always respect my Navajo elders.

KK: Very cool. Do you think that living in a hogan, growing up in a hogan, did that affect you mathematically? Do you think it sort of made you want to be a mathematician? Were you aware of it?

HF: I believe so. We did a lot of our own construction, nothing so much that would be store-bought. If you want to play with toys, you’d have to create that toy on your own. So that spatial thinking, driving our animals from different locations to different spots, and then bringing our sheep back at a certain time. You’d calculate this distance, you’d estimate distance. You’d do a lot of different relationships interacting with nature, how it releases patterns. You’d get to know the patterns, and the number sense, the relationships. I really, truly believe that my culture gave me that background to engage myself to study mathematics.

KK: Wow.

EL: Yeah, and now you’re making sure that you can pass on that knowledge and that love for mathematics to younger people from your community as well.

HF: That’s my whole passion, is to strengthen our math education for my Navajo people. Our Navajo reservation is as large as West Virginia.

EL: Oh, wow, I didn’t realize that.

HF: And there’s no leader that has stood up to say, “I’m going to promote math education.” Right now, in my people, I’m one of the leaders in promoting math eduction. It’s strengthening our math K-12 so we build our infrastructure, we build our economy, we build better lives for my Navajo people, and that we build our own scientists, we build our own doctors and nurses, and we want to promote our own students, to show interests or take the passion and have careers in STEM fields. We want to build our own Navajo professors, Navajo scholars, Navajo researchers. That all takes down to math education. If we strengthen the education, we can say we are a sovereign nation, a sovereign tribe, where we can begin to build our own nation using our own people to build that nation.

EL: Wow, that’s really important work, and I hope our listeners will go and learn a little bit more about the Navajo math circles and the work you do, and other teachers and everyone are doing there.

HF: It’s wonderful because we have so many social ills, social problems among my people. There’s so much poverty here. We have near 50 percent unemployment. And we want my people to have the same access to opportunity just like any other state out there. And the way, from my perspective, is to promote math education, to bring social justice and to have access to a fair education for my people. And it’s time that the Navajo people operate their own school system with their own indigenous view, create our own curriculum, create our own math curriculum, and standardize our math curriculum in line to our elders’ thinking, to our culture, to our language, and that’s just all for my Navajo people to understand their self-identity, so they truly know who they are, so they become better people, and they get that strength, so that that motivation comes. To me, that’s what my work is all about, to help my people as a way to combat the social problems that we’re having. I really believe that math kept me out of problems when I was growing up. I could have easily joined a gang group. I would not have finished my education, my western education, but math kept me out of problems, out of trouble growing up.

KK: You’re an inspiration. I feel like I’m slacking. I need to do something here.

EL: Yeah. Thank you so, so much for being on the podcast with us. I really enjoyed talking with you.

KK: Yeah, this was great, Henry. Thank you.

HF: You’re welcome.

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