A film by
Navajo Math Circles follows Navajo students in a lively collaboration with mathematicians. Using a model called math circles, the students stay late after school and assemble over the summer at Diné College in Tsaile, Arizona, to study mathematics. The math circles approach emphasizes student-centered learning by putting children in charge of exploring mathematics to their own joy and satisfaction.
The documentary reveals the challenges in education on the Navajo Nation and looks at a new approach for improving the mathematical skills of students. The Navajo Math Circles project summons applications of math in Native culture to provide tools for increasing math literacy, and highlights the special connections between Navajo culture, natural beauty, and mathematics. The film shows how math circles help raise the hopes of parents, students, and teachers for a brighter future.
Navajo Math Circles opens with a scene inside the hogan built by Navajo mathematician Henry Fowler on his mother’s homestead at Tonalea, Arizona. "My mother’s eyes, it was my mother’s eyes that introduced me to Diné values, our customs, our way of life," Henry says. The spirit of those traditional values informs the stories and experiences of the Navajo middle and high school students at the core of the film. The film documents the implementation of Henry’s vision of merging mathematics and science education with traditional Navajo culture at reservation schools.
We first meet the students at the Baa Hozho Math Camp organized at Diné College in Tsaile in July 2014. They are immersed in solving problems posed to them by visiting mathematicians engaged in a bold experiment organized by Henry. He hopes that by introducing the math circles model to the Navajo Nation, he can produce a new generation of leaders who will create a more prosperous and successful Navajo future.
With a strong knowledge of his Navajo heritage and language, Natanii Yazzie embraced the math circles concept as soon as he encountered it. He explains, "I thought I was dumb, but since I went to Math Camp, I think I am smart, and can succeed." Following a discussion with mathematician Matthias Kawski from Arizona State University, Natanii tells us that "it can’t just be when you find the answer then you’ll be happy. It’s the process that makes you enjoy math."
Matthias Kawski extols the transformative nature of math circles. "We have open-ended questions. Open-ended questions are totally new to most of the kids. Usually, you have to have an answer within 20 seconds, 30 seconds. That’s what math is. Math circles are the opposite. We start with some simple questions and we ask more questions and more questions. We get some answers along the way. The answers actually don’t matter. We’re opening whole research programs and that is something totally new to the kids, and once they like it, it’s just amazing."
Another student in Matthias’s workshop is 13-year-old Irvilinda Bahe, who thinks that the way numbers fit together, and what you can do with them "is cool." After she introduces herself, we see her walking through a desiccated field where her grandfather is trying to plant corn. He explains the conditions and challenges to Irvilinda. She is one of several bright young students at the Math Camp, who also shows a serious interest in Navajo culture. Later, on a bluff overlooking the town of Chinle, her mother Carmelita Charley tells us how proud she is of her daughter’s scholarly abilities. "She is our dictionary," says Carmelita.
We return to Henry Fowler, who illustrates how his own knowledge of numbers and adding stem from childhood experiences counting stones and sheep, and from watching his mother measuring for a pattern on a rug she was weaving using her handspans. When he started teaching at a Navajo school, he discovered that the students disliked math. "That’s when I started creating my own cultural math stories for them, designing my own cultural math curriculum for them, to engage them," Henry says.
Then in 2012 he met Tatiana Shubin, one of the pioneers of math circles in the United States. A professor at San José State University, Tatiana had conducted the first math circle on the Navajo Nation earlier that year at St. Michael High School, in St. Michaels. Sister Joan of Arc, the school’s long-time math teacher, immediately embraced the approach. Soon there were math circles in five Navajo schools.
When Tatiana brought the math circles model to Henry Fowler’s attention, the connection clicked, and they very quickly started circles at more schools. They also started to organize a two-week summer camp at Diné College.
At the film’s core are the stories of individual students and how they connect to math camp and math circles. Albert Haskie from Lukachukai relishes the Math Camp experience because he feels that he is among students who don’t treat him as being different. He likes recreational math and the Rubik’s cube, but his real ambition is to become a manga artist. Albert’s parents, a jeweler and sociologist, are committed to his education whatever he decides. They drive 272 miles each day, back and forth to drop off and pick up their sons at a school they feel will prepare them for a good college.
Another student, "Buddy" Joe, who likes exploring the patterns he discovers in math, sees math as an expression of divine creation. He is a young missionary in his grandfather’s Nazarene church at St. Michaels. Buddy looks forward to studying medicine.
Charmayne Seaton was 15 when she participated in a math circle and met Tatiana. She now sees her as a second mother. She appears in scenes from both the 2014 and 2015 Math Camps. She used to want to escape from the family’s spread at Ganado, but since attending Math Camp, her goal is to get into a good university, then come home to help her people. We see her daily routine: walking home up a mile-long dirt road from where the school bus drops her off, and tending to the family’s sheep and horses. It is a life she must leave behind in order to complete her education. She is dedicated to "finishing strong." As the film ends, Charmayne gives voice to the hopes of her generation of Navajo students. "If you have a goal in mind, and you really want it, you can do it. You just have to believe in yourself and you just have to push."
Between scenes of the students at their homes or with their parents and siblings, we return to Math Camp, where they are engaged in problem-solving, or explaining their solutions at a weekly competition called the math wrangle. Some of these problems and their solutions are illustrated in animated clips produced by Andrea Hale.
Among the highlights are visits from prominent mathematicians, like Joe Buhler, who is among the world’s leading researchers in discovering mathematical patterns in juggling. At Diné College he leads a juggling lesson for the whole camp, then a workshop that ranges from the Pythagorean theorem to the 1994 solution to Fermat’s last theorem.
One unexpected development occurred when producer George Csicsery discovered math camper Briana Littleben shooting a math camp music video with her iPhone. Briana explained that her music video served as a memento of the bonding between the students at the camp. She agreed to have some of her video included in Navajo Math Circles and went on to shoot some footage of herself running along her favorite trails for the film.
The Navajo sense of mathematics is established as Fowler and others explore the connections between math and traditional approaches to architecture, weaving, herding, games, and the concepts of familial affiliations through clan structure.
The person responsible for creating the Navajo cultural component of the Math Camp at Diné College is Dawnlei Ben, who grew up in Canyon de Chelly with no running water or heating. "I’m okay with that. We have cornfields, apples, plums, peaches, apricots on our land," she says. Dawnlei was raised traditional and wants to impart what she learned to children, so the culture can survive. She has taught Navajo remedies. On the Canyon de Chelly excursion she put mint water on the heads of overheated hikers. In the film she teaches the students the game of Tsidil (sticks and stones), and describes making the markers and wooden die by hand exactly as her grandmother taught her to do. Dawnlei’s father, Stanley Ben, adds that her grandmother was a treasure trove of traditional knowledge about nature, survival, and stories. An example he explains in detail correlates the number of logs used in the construction and shape of the female hogan, with music and the duration of human pregnancy.
Through this story and other examples we see how mathematical concepts are incorporated in Navajo art, architecture, cosmology, and thought. The teachers and students in the film often speculate on how the vistas and configurations of the land inform Navajo concepts of geometry and shape. These thoughts are echoed in sequences filmed in Monument Valley area and during a student excursion into Canyon de Chelly.
Behind the beauty lie stories of deep poverty and deprivation. Bob Klein, one of the Math Camp leaders and a mathematician at Ohio University, has studied education in impoverished regions around the United States. For him the Navajo Math Camp was a transformative experience. "The Navajo students who take the daily trip to Math Camp are courageous," he says. Math Camp student Briana Littleben, who lives at Rock Point, is a perfect example. The 18-year-old senior at Many Farms High School feels caught between two worlds. While interested in preserving her Navajo culture and values, she knows she must go away from home to reach her goals. "I would want to live here," she says, "but at the same time as I’m growing up I have to go to college, which is away from home. And then I have to get a job. There’s no jobs out here, so I probably have to go outside of the reservation to succeed in life."
Another student trying to negotiate a path between two cultures is Darenda Black, who lives with her mother Sandra Etsitty and three little sisters in Flagstaff. Sandra sent her daughter to Math Camp specifically to be with other Navajo children so she could learn about the culture and speak the Diné language. The film follows Darenda to her stepfather’s home at Cortez Spring, and we hear Anson Etsitty tell about teaching Darenda the traditional ways.
Robert Megginson, an Oglala Lakota mathematician at the University of Michigan, and one of the film’s senior advisors, makes a brief appearance to discuss the shortage of Native Americans in mathematics. "The majority of folks in this country have not viewed us as being part of the folks who contribute to the mathematical enterprise. That’s certainly it, but you know we also have not viewed ourselves as being part of the mathematical enterprise and we’re working on that problem. We can in some sense blame other people for excluding us, but it’s also the case that for whatever reason, we have not viewed this as something that we do," Megginson says.
Henry Fowler and Tatiana Shubin are well aware of the bleak historical context against which the Navajo Math Circles project must take root. If they are successful, the model can be tried among other Native groups. Tatiana, stresses that the Baa Hozho Math Camp at Diné College is the "only one attempting a fusion of real mathematics and real Navajo culture, and maybe even the only one blending mathematics and any Native American culture." Mathematicians who come to the reservation because of the Navajo Nation Math Circles (NNMC) project hope to further identify deep math ideas hidden in traditional Navajo art such as rugs, sand painting, and pottery. They hope that revealing these connections will help to promote interest in math, not only among students and teachers, but throughout the entire Native community.
One key for improving math education on the Navajo Nation is to improve the math skills of local teachers. Two teachers to embrace the math circles idea and the innovative concepts of Henry Fowler are shown at their respective schools. Corvina Etsitty is a math teacher at Many Farms High School, and she talks about the math circles approach opening up her students, helping them become more active in asking questions. Lashanna Descheny teaches at Rock Point Community School, where she has introduced some of Henry’s ideas for using examples from daily life in Navajo culture to teach math.
James Tanton is a research mathematician interested in bridging the gap between the math experienced by students and the creative mathematics practiced and explored by mathematicians. He is Mathematician in Residence at the Mathematical Association of America (MAA) in Washington D.C. Filmed leading a teacher’s workshop at Diné College, James shares his hope for the future: "I would love the next generations to feel confident to try something when they’re solving problems. I mean if you think about life, life is nothing but one challenge after another. This is a vague issue you have to deal with; you don’t even know what the question is in life. Life is full of problems. There are no answers in the back of the book; there are no clarifications in the textbook for life. All you can do is just flail, just try something, just do something. And when you’re completely stymied, you have to do something nonetheless. And I actually think mathematics teaches exactly that."
In April of 2015, Duane Yazzie, a young Hopi-Navajo middle school teacher, started a regular math circle—meeting twice a week—at TseHootSooi Elementary School in Fort Defiance. It is the first regularly run math circle organized by a Navajo teacher.
Will Navajo Math Circles usher in a new generation of accomplished Native American mathematicians? Will there be further innovations to math education in Native communities? The concluding scenes are with the students talking about the future. Has participating in math circles changed their life goals, their plans for college and careers? There are glimmers of hope in their answers, hinting at how effectively mathematics and the beauty of Navajo culture have found each other in the Navajo Math Circles program.
Navajo Math Circles was filmed in 2014 and 2015 with George Csicsery directing and cinematography by Ashley James. Asali Echols edited the film and five extra features. Andrea Hale produced animated maps to help audiences pinpoint locations on the Navajo Nation and animated clips illustrating some of the math problems in the film.
Composer Alex Lu created an original music track, combining elements from songs performed in the film by two students and some parents, and flute music by Darryl Travis recorded in the field. Composer Tyrone Jenkins produced a rap track for clips from the music video created by Briana Littleben. E. Larry Oatfield mixed the sound track. Ed Rudolph and Loren Sorensen completed color correction and online editing.
Production of Navajo Math Circles was supported by the Mathematical Sciences Research Institute (MSRI) through a generous grant from the Simons Foundation, and by Vision Maker Media (VMM), and by the Corporation for Public Broadcasting (CPB).
Through problems and discussion, math circles introduce students to math outside of the regular curriculum. The goal is to excite students about the mathematics, showing them that it is not about quick answers and memorization. One thing all math circles have in common is that the students enjoy learning mathematics, and the circle gives them a social context in which to do so.
Math circles originated in Hungary more than a century ago. They soon spread throughout Eastern Europe and Asia, and since then have produced many of the great scientists from those parts of the world, in mathematics and in other disciplines.
The Berkeley Math Circle was founded with the help of MSRI in 1998, and dozens of other circles have followed in the intervening years, spanning the country. There are now over 130 Math Circles that belong to MSRI’s "National Math Circles Association" (see mathcircles.org.
Dr. Tatiana Shubin, professor of mathematics at San José State University, is a native of the Soviet Union. She grew up in the USSR Central Asian republic of Kazakhstan, where she studied with and worked with indigenous peoples such as Kazakhs, Kyrgyz, and Uzbeks. Tatiana was herself was in a math circle while at school. She spent her sabbatical leave in the fall semester of 2012 living on the Navajo reservation, where she started five math circles for the reservation students at St. Michaels, Fort Defiance, Many Farms, and Chinle, and several workshops for middle and high school math teachers. She familiarized them with the math circles model and encouraged them to start math circles at their own schools.
Dr. Henry Fowler, a member of the Navajo Nation and professor of mathematics education at Diné College in Tsaile, learned about math circles from Tatiana, and quickly embraced the idea. He is the project’s staunchest supporter and the main local driving force for its expansion. Henry’s passion is promoting math literacy. He advocates social justice through mathematics. He adapted the math circles model introduced by Tatiana into a highly successful summer program involving several leaders—all professional mathematicians—and 28 students. Navajo Math Circles documents the second and third of these math camps designed for Navajo students.
George Csicsery’s first documentary about a mathematician, N is a Number: A Portrait of Paul Erdős (1993), is a classic in the field, and has been followed by a series of feature-length, one-hour, and short films about mathematicians and math education. In 2009 George Csicsery was awarded the Joint Policy Board for Mathematics (JPBM) Communications Award for bringing mathematics to non-mathematical audiences. See zalafilms.com for a full list of his films. His background as an ethnographic filmmaker provides an additional skill for working with Navajo subjects.
Native American mathematician Robert Megginson, Professor of Mathematics at the University of Michigan, is widely known in the Native American community. He was an advisor the film, as was Hugo Rossi, Professor Emeritus at the University of Utah, who worked with Zvezdelina Stankova to start the Berkeley Math Circle and has remained intensely involved with the math circles movement.