Indigenous peoples have rich ways of knowing that have been passed down for generations. Specifically, mathematical ways of knowing are embedded within a nation’s and community’s language, culture, and place. Turtle Mountain Community College (TMCC) now teaches three courses on Ojibwe mathematics. Local fluent elders were and are essential in the development of the courses and the content. The three courses seek to revitalize the mathematical ways of knowing embedded within the Turtle Mountain language, culture, and place so they can once again be widely known and normalized within the community. These courses are part of a newly accredited bachelor’s degree program at TMCC in secondary math education. The program entered its first cohort in the fall of 2023. The program follows the Indigenous math education model, pursuing balance and harmony between Western and Indigenous math, as well as Western and Indigenous ways of teaching and learning.

OJIBWE MATH I, II, AND III

As these courses are new to TMCC’s course catalog, students have been uncertain of what to expect when entering Ojibwe Math I. Faculty teaching the course begin day one by sharing an overview of the three-course sequence. The students are then introduced to the Indigenous math framework that is continually referenced throughout all three courses. These courses introduce students to the basic differences between Indigenous math and Western math.

The first topic of discussion after introductions—Indigenous mathematizing—emphasizes a core concept for all three courses. Although the term “mathematizing” may be new to many people, Indigenous knowledges have always been applied to mathematical concepts. Mathematizing refers to the process of mathematical thinking, doing, living, and being (Aikenhead, 2018; Stevens, 2021). In contrast to Western math, Indigenous mathematizing isn’t isolated from context, other school subjects, spirituality, place, language, or culture. Instead of math that’s studied inside, on paper, at a desk by yourself, mathematizing takes place outside and in the community through specific activities. Western math, contrarily, focuses strictly on intellect and assumes that math is culture and value free, universal, decontextualized, pure, and objective (Aikenhead, 2018; Stevens, 2021). That is to say, Western math assumes the adjective “Western” is irrelevant and unnecessary because it is universal and contains all mathematical knowledge.

These contrasts between Indigenous and Western math are also evident in language. Indigenous languages are more verb/action-based, in contrast to English which tends to be noun/definition-based. This leads to a more verb/action-based understanding in Indigenous math in contrast to a more noun/static-based understanding in Western math. Despite upper-level Western mathematics still being advanced through research, most people experience Western math as a static body of knowledge to memorize and reproduce without substantial connections to other areas, to other ways of knowing (emotional, spiritual, etc.), and to the place and culture the students are in. Angidaasowinan is the Turtle Mountain Ojibwe word for mathematics. The root is the verb angidaaso meaning “s/he counts” (-win implies the noun form of the action, and –an implies plural). Thus there isn’t a direct translation between mathematics and angidaasowinan, leading to misunderstanding between the more noun/definition-based perspective of math concepts and the verb/action-based perspective of mathematical activity.

Students in these courses are introduced to alternative names for the course including Angidaasowinan Bezhig, Niizh, Niswi, Ojibwe Mathematizing, Michif Mathematizing, D/Lakota Mathematizing, or the mathematizing of their specific ancestors. These alternative names demonstrate the language- and place-specific nature of Indigenous math. Each student personalizes the course by connecting the student learning outcomes to their own language and culture. This is a practice that aligns with some of the key themes in the book Indigenous Statistics (Walter and Anderson, 2013), which describes Indigenous as being both modern and place specific.

Figure 1: The Indigenous math framework for the Ojibwe math courses at TMCC has four components with spirituality intertwining them all.

The Indigenous math framework for the Ojibwe math courses has four components, each of which demonstrates a commitment to being place based and language specific. The four components are nature/place/land, community, heritage/tradition (including games, arts, engineering, etc.), and language. The math, mathematical thinking, and mathematical activity embedded within these four components makes up the core of the Ojibwe Math I, II, and III curriculum. Spirituality is intertwined through all four components and demonstrated by viewing the four components as part of the maskikiinan tiitibisenan (medicine wheel) teachings (see figure 1).

Starting in the East, the nature/place/land component emphasizes that math is both from nature and is a powerful language to describe nature. Land/nature is our first and greatest teacher. Nature teaches us that math is universal—in the sense that math comes from nature—and that math, like land, is unique to each place. Further, nature, place, and land each have their own nuances of meaning within the English language. Together, they contain mathematical knowledge to be experienced and observed in a specific location. For example, many Indigenous peoples observed patterns of the sun, moon, and stars to determine the solstices and equinoxes to guide the timing of ceremonies. More specifically, the sun rises on the eastern horizon, moving up and down (left and right, north and south) the horizon each day. As the sun moves north (up) the days are getting longer, until it slows down, appearing to stop, and then changes directions back to the south (right). This marks the summer solstice. This annual pattern/cycle was seemingly well known by most Indigenous peoples. In Western math, this is called the sine wave, where the x-axis is time and the y-axis is the place on the horizon. Many Indigenous nations were able to observe the stars, sun, and moon with even greater accuracy than just the summer solstice, leading to decisions about the placement of architectural structures on land to align with celestial movements, such as those at Chichen Itza and Chaco Canyon. The Ojibwe math courses seek to include local stories about nature and its spiritual significance as well.

Another example of Indigenous math is seen in describing the aquifer that provides the water for the Turtle Mountain reservation and surrounding area. In this way, math is used to describe nature and is connected to science, astronomy, language, history, politics, spirituality, etc., in contrast to the Western math value of isolation/ purity from other subjects. Aquifer math also demonstrates the community component of the Indigenous math framework. Using math to better understand the aquifer is valuable and useful to the community. In all three courses, students are reminded regularly that their learning is not strictly for themselves (as Western math would value), but for the community. This is put into practice through the final project for all three courses: a free, public, community presentation where students share their learning on specific aspects of Indigenous (Ojibwe, Michif, etc.) mathematizing. Everyone at the college is invited, along with friends, family, and local business and tribal leaders, as well as anyone interested in joining virtually. Faculty and teachers from other Indigenous communities have attended. And you too are invited to future community presentations—just reach out by email and you can join us!

Next, mathematical thinking is embedded within traditional games, art, engineering, architecture, etc. The mathematizing within these activities is not written down on paper at a desk inside by individuals working independently. Instead, traditional mathematizing is embodied through experiential activities learned in the community through story connected to spirituality and passed down generationally. For example, traditional architecture is associated with stories and spiritual meaning to the structures alongside their physical capabilities. Today, TMCC’s main campus building has a large medicine wheel engraved on the ground, central to the whole campus. A tipi structure is the highest point of the building, and the Seven Teachings are engraved within them.

Two students in a lab analyze materials from the local aquifer. At TMCC, students make connections between math, science, geology, water rights, politics, community needs, language, nature, and spirituality.

Lastly, the language component emphasizes mathematical thinking already embedded within the number system and other quantifying (and time-describing) vocabulary. For example, the Ojibwe number system embeds exponential thinking and grammatically uses numbers as verbs in certain contexts. This practice is distinct from Western math, where numbers are seen strictly as adjectives/nouns. The Indigenous language around time demonstrates a distinct perception of time and measuring. Language, intertwined with culture, identity, and worldview, help describe Indigenous mathematizing as verb-oriented, contextualized, embodied, and specific to each language.

One student shared that when they first heard of this course they were uncertain what to expect. They shared that both language learning and math can be intimidating, so putting them together seemed a little crazy. Yet, this “putting together” is one of the core practices in the Ojibwe math courses (and the entire secondary math education program), which emphasize and teach that math and language fluency can grow together. The Ojibwe math courses (and the larger program) can be and are part of the language revitalization efforts happening in our community.

Ojibwe Math I and II hit all four components of the Indigenous math framework. Further, additional student learning outcomes describe the varying strategies for talking about and/or revitalizing Indigenous math. Indigenous math needs no justification from Western math. However, Indigenous mathematizing and Western views of math are intertwined and overlapping. Along with the content within the Ojibwe math courses, the pedagogy aligns with Indigenous ways of teaching and learning. “Experience then describe” is a common philosophy within the course. For example, in a unit on hand games, the class begins with learning the rules and then playing the game. After multiple class periods of playing and experiencing the game, there are class discussions on the Indigenous mathematizing happening within the game and a Western math description of components of the game. In this way, the cultural game is sustained for future generations, the Indigenous mathematizing is experienced through playing, and the Western math connections to hand games are described.

Ojibwe Math III looks at Indigenous research methodologies as a way of continuing to develop and learn about Indigenous mathematizing. There are still so many unknown details about the mathematical ways of knowing embedded within Turtle Mountain’s (or any nation’s) language, culture, and place. This course gives students the opportunity to dive into the details on one particular context or topic. Further, this course enhances the Ojibwe Math I and II courses, which are always in development as more Indigenous math is researched and revitalized.

The course structure was developed through many inputs, including the Indigenous Mathematicians Network, the Indigenous Math Education Network, the D/Lakota Math Connections project at Sitting Bull College (SBC), and SBC faculty David Sanders’s dissertation (2011) on applying the six universal math activities (counting, designing, locating, measuring, playing, and explaining) toward a math education for tribal self-determination (Luecke and Sanders, 2023).

The greatest guidance in course development came from Cecelia Myerion and Velda Belgarde, two fluent elders in the Turtle Mountain community. They not only shared mathematical vocabulary but helped shape the foundational beliefs and pedagogy for the courses. We spent many hours together in person and virtually discussing aspects of the Ojibwe math classes—from the role of the instructor to vocabulary, and from local stories to perceptions of math in general. Alongside lots of Ojibwe vocabulary, Myerion and Belgarde shared that there is so much more depth to Ojibwe views and experiences with mathematics than just vocabulary. They discussed the four directions, four seasons, four senses, spirituality within math, the role of a math instructor, stories of math in context of certain activities, and their ideas for the Ojibwe math courses and Indigenous math framework. They both shared mathematical thinking as embedded within activities, the need for students to have experiential learning, and the need for Indigenous math to be connected to the community and the land.

Ojibwe math students, a guest presenter, and some of their children learn and play hand games together. The cultural activity is an excellent way to teach and illustrate Indigenous mathematizing.

Currently, neither math (such as college algebra) nor Ojibwe language is required for these courses. They are designed to be easy entry opportunities for anyone to begin learning about Indigenous mathematizing and overall language learning. At TMCC, the elementary education program now includes Ojibwe Math I as part of the curriculum. Further, the courses align well with TMCC’s new Indigenous learning outcomes. The long-term goal is that what is taught and learned in Ojibwe Math I will become common knowledge in future generations taught at the local elementary and secondary level. The plan is for future Ojibwe math courses at the college to go even deeper into each of the four components of the Indigenous math framework.

SECONDARY MATH EDUCATION BACHELOR’S DEGREE PROGRAM

The Ojibwe math courses (I, II, and III) were designed and developed as one component of the Secondary Math Education bachelor’s degree program. The program was accredited by the Higher Learning Commission (HLC) and the North Dakota State Board for Teacher Licensure in 2022 and is currently in its first cohort. The program joins the established elementary, secondary science, and secondary English bachelor’s programs in the TMCC Teacher Education Department. The Turtle Mountain community has a “grow your own” philosophy that creates pathways for local students to earn their bachelor’s degree at TMCC and then return as elementary and secondary teachers to the schools in which they grew up. The program’s mission is to prepare future math teachers with content and pedagogical knowledge from a balanced perspective of Western and Indigenous ways of knowing— two-eyed seeing.

Figure 2: The Indigenous math education model for the Secondary Math Education bachelor’s program pursues harmony and balance between Western and Indigenous ways of knowing and being, as well as between math content and pedagogy.

This balanced approach between content and pedagogy, Western and Indigenous, is demonstrated in the Indigenous Math Education program model (see figure 2). The circles in the Venn diagram are labeled Western math, Indigenous math, Western ways of teaching and learning, and Indigenous ways of teaching and learning, respectively. Note that each individual circle is distinct (described previously) and can stand on its own without any justification from another circle. Further, each individual circle is equivalent in size, representing their equivalent value (in contrast to colonial thinking that overvalues Western math and devalues Indigenous math). Lastly, notice that all of the circles are connected, representing the interconnected and interdependent nature of these teaching model components.

The program courses align with the four circles. The Ojibwe math courses are part of the Indigenous math circle (see figure 2). The sunrise location on the horizon exemplifies the distinctiveness and interconnectedness of Western and Indigenous math. For accreditation, the HLC and state board for teacher licensure did not consider the right side of this model. However, the Indigenous circles follow directly from part of TMCC’s mission to bring cultural and social heritage to bear throughout all the curriculum and to provide leadership to the community (Erdrich, 1990). Further, all students in the program and their future middle school and high school students will experience the complementary strengths of each circle to maximize the mathematical learning and ways of knowing for all.

MATH FOR THE SEVENTH GENERATION

Together, the Secondary Math Education bachelor’s program, Indigenous math education model, Ojibwe math courses, and the Indigenous math framework are an avenue toward revitalizing Indigenous math, tribal self-determination in math education, and Native ways of knowing. The depths and riches of Indigenous mathematizing are currently being rediscovered and adapted with the valued guidance of elders and the Indigenous mathematics community. The fluent elders who were part of designing the Ojibwe math courses inspired us to consider what could be in these courses or courses like them 10, 50, or 100 years from now, and in seven generations. There is so much more to continue to learn and normalize once again within our communities.

Danny Luecke (Choctaw) teaches Ojibwe math courses at Turtle Mountain Community College.

REFERENCES

Aikenhead, G. (2018). Indigenous Perspectives in School Mathematics: From Intellect to Wisdom. In A. Kajander, J. Holm, & E. Chernoff (Eds.). Teaching and Learning Secondary School Mathematics. Advances in Mathematics Education (pp. 39-49). New York: Springer.

Erdrich, L. (1991). Education as a Spiritual Journey: Like All Tribal Colleges, Turtle Mountain Community College Emphasizes Respect for Native Culture.” Tribal College: Journal of American Indian Higher Education, 2(3), 8-9.

Luecke, D., & Sanders, D. (2023). D/Lakota Math Connections: An Epistemological Framework for Teaching and Learning Mathematics with Indigenous Communities and Students. Frontiers in Education, 8. Forthcoming.

Sanders, D.W. (2011). Mathematical Views within a Lakota Community: Towards a Mathematics for Tribal Self-Determination (Doctoral dissertation). University of Colorado at Boulder.

Stevens, P. (2021). A Woodcutter’s Story: Perceptions and Uses of Apache Mathematics on the San Carlos Apache Reservation. Anthropology and Education Quarterly, 52(4), 430-450.

Walter, M., & Andersen, C. (2013). Indigenous Statistics: A Quantitative Research Methodology. New York: Routledge.