By Lina Sorg
In mathematics education, “ethnomathematics” refers to the study of the relationship between mathematics and culture. It aims to improve students’ understanding of mathematical concepts while simultaneously cultivating their appreciation of culture and fostering a stronger appreciation of the relationship between the two. During a minisymposium at the 2022 SIAM Conference on Applied Mathematics Education, which took place earlier this week in Pittsburgh, Pa., in conjunction with the 2022 SIAM Annual Meeting, Kamuela Yong of the University of Hawaiʻi – West Oʻahu overviewed his incorporation of Native Hawaiian culture—specifically Polynesian navigation techniques—into the precalculus classroom.
Yong is the first Native Hawaiian to earn a Ph.D. in applied mathematics. He is also the co-founder of Indigenous Mathematicians, an organization for all Indigenous people in mathematics. Over the years, he realized that he was fielding a lot of questions about ethnomathematics and place-based education — topics with which he was not especially familiar. After attending a National Science Foundation workshop about the incorporation of place-based education in the classroom, Yong decided to redesign the entire curriculum of his precalculus course to reflect Polynesian navigation.
“Navigation” thus became Yong’s big concept around which the whole course was based. His intent is the relationship between trigonometry, navigation, and everyday lives. “That’s where the place-based knowledge is,” he said. “Things are very tangible. Even if it’s not their culture, it’s fascinating because [students] can really just understand.” Finally, the course’s content and skills include the compass, angles, direction, and movement.
Yong breaks his curriculum into four different units:
- Unit 1: Introduction to Trigonometric Functions
- Unit 2: Applications of Trigonometric Functions
- Unit 3: Analytic Trigonometry
- Unit 4: Polars/Vectors/Trigonometric Equations.
Each unit features its own breakdown of big concepts, intents, and content and skills. For example, unit 1 focuses on understanding the compass, angles, and triangles. Students learn how angles and triangles relate to trigonometry and explore the relationship between trigonometry and navigation in the context of direction and movement. In unit 2, students analyze triangles and learn about inverse trigonometry, the law of sines and cosines, functions of angles, and trigonometric graphs. Unit 3 centers on the application of tools from the previous two units, and unit 4 investigates direction and movement with an emphasis on polar coordinates and polar graphs, vectors, and dot/cross products.
In the classroom, students frequently ask why? “These are the lessons that speak to their heart,” Yong said. “What hooks them in, grabs their attention, and builds off of what they’ve known before?” They also want to know what they are learning, which materializes from direct instruction and a variety of educational strategies. Hands-on classroom activities show students how they will apply and practice the material. Finally, they create, produce, and showcase their newfound knowledge to answer any queries of what now?
Figure 1. The Hawaiian star compass (left) and the unit circle (right) share many similarities, including symmetry and quadrants.
Yong answered the why, what, how, and what now for each unit. He began with unit 1, which examines the many similarities between the Hawaiian star compass and the unit circle (see Figure 1). Both the compass and unit circle have symmetries and different quadrants; in the star compass, each quadrant represents a separate wind. The class learns how to read both entities. “We can start relating these terms to students as something that they can visualize, feel, and stand outside and see,” Yong said. “When I teach students how the star compass works, all of the sudden they make the connection that the unit circle is not so big.” Unit 1 culminates in the creation of one’s own compass or unit circle from a paper plate.
In unit 2, students appreciate the use of right triangles in everyday life. They learn to determine the distance to the other Hawaiian Islands and identify the islands that are visible from the shoreline. They also calculate the distance of the horizon—i.e., how far they can see—when standing on the shore and looking out to sea, even accounting for the Earth’s curvature. “That’s something that is really important,” Yong said. “When are you going to see land?” He then shared the following example from his classroom:
The Hinemoana, a waʻa from Aotearoa, plans to sail from Auckland to Rarotonga. It is known that the destination is 937 nautical miles east and 839 nautical miles north of its current location. What house do you need to sail towards and what distance will you need to sail? If the waʻa travels at 5 knots, how long will it take to reach the destination? Note that 1 knot = 1 nautical mile/hour.
Figure 2. Kamuela Yong introduces a current into a navigation problem about sailing from Auckland to Rarotonga to teach his students about oblique, non-right triangles.
By working on problems like this, students become familiar with inverse trig functions and begin to solve them. “I’m teaching them the navigation components,” Yong said. “That’s what gets the students hooked, then I introduce the math behind them.”
One variation of the aforementioned problem involves the top of a mountain, and Yong incorporates the legend of the demigod Maui for additional cultural context:
According to moʻolelo (legend), the sun traveled very fast across the sky, leaving people with days so short there was not enough time to carry on with their daily lives. Determined to slow the sun, Maui climbed to the summit of Haleakalā, which stands 10,023 feet, to snare the sun. Assume the radius of the Earth at Haleakalā is 3,961.527 miles.
- How far is the horizon when Maui stands at the summit?
- How much faster would Maui see the sun emerge over the horizon than someone standing at the seashore?
Figure 3. Sail plan versus actual course for the final project, which involves sailing from Tahiti to Hawaii.
As the course progresses, Yong returns to the original sailing navigation problem from Auckland to Rarotonga and introduces a current that necessitates the use of oblique, non-right triangles; the subsequent calculations exemplify the law of sines (see Figure 2). Other scenarios use sailing to introduce vectors. “If you’re sailing from one place to another, that’s a vector,” Yong said. “It has both a magnitude and a direction.” He incorporates his own knowledge about latitude sailing to teach the sum of vectors.
The sum of vectors factors into the students’ final project, which asks the following question: How would you sail from Tahiti to Hawaii? Yong provides maps of the winds and currents to help them navigate a path; if they drift too far off course, the wind and current patterns will prevent them from getting back on track (see Figure 3). To craft this problem, Yong consulted resources from various Hawaiian societies, including hokulea.com. Because no one in his immediate family sails, he worked with the sailing and navigation communities and even completed an internship with the Kānehūnāmoku Voyaging Academy to ensure that he was accurately representing the culture. “When I immersed myself, I started seeing even more connections,” Yong said. “I saw more ways that I could talk about connecting actual history, rather than just the textbook approach.”
Ultimately, Yong hopes that his navigation-oriented precalculus course will spread awareness of Polynesian culture and increase participation from both Indigenous and non-Indigenous students. In the future, he wants to get students out of the classroom and immersed in some of the relevant cultural navigation activities firsthand.
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Lina Sorg is the managing editor of SIAM News. |