Math in Waʻa: Examples of math in waʻa and navigation

Navigators in Micronesia utilize the paafu mat or map (shown in Figure 1.2.1). It is often misunderstood and misinterpreted as a Star “Compass” due to its use of stars and constellations for direction finding. However, paafu serves a different purpose and is not equivalent to the cardinal directionality marked by compasses (North, South, East, and West). Instead, it is a learning and teaching tool designed to teach the locational positions of islands, locales, or canoes relative to other places. This is achieved by observing the rising and setting points of stars and constellations, which act as markers for different locations.

A constellation is a cluster of stars whose shapes and meanings reflect and carry cultural significance. In modern society, a well-known cluster of stars in the southern hemisphere, shaped like a cruciform, is commonly referred to as the “Southern Cross.” However, for the Polowatese and other islanders from the Central Carolines region in the western Pacific islands, this same constellation resembles the triggerfish, and so it is named accordingly.

In the Central Carolines, the general location in the celestial sky where stars appear to rise after sundown is referred to as “tan.” This term is often mistakenly translated as “east” due to the modern association of stars (like the sun) with “rising” in the east. However, it’s important to note that “tan” means “rising” and not “eastward.”

Figure 1.2.2 orients the cardinal direction known as “east” at the top of the page, and so the top half of this diagram is also identified as “tan” – where stars appear to rise. The diagram illustrates the apparent path of stars across the sky each night and day (though most stars are not visible during the day) and throughout a year. In this system, the rising and presence of specific stars mark months, and these same stars will eventually set at a point horizontally opposite to where they rose. This point corresponds to the cardinal direction known as “west,” which is referred to as “tolon” – the area where the stars “set” or go down.

In the paafu “map” shown in Figure 1.2.2, a canoe is placed at the center, and the star called Mailap (Altair) marks due east. The time and location when Mailap rises are referred to as “Tan Mailap” (rising Mailap), while the time and place it sets are referred to as “Tolon Mailap” (setting Mailap). The map’s orientation places east, or the rising points of the navigation or paafu stars, at the top of the circle, and west, or the setting points of these stars, at the bottom of the circle. As a map, paafu uses the rising and setting points of stars to mark places around a given locale, which is placed at the center of the circle. Table 1.2.3 displays the star and constellation names, provided in both Polowat and according to the International Astronomical Union, listed in the order of their rising during the third week of March in Polowat.

Paafu can also be used to identify the direction in which moving objects, such as canoes, or creatures like birds, fish, and humans, are heading or coming from. This version of paafu utilizes the Polowat dialect of the Chuukese language, as used by members of the Weriyeng School of navigation.

With the aim of reviving wayfinding in Hawaiʻi, Nainoa Thompson journeyed to the island of Satawal in the Federated States of Micronesia to learn from master navigator Mau Piailug, affectionately known as Papa Mau. Using this knowledge, Thompson adopted the paafu method, leading to the creation of the Hawaiian Star Compass, also referred to as the Kūkuluokalani (Figure 1.2.4).

In the star compass, featuring the figure of an ʻiwa or great frigatebird at its center, Thompson divides the visual horizon into 32 equidistant points around a circle, referred to as houses. Each house in the Hawaiian Star Compass represents a specific space on the horizon (${11.25}^{\circ }$) where celestial bodies such as the sun, stars, moon, and planets rise and set. In the same way that we use addresses to locate homes, each celestial body has its own address represented by these houses.

The four cardinal points align with particular houses. Stars rise from the horizon called Hikina (“To Arrive”) or East and set on the horizon called Komohana (“To Enter”) or West. If you face Komohana (West) with your back towards Hikina (East), your right will point towards ʻĀkau (“Right”) or North, and your left will point towards Hema (“Left”) or South. The Hawaiian Star Compass is oriented with North at the top.

The star compass is divided into four quadrants, each named after winds in Hawaiʻi. Koʻolau is the Northeast quadrant named for the trade winds; Hoʻolua is the Northwest quadrant, Kona is the Southwest quadrant; and Malanai is the Southeast quadrant.

Each house on the star compass is given a name. The corresponding houses in the east and west share the same name. Starting from the east or west and moving northwards and southwards, the first house on either side of Hikina (East) and Komohana (West) is called Lā (Sun). It is followed by ʻĀina (Land), Noio (Tern), Manu (Bird), Nālani (Heavens), Nā Leo (Voices), and Haka (Empty). The 32 houses in the Hawaiian Star Compass correspond to the points of the 32-wind compass rose (Table 1.2.5).

Celestial bodies move along parallel paths across the sky from East to West, rising and setting in the same house, remaining in its hemisphere. For example, if a star arrives in the Koʻolau (northeastern) quadrant in the star house ʻĀina, it will arc overhead, staying in the northern hemisphere, and enter the horizon in the same house it arrived in, ʻĀina, but in the Hoʻolua (northwestern) quadrant (see Figure 1.2.6). Similarly, if a star arrives in the Malanai (southeastern) quadrant in the house Lā, it will remain in the southern hemisphere as it arcs overhead and enters the horizon in the house Lā in the Kona (southwestern) quadrant.

The star compass also serves as a guide for determining direction based on wind and ocean swells. As the wind and swells move, they intersect the star compass diagonally. For example, if a wind blows from the house Noio in the Koʻolau (northeast) quadrant, it will blow in the direction of the Kona (southwest) quadrant and eventually exit in the same house, Noio.

Directions are very important. From Hawaiʻi, flying 3,000 miles in the East-West direction could take you to North America or to Asia, depending on which direction you are flying towards. In the Star Compass, the celestial objects travel along the Hikina-Komohana (East-West) direction. In math, we can call this direction the $x$-direction. To make sure we know which way to travel in the $x$-direction, we will say positive means to the right (or in the direction of Hikina) and negative means to the left (or in the direction of Komohana). Similarly, instead of ʻĀkau-Hema (North-South), we can call this direction the $y$-direction with positive values pointing in the direction of ʻĀkau (North) and negative values in the direction of Hema (South). GPS, which we rely heavily on needs to know where in the world objects are. Assigning each point on Earth a value for its position in terms of East and West as well as North and South, we can learn about our coordinates. In math, each point is represented by both an $x$- and $y$-value, written as an ordered pair, with the $x$-value first, followed by the $y$-value, separated by a comma, and both inside parentheses. For example, if the $x$-value is -5 and the $y$-value is 4, the ordered pair is $(-5,4)\text{.}$ Stand at the center of the Star Compass. If you have a physical Star Compass, this would be a good opportunity to visit that location. If you do not have one, you can create one in your classroom. To create one in your classroom, use two long pieces of string to represent the $x$- and $y$-axes. Use objects such as rocks, or even water bottles to represent each house and place them evenly spaced in a circle. Now that you are at the Star Compass, stand in the middle, this is called the origin and both the $x$- and $y$-values are zero, or written as an ordered pair $(0,0)\text{.}$ walk 5 steps in the hoouse Hikina. In terms of math, is this the $x$>- or $y$-direction? Is it in the positive or negative? Ans: Positive $x$-direction. Now take 2 steps towards ʻĀkau. Is this in the $x$- or $y$-direction and is it positive or negative? Ans: Positive $y$-direction. So our coordinate should be $(x,y)=(5,2)\text{.}$ Looking at the Star Compass around you, which house and quadrant are you in? Ans: ʻĀina Koʻolau. Repeat these for the coordinates $(x,y)=(5,-2)\text{,}$ $(x,y)=(-5,2)\text{,}$ and $(x,y)=(-5,-2)\text{.}$ What are the associated houses and quadrants? Ans: ʻĀina Malanai, ʻĀina Hoʻolua, ʻĀina Kona, respectively. What do you notice about these numbers? These numbers are all the same, except the negative and they all have the same house, just different quadrants (both wind quadrants and cartesian quadrants) Now we introduce to Cartesian quadrants: I, II, III, IV. In math we also have quadrants and define Quadrant I to be the quadrant on the top right, or when x and y are both positive; moving in a counterclockwise direction, Quadrant II is where x is negative and y is positive; Quadrant III is where x and y are negative; and Quadrant IV is where x is positive and y is negative. Which quadrants from the Hawaaiian Star Compass correspond to which quadrants on the Cartesian plane?

Identify a point in Noio Kona.

Keeping in mind what we just did, without looking, identify a point in Noio Hoʻolua, Noio Koʻolau, and Noio Malanai. You should be able to use the negatives to find those points.

Insert grid

Draw a waʻa on the coordinate plane and then determine the coordinates Distance problem on coordinate plane (vertical and horizontal only) using absolute values (can use fractions or decimals) Map the Moananuiakea voyage and have the students calculate the distance (superimpose map on coordinate plane)