Native American mathematics

Summary: Native Americans developed numbering systems and had a clear sense of dimension, geometry, and probability.

The term “Native American mathematics” is deceptive because there is no single culture for all Native Americans. Rather, each of more than 400 Native American tribes has its own distinct culture, with each mathematical element being specific to that culture. Nonetheless, in an examination of mathematical aspects, it is possible to discuss some commonalities across the many tribes, producing evidence of multiple number systems, arithmetic operations, geometry, and probability.

Number Systems

Native American numbering systems often used a simple grouping system that corresponded to different parts of the human body. For example, the idea of “tens” is contained in the numbering system of the San Gabriel Indians in California, where “all my-hand finished” represented the number 10, “all my-hand finished and one my-foot” represented the number 15, and “another finished my-foot the side” represented the number 20. It is inferred that they used single fingers on each hand to represent any number less than 10. Often, a Native American tribe would have names for large numbers, but had little use for such in their daily lives. For example, Michael Closs, a cultural historian, describes a Copper Eskimo elder while relating a story about two men who, trying to settle an argument, begins to count the hairs on a wolf and a caribou. The story ends with the count unfinished, as both men die of starvation. And, the story concludes with the phrase: “That is what happens when one starts to do useless and idle things that can never lead to anything.”

Though using groupings of 5 and 10 as the structure for their number systems, the idea of a number base is not always evident. Also, some evidence exists for the use of 2, 4, and 20 as the structuring element. For example, the Yukis tribe in northern California used a combination of the quaternary (base four) and octal (base eight) systems. In turn, their counting mechanism depended on referring to the four spaces between the fingers on both hands, not the fingers themselves.

In a study of North American Native Americans, researchers documented the use of 307 different number systems; 33% were base 10, 33% were base 5, 23% were base 2, 10% were base 20, and the remaining 1% were base 3.

In any discussion of Native American mathematics, it is necessary to include the Aztecs, Incas, and Mayans. For example, the Aztecs’ number system was based on the number 20, with the numbers 400 and 8000 given special significance. In contrast, the Incas used a slight variation of the base 10 system, and even had specific words for the numbers 1–10, 100, 1000, and 1,000,000. Finally, the Mayans, the most mathematically sophisticated of the three, had a vigesimal system using the number 20 as its base. The Mayan system also included special notations for multiples of numbers and used a special symbol glyph to represent zero.

Arithmetic

The idea or use of arithmetic operations was not something needed by early Native Americans, who depended on a hunting-gathering culture. Historians suggest that any signs of significant arithmetic are due to a tribe’s interactions and trade with the early fur traders or buffalo hunters. For example, the language of the Navajo does not include words for “multiply” or “divide,” yet that should not imply their inability to perform either process computationally or using real items.

Evidence of addition is found in words used to denote different numbers, using a process of addition by juxtaposition. For example, Alaskan Natives living near the Yukon River essentially used the words “five one” and “five four” to denote the numbers 6 and 9, respectively. In direct contrast, the Miluk Coos, an Oregon tribe, used subtraction by juxtaposition, where “four ten” and “one ten” denoted the numbers 6 and 9, respectively. Some historians claim that 40% of the Native American tribes used some version of this subtraction process, especially for numbers close to multiples of 10.

Evidence of multiplication is found among Pawnee tribes, in a very creative fashion. Their term “50 persons” represented the number 1000, based on their use of the word “man” for the number 20, knowing that “man” had 10 fingers and 10 toes. Thus, “50 persons” was equivalent to 50 sets of 20 fingers and toes, or a total of 1000.

Measurements

The measurements invoked by Native Americans were context-sensitive and personal in nature. No standard units were established and used widely either within a tribe or across tribes. In most instances, the measurements used were specific to the context and informal. The Ojibwa tribe is a good example. For short lengths, their units were finger widths, hand spans, forearm lengths, and arm spans, while their longer lengths might reflect a changing position of the sun or even mention the unit “number of sleeps” involved in traversing a long distance.

Geometry

Native American geometry is evident in the colorful decoration and intricate patterns found on knife cases, moccasins, blankets, pouches, baskets, and pottery. At first, many of these patterns were created using porcupine quills but eventually the shift was made to using glass beads.

When creating a pattern, the different Native American tribes differed in their use of geometrical structures. In some instances, a tribe’s members created irregular floral patterns, while other tribes used a geometry based only on straight lines, allowing them to create blocks, crosses, and triangles. The types of triangles ranged from isosceles to equilateral to right, with common traits being tall isosceles triangles or pairs of reflecting congruent triangles. Occasionally, circles and spirals appear as part of a design.

Many studies have focused on Native Americans’ use of symmetry in strip patterns using beads. Of the seven possible symmetry groups, the most popular pattern is labeled “pmm2” in standard transformational schemes, which means that the pattern has horizontal, vertical, and rotational symmetry. Figure 1 shows this pattern.

Also, in the process, the creator of the visual pattern possibly used counting or even some computing skills (for example, skip counting by threes to form a border). It is possible that creators of some of the patterns included elements of measurement (perimeter or area), number theory (multiples and divisors), and fractions (common, decimal, and ratios).

Tiling patterns are evident in the creation of blankets, going beyond strip patterns. An example is the section from a Navajo blanket in Figure 2.

Some historians claim tiling elements are also found in some of the Native American petroglyphs carved on the surfaces of caves, cliffs, and large stones.

Finally, Native Americans had a clear sense of dimension, using objects to represent the three possibilities. A stick represented dimension one, an animal skin represented dimension two, and an apple or walnut represented dimension three. However, in their paintings on flat surfaces, the idea of dimensional perspective is not utilized.

Probability

Elements of probability are found in some of the children’s games played by various Native American tribes. For example, consider the Apaches’ “Throw Sticks” game involving two or more people. In one version, three sticks are decorated with colorful designs on one side only, called the “face.” The sticks are held in one hand and then dropped on the ground. The scoring is as follows: 10 points for three faces up, 5 points for two faces up, 2 points for 1 face up, and 1 point for no faces up. The score is kept by moving small sticks or “horses” around a circle of 30 stones. Play continues until someone travels the full circle. Elements of probability, such as likelihood, events, and dice-like actions, are all evident in this game.

Native Americans also played dice games, using dice made from bone, peach stones, deer horn, beaver teeth, or walnut shells. As most of these dice were two-sided, one side was colored to distinguish the two sides. When sets of dice were thrown, the scoring was based on the number of a given side appearing. Because the “dice” were crudely made, the chances of each side appearing are not equal. This observation actually validates the claim that Native Americans had a good sense of probability, because the higher score values were assigned to the least probable events.

Bibliography

Barta, Jim. “Native American Beadwork and Mathematics.” Winds of Change (Spring 1999).

Closs, Michael. A Survey of Mathematics Development in the New World. Ottawa, Canada: University of Ottawa, 1977.

——— ed. Native American Mathematics. Austin: University of Texas Press, 1986.

Culin, S. Games of the North American Indians, Volume 1: Games of Chance. Lincoln: University of Nebraska Press, 1992.

Hughes, Barnabas, and Kim Anderson. “American and Canadian Indians: Mathematical Connections.” Mathematics Teaching in the Middle School 2, no. 2 (November–December 1996).