Culturing Calculations

1. Culturing Calculations MSU Billings Summer Institute 2010 Transforming Perceptions Tuesday, June 8th 1:00 – 2:45 p.m. Reno Charette, MSUB American Indian Outreach

2. What You Will Learn • Although mathematics is culturally based, modern educators are just now beginning to understand how American Indians calculated within their culture. Learn a few strategies on how to “culturize” mathematical instruction in a relevant manner for American Indian students. Reno Charette, MSUB American Indian Outreach

3. Mathematics is Cultural • Ubiratan D’Ambrosio stated that “mathematics is a cultural system [that] differ[s] essentially from [one] cultural group to another.” He further states that groups formed their own patterns of behavior, symbols, and codes which pave the way for formulating a culture’s mathematics. For example, diverse mathematizing distinctly separates the civil engineer from the dessert chef. Reno Charette, MSUB American Indian Outreach

4. What Characteristics Define Mathematics? • According to Charles G. Moore: • “Mathematics…reflects the active will, the contemplative reason, and the desire for aesthetic perfection…It’s basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.” Reno Charette, MSUB American Indian Outreach

5. Ethnomathematics Definition by Gloria Gilmer, Math-Tech Inc. • The prefix 'ethno' refers to identifiable cultural groups, such as national-tribal societies, labor groups, children of a certain age bracket, professional classes, etc. and includes their ideologies, language, daily practices, and their specific ways of reasoning and inferring. • 'Mathema' here means to explain, understand and manage reality specifically by ciphering, counting, measuring, classifying, ordering, inferring and modeling patterns arising in the environment. • The suffix 'tics' means art or technique. • Thus, ethnomathematics is the study of mathematical techniques used by identifiable cultural groups in understanding, explaining, and managing problems and activities arising in their own environment. Reno Charette, MSUB American Indian Outreach

6. “Indian Thinking” • “A ‘circular’ approach toward life is inherent in Indian cultures since time immemorial. The native world is one of cycles, and observing the cycles provides an order to life and community.” Donald L. Fixico. The American Indian Mind in a Linear World. p. 42 • “Indian Thinking is ‘seeing’ and ‘listening.’ Listening for sounds of what is most relevant about the interaction between two things is a part of the realization of relationships and learning their importance about life….’seeing’ in this way involves trying to understand the significance of relationships.”ibid. p. 4 Reno Charette, MSUB American Indian Outreach

7. Making Connections • Understanding relationships from seeing connections to familiar environments. • Listening for the essential understanding of relationships and their multiple meanings. • Silence is imperative in listening for reflection and self-examination as information is related to the seer. Donald L. Fixico. The American Indian Mind in a Linear World. p. 5. Reno Charette, MSUB American Indian Outreach

8. American Indian Mathematics • Language • Counting, measuring • Architecture • Building structures with a plan, process, order • Hunting & Gathering • Estimating, tracking • Food processing • Hydrating to reduce weight • Manufacturing • Clothing from available resources • Tools • Games • Chance, accuracy, speed Reno Charette, MSUB American Indian Outreach

9. Language of Mathematics • Base Ten Counting • Descriptive: Infolded four times • Sign language • Artistic expressions of numerations • Pictographs • Ledger drawings • Tipi Paintings Reno Charette, MSUB American Indian Outreach

10. Base Ten Counting in Cheyenne Reno Charette, MSUB American Indian Outreach

11. Cheyenne Cardinal Numbers Reno Charette, MSUB American Indian Outreach

12. Cheyenne Ordinal Numbers Reno Charette, MSUB American Indian Outreach

13. Numerical Picture-Writing Pictograph Winter Count Reno Charette, MSUB American Indian Outreach

14. Architecture • Tipi structure • Symbolism • Air & heat controls • Strength of structure • Transportable • Symmetrical Reno Charette, MSUB American Indian Outreach

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16. Crow Tipi Floor Plan Reno Charette, MSUB American Indian Outreach

17. Crow Tipi Poles – Vera He Does It-Half • Four base poles represent the cardinal directions and seasons of the year. • Northeast – the force that controls the day coming over from the east. • Southeast – the eternal summer. • Southwest – the point where people leave the world. • Northwest – the eternal winter where the weather comes and freshens the earth. http://images.google.com/imgres?ingurl=http:www.forevermontana.com/images/Pole_History_copy.jpg

18. Crow Tipi Poles – Vera He Does It-Half • Facing east – the two door poles represent the spirits of the lion is on the left and the bear is on your right, both are protectors. • Two flap poles – the smoke flaps represent the spirits of the owl on the left and the right represents the coyote who are on guard over the tipi night and day. • Chief pole – represents the owner of the tipi.

19. Crow Tipi Poles – Vera He Does It - Half • Two helper poles and the secondary poles, represent elements of forces in the life of the owner and are personal and connected with nature. They reflect harmony and are important for a good life. http://images.google.com/imgres?ingurl=http:www.forevermontana.com/images/Pole_History_copy.jpg

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21. Hunting & Gathering • How much is enough? • Reading animal signs to determine measurable results. • Strategic use of harvest cycles. • Sites of highest production • Plant foods • Game animals Reno Charette, MSUB American Indian Outreach

22. Exploration • A young Native boy is learning to track animals. When animal tracks are discovered, how does he read the signs to know if he is on the trail of many animals or only a few? Reno Charette, MSUB American Indian Outreach

23. Food Processing • Storage methods • Packed in leather bags sealed with animal fat • Dehydration • Meat cut thinly and dried in the sun • Berries mashed into patties and dried in the sun • Grinding • Pemmican • Dried turnips • Dried berries Reno Charette, MSUB American Indian Outreach

24. Pemmican Recipe • 2 cups buffalo, elk, or deer jerky, shredded. Use an old fashioned hand grinder to finely shred the dried meat. • 1 cup dried chokeberries or tart red cherries, or dried cranberries, (other dried fruits can be substituted) chopped or finely ground with the hand grinder. • Honey – enough to bind the dry ingredients together. • Optional: add crushed nuts of any kind. • Mix all the ground dry ingredients together. Add a little honey at a time until the ingredients are moist enough to mold well and hold their shape. Roll into small golf size balls and wrap in cellophane. Store in the fridge or freezer. Reno Charette, MSUB American Indian Outreach

25. The Prairie Turnip produces a spindle-shaped tuber about four inches below the ground. This tuber, although nutritionally similar to a potato, differs in taste and texture due to different types of sugars and starches. The white edible portion is exposed by removing a coarse brown husk. If the thin portion of the root is left attached, the tubers can be woven together into an arm-length bundle for easy drying and transport. When air dried, the tubers can be stored indefinitely. • Timpsula: Prairie Turnip Psoraleaesculenta - also known as the prairie wild turnip, Indian breadroot, and several other names…The Prairie Turnip was probably the most important wild food gathered by Indians who lived on the prairies. In 1805 a Lewis and Clark expedition observed Plains Indians collecting, peeling, and frying prairie turnips. The Lakota women told their children, who helped gather wild foods, that prairie turnips point to each other. When the children noted which way the branches were pointing, they were sent in that direction to find the next plant. This saved the mothers from searching for plants, kept the children happily busy, and made a game of their work. Prairie turnips were so important, they influenced selection of hunting grounds. Women were the gatherers of prairie turnips and their work was considered of great importance to the tribe. http://woodenknife.com/campfire/newsletter.asp?ID=4 Reno Charette, MSUB American Indian Outreach

26. Exploration • In traditional times, an Indian mother had to gather enough turnips to feed her family for a year. If she had two children and a husband, how many parfleches of dried turnips would she need to prepare? • Measuring: one handful per meal over 9 to 10 month period for an adult, and about half that for a child. • Caution: Turnip’s are eaten in small quantities to avoid constipation. Reno Charette, MSUB American Indian Outreach

27. Manufacturing - Clothing • One hide to three hide dresses • Introduction of the horse • Introduction of guns • Measuring to ensure proper fit • Beading is mathematical • Peyote stitch with uneven numbers • Counting out geometric designs on a loom, lazy stitch, or double needle styles. Reno Charette, MSUB American Indian Outreach

28. Manufacturing - Tools • Tipi furniture: Back rests, pillows • Indian suitcase: Parfleche, bags, quivers • Hunting weapons: bows, arrows, knives • Tack: Double horn saddles, bridles, travois • The original occupant safety seat: Cradleboard • Cookware: Animal bladders, grinding stones • Sewing tools: bone needles • Hide preparation: Elk horn scrapers Reno Charette, MSUB American Indian Outreach

29. Women’s Tools Gardening hoe made from a scapula or shoulder blade bone. Elk horn scraper used to flesh hides. Reno Charette, MSUB American Indian Outreach

30. Men’s Tools War clubs Kickapoo lance point Bow and arrows Reno Charette, MSUB American Indian Outreach

31. Mathematics of Building Furniture Indian Style

32. Tipi Backrests • Trapezoid • A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. The sum of the angles of a trapezoid is 360 degrees.

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34. What Else can we use a trapezoid for? Reno Charette, MSUB American Indian Outreach

35. Tipi Liners Reno Charette, MSUB American Indian Outreach

36. Date: 09/23/97 at 04:55:18 From: Doctor Pete Subject: Re: Properties of circles A circle has 360 degrees, but it also has 400 gradients and approximately 6.2831853 radians. It all depends on what *units* you measure your angles with. Allow me to explain. Say you think 360 is a terrible number, and you think that you want a circle to have 100 "somethings" in it. Well, you divide up the circle into 100 equal angles, all coming out from the center, and then you call one of these angles a "deeg." Then you've just defined a new way to measure a circle. 100 deegs are in a circle. This invented unit, the deeg, is much like the degree, except the degree is smaller (why?). They are both angles. Just as 1 inch = 2.54 centimeters, although the centimeter is smaller, the inch and centimeter are both units of length. So the ancient Babylonians (not the Greeks), decided that a circle should contain 360 degrees. In one degree there are 60 minutes (though they have the same name, one minute-angle is not the same as one minute-time). Furthermore, in one minute there are 60 seconds (again, one second-angle is not one second-time, though they are the same word). http://mathforum.org/library/drmath/view/58395.html Reno Charette, MSUB American Indian Outreach

37. Standards OPI – Math Standards Indian Education for All Essential Understanding 3 The ideologies of Native traditional beliefs and spirituality persist into modern day life as tribal cultures, traditions, and languages are still practiced by many American Indian people and are incorporated into how tribes govern and manage their affairs. Additionally, each tribe has its own oral histories, which are as valid as written histories. These histories pre-date the “discovery” of North America. • Content Standard 4 - Students demonstrate understanding of shape and an ability to use geometry.

38. Games • Northern Cheyenne Plum Stone Seed Game • Pin and Hook games Reno Charette, MSUB American Indian Outreach

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40. Resources • Native American Mathematics. Michael P. Closs. University of Texas Press, Austin. 1986. • Picture Writing of the American Indians. Garrick Mallery. Dover Publications, Inc. New York. Vol I and II. 1972. • The American Indian Mind: In a Linear World. Donald L. Fixico. Routledge, New York. 2003. Reno Charette, MSUB American Indian Outreach