tham larutluc: culture and ethnomathematics
TRANSCRIPT
Tham Larutluc: Culture Tham Larutluc: Culture and Ethnomathematicsand Ethnomathematics• Welcome!
Tham Larutluc: Culture Tham Larutluc: Culture and Ethnomathematicsand Ethnomathematics
• Welcome!
• Bienvenidos
• Boas vindas
• Welkom
• Hoşgeldin
• Accueil
• Willkommen
Tham Larutluc: Culture Tham Larutluc: Culture and Ethnomathematicsand Ethnomathematics• Thomas E. Gilsdorf, NASGEm, TODOS, and
• University of North Dakota, Grand Forks, ND.
• thomas.gilsdorf @ und.nodak.edu
•
• I can send you copies of the talk by email.
Tham LarutlucTham Larutluc
• Can you figure out what the title means?
• Our tendency is to read from left to right. Try reading from right to left.
• Yes, “Cultural Math”
• The paradox of ethnomathematics: We want to describe the mathematics of another culture, but we must use our own interpretation of mathematics in order to do that description!
The “Forest Phenomenon”
From the outside, we only see leaves and tree trunks.
If we go inside the forest, we begin to see more details.
The longer we stay in the forest, the more we learn about subtle details,
special events (e.g., dawn, dust, rain, shine, etc.).
To completely understand the forest, we have to become part of the forest
environment.
The same ideas apply to understanding another culture.
Cultural Tendencies• To completely understand the mathematics of another
culture, we would need to become genuine participants of that culture.
• For many cases, it is not possible or reasonable to become part of the other culture, so what can we do?
• An important step in understanding the mathematics of other cultures is recognizing our own cultural tendencies.
• Everyone grows up being part of at least one culture.
• The long-term effect of living in a culture is that tendencies are subtle and strong.
• So, our goal can be to recognize our own cultural tendencies and learn to separate them from the culture we are learning about.
Cultural views
• Preliminary questions: Write down your responses to:
• Can you remember exactly how much the stock market or financial markets changed three days ago?
• How many pens and pencils do you have in your desk (office, school room)?
• If you have a kilogram of grapes, how many grapes is that? How many grains of rice are in an one pound package?
• What does it mean for a culture to have a writing system? Can a culture develop mathematics without developing a writing system?
Western views of math
• Four main views- relevant to mathematics.
• 1) An emphasis on number. Importance of numbers in conveying information.
• Example: News! Business pages, sports pages, weather, general news. A lot of information expressed as numbers.
• How is this a cultural view? Because knowing such detailed information is generally not necessary for daily survival.
• Example: Economic news could be expressed as “going well”, “doing poorly”, etc., instead of with numerical details.
• Think about the question of “financial markets three days ago.”
Western views of math
• 2) An emphasis on precision: Making precise measurements or counts is considered superior to making general estimations.
• A hunting/gathering or livestock tending culture in which precise records of “how many” are not kept, is often considered as a culture “unable” to keep accurate records.
• However, having precise information is not necessary in many contexts.
• Think about how many grapes in a kg. or grains of rice in one pound. How accurately do you need to know the number of pens and pencils you have?
• Example: “The search for precision, . . ., is characteristic of mathematics.” Assumes precision is always best.
Western views of math
• 3) Assumptions that all cultures eventually follow a path of (mathematical) development similar to how mathematics has developed in Western culture.
• This assumption is often used to conclude that most other cultures have not “evolved as far in math” as Western culture.
• However, cultures are not obligated to continually develop more “sophisticated” math, especially math that would not be useful to the culture.
• Example: “There is no way to know when the world adopted this [decimal] system . . .”
• The assumption is that all cultures eventually use the decimal system for counting.
Western views of math
• 4) An emphasis on cultures that have developed writing.
• Cultures that have not developed writing are usually considered to not have developed mathematics.
• M. Ascher: More than 90% of all known cultures never developed a system of writing!
• Many cultures develop complex systems of symmetry (art), measurement (e.g. navigation), etc., without developing writing.
• Example: “ . . . The use of marks and notches as early ways of recording numbers, . . . written number systems gradually evolved from these primitive efforts.”
• Notice the use of “primitive” generally considered derogatory by anthropologists. Notice the assumption that all cultures develop as has occured in Western culture.
Observations• We need to be careful not to assume the math of
another culture must be similar to Western math, or that it would eventually become like Western math.
• People from other cultures may interpret, understand, and perceive mathematics in ways very different than in Western culture.
• It is best to view math from a very broad perspective- like “Bishop’s Six”.
• Let us look at textile art as an example.
Traditional textile Art
• A closer look: What is going on?
• The artists learn their craft over a long period of time, starting as children (girls).
• Their products often have important cultural meaning such as location, religious or social purpose, etc.
• Their products often have economic value.
Traditional Textile Art:
Weaving and Embroidery
•The artist makes the product entirely from memory. No notes (i.e., writing) or diagrams are used!
•The artist must keep track of many thread counts.
•Measurements must be precise, but no rulers are used.
Weaving: Inca culture
• The next slide has a diagram of thread counts.
• Notice the thread count values!
• There are Inca Women who specialize as weavers (“mamas”).
• “Mama” refers to an expert weaver, not “mother”.
An Incan Shawl
Let’s learn about Symmetry
• Vertical: Y or M, not K.
• Horizontal: E or D, not A.
• Example: Complete \ so that it has vertical symmetry.
• We can form: \/, or, /\.
You try: “Triangle on a stick”
• Complete the figure so that it has vertical and horizontal symmetry.
How did you make the
figures?• Knowing how to change the
orientations indicates an understanding of symmetry patterns.
• The artist (weaving, embroidery), understands and uses these mathematical concepts!
• Descriptions are not as we might expect in Western math.
• Examples next . . .
Mazahua Art Example
Western Perceptions of the Mathematics of
Women• Western views of mathematics go beyond the four views expressed earlier.
• Let us look at a few examples of cultural mathematics done by women. Cultures: Aztec, Maya, Inca, Otomi, Mazahua.
• We have already seen textile art, as mathematics done (mainly) by women.
• We will look at past records: Archaeology, codices. First, the Aztecs.
Aztec Women and Math
• Brumfiel, Hendon, others: Aztec women were identified (gender roles) by weaving, cooking.
• Women of all social levels wove.
• Most market activities done by, and managed by, women. That is, women controlled the financial aspects of the family.
• Aztec women knew mathematics, and often rose in social status based on their skills at weaving and market practices.
• Their skills were crucial to the success of the Aztecs.
A Mendoza Codex Image
Mayan Culture
• The Importance of girls learning to be weavers.
• Weaving is described in codices, artifacts, descriptions of deities.
• Mayan women still consider weaving a divine gift.
Mayan women & weaving
• Mayan Moon Goddess:
• Her hair has a headdress of cotton cords, representing her connection to weaving.
• Aged Goddess I (not shown here): a Mayan goddess of weaving.
Female Mathematicians: Mayans
• The Mayans developed very advanced mathematics.
• See the following slide of a female Mayan mathematician.
Female Mayan
Mathematician, from a
ceramic artifact
Incas. See Guaman Poma drawing below.
Mamas are treated with respect in Inca
culture.
Pre- Inca Weavers (Wari)
• Major archaeological find in 2008, Lima, Peru:
• A mummified woman wrapped in textiles, with weaving tools. She was a “chief of weavers”. (Wari culture- Pre Inca).
Observations
• In these examples we see that women developed mathematical skills through textile art, managing finances, and being mathematicians.
• Their skills were recognized as important activities (Brumfiel, etc.).
• Now, let us compare this with how mathematics by women has been viewed in Western culture.
How is the mathematics by
women viewed in Western
culture?• How did the view of women and
mathematics change with the arrival of the Spanish?
• Western views of female mathematicians in the 1500s:
• Women were prohibited from studying math (until almost 1900!).
• Textile art, gardening, cooking, languages, etc.: “Not mathematics”.
Before: Another Mendoza Codex
Image
Tributes collected by Aztecs from locations including Mixquiahuala and Ixmiquilpan (Otomi culture: related to Mazahua).
Notice weaving and artwork- done mainly by Otomi women.
The Otomies were known and respected as textile artists (still today). The Aztecs considered these textiles valuable.
After: A Post-Conquest Codex
from Mixquiahuala: What do you
see?
Post Conquest Mesoamerica and
Andes• In most parts of the Americas, Europeans
assumed that indigenous mathematics either did not exist, or was created in the same way as in Western culture.
• Indigenous women were treated as servants.
• Activities such as weaving no longer considered valuable.
We’re smarter than that now, right?
• Observe that women doing math is still discouraged.
• After strong efforts to encourage women to become mathematicians, still only about 30% of PhD’s in math (USA) are earned by women.
• Activities by women still considered “non-mathematical”.
• Example: Crocheting Adventures with Hyperbolic Planes, by Daina Taimina, voted “Oddest book title of the year” (2010).
• The title is referred to as “completely bonkers”.
THANK YOU!
• Di Hamadí (Otomi)
• Mite icnehlmati (Nahuatl- Aztec)
• Merci beaucoup
• Vielen dank
• Muchas Gracias
• Muito obrigado
Thanks again!
Thank you for your time and attention.
Questions, comments, etc.:
[email protected], or,
Have a safe trip back home after the conference!
Fun Books to read
• Ascher, Marcia. Mathematics Elsewhere. Princeton Univ. Press, 2002.
• Ascher, Marcia. Ethnomathematics: A Multicultural View of Mathematical Ideas. Brooks/Cole, 1991.
• Bishop, Alan. Mathematical Enculturation, Kluwer Academic Press, 1988.
• Closs, Michael P., ed. Native American Mathematics. Univ. of Texas Press, 1986.
More fun books (& magazine) to read
• Ventura, Carol. Maya Hair Sashes Back- Strap Woven in Jacaltenango, Guatemala, Second Ed. CV, 2003, write to cventura @ tntech.edu for more information about purchasing.
• Powell, Arthur B. and Marilyn Frankenstein. Ethnomathematics: Challenging eurocentrism in mathematics education. State Univ. of New York Press, 1997.
• Archaeology , January/February 2009.
Codices Comments
• Codices- a type of book.
• The image from the Codex “Matricula” (another name for the Mendoza Codex) shows tributes collected by Aztecs from various groups, including Mazahua and Otomí.
• Note the importance of weaving.
Guaman Poma’s “letter”
• Written in the early 1600s.
• Guaman Poma was half Inca, half Spanish.
• 1170+ pages, with 400 illustrations.
• A really long “letter”!!
• Guaman Poma’s work is considered a vital resource on the history of Inca culture.
Mayan women & weaving
Mathematics and women:
Aztecs, Otomies, Incas,
Mazahuas• For this talk, we will look to past
records.
The Mendoza CodexThe Mendoza Codex• The square figures with patterns
represent woven or embroidered products.
• The “feather” or “pine branch” represents the number 400.
• 400 of each type of cloth was collected as tribute.
• Notice patterns on the images showing geometric complexity.
• Almost all products made by women.
• The Aztecs considered them valuable.
How are female textile
artists treated in their own
cultures?• How can we find out?
• For this talk (Incas, Aztecs, Mazahuas, Otomies, Mayas), we’ll look at past history.
• Codices, archaeology, etc.
• The next slide: The Mendoza Codex of tributes collected by the Aztecs (Central Mexico).