ethnomathematics

culture & history of measurement

[damitr-deactivated20180215]

The universal language of mathematics, expressed in different forms. Though I would have liked to include illustrations from other cultures: Mayan, Indian, Egyptian and others.

A Globe Girdling Theorem

The Pythagorean Theorem, first expounded more than 2,000 years ago, was familiar all over the civilized world by 17th century. At top left is a Greek text of Euclid’s proof, and with it its five translations. Although the Chinese text is only 350 years old, the Chinese were actually familiar with the theorem at about the time of Pythagoras.

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the Lebombo Bone is the oldest tally stick (and oldest mathematically related artifact!) dating to around 35,000 BC.  Originally a fibula of a Baboon, it has 29 notches carved into it, possibly to keep track of the lunar cycle or a woman's menstrual cycle, probably one of the earliest systems of calendar keeping that remains today.  It was found in Africa in present day Swaziland.

The Ishango Bone dates from before 20,000 years ago, found near Lake Edward, is a bit more cryptic in its message.  It has a piece of quartz on one end probably used for writing, and it bears a series of mostly-odd numbers etched into it, except its center column, which appears to group integers with their doubles, 4 and 8, 5 and 10.  Perhaps doubles and halves were early explorations into multiplication and division, and odd numbers being the curious ones which don't half evenly.

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African Windscreens are made from two directions of overlapping straw woven together. Long rows take less straw and less time, but are less protective against the wind. Since wind speed increases the higher you are off the ground, the lengths of weaves decrease logarithmically as the height increases.  This is an excellent example of logarithmic engineering as distinct from any integer system of calculation.

from a windscreen maker near Bamako, Mali.  Courtesy Ron Eglash.

He includes a tutorial to learn windscreen weaving patterns.

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Astrolabe

Astrolabes were developed in Greece around 150 B.C., and further developed in 8th century Islam. Spherical astrolabes [pictured above], also from the Islamic area, date to the 10th century.

Astrolabes can: - determine the angle from the horizon of a celestial body - determine the time of day using your latitude, or determine your latitude (ships at sea) using the time of day - anticipate the movement of the moon and planets - determine one’s horoscope

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Why don't the English words for "eleven" and "twelve" take the form #-teen like the rest of their neighbors?

When Glen Lean was studying counting systems in Indonesia, he formalized a system to categorize the language of numbers, and was able to depict migration patterns during the settling of the islands.

Cyclic Patterns: this language (above chart) has unique words for 1-4, as well as 20. So it is said to have a cycle of 4, and a super-ordinate cycle of 20

Operative Patterns: symbols like + and ×, which are spoken when speaking the name of the number.  An analog in English would be to say the number 9 is to say "4 times 2 plus 1"

Number Morphs are the numerals which have distinct words (1, 2, 3, 4, 20).  This sequence of number morphs is called the Frame Pattern.

developed by Z. Salzmann and Glen Lean

afterthought- often in Indian-European base-10 systems' number morphs are separated from each other exponentially, all number morphs after 10 occur at 10ⁿ.  It's sufficient to just describe it by its Number Base.

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Oksapmin's 27-body part counting system (NW Papua New Guinea)

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The Odometer

a page from Leonardo DaVinci's Codex Atanticus

"So that we were not tired and lose time measuring it with the chain or the rope, but found on a moving vehicle we could determine with precision the already mentioned intervals, with the turning of the wheels." Heron of Alexandria, Dioptras, paragraph 34.

The origins of the invention are unclear, possibly credited to Archimedes (287 BC – 212 BC), but the earliest account of the odometer comes from Vitruvius (~ 75 BC - 15 AD) a Roman architect and engineer. He describes a simple machine, a pin on the axle of the wheel progresses a large gear by one notch every revolution, which in turn releases one pebble to fall into a container. Distances can be measured by counting pebbles.

In Greece, Heron's (~ 10 AD - 70 AD) account of an odometer incorporates another layer of precision, progress between pebbles is depicted by a needle turning against a protractor, marked according to the number of cogs operating on the needle's axle.

In China, accounts of the odometer date back to the first century, and was given a name in the third century, 'li-recording drum carriage" ('li' is a Chinese unit of distance, ~415 meters at this time). An account from the Song Shi describes one such carriage from the Song period, possessing multiple gears and precisely fabricated wheels so that after one li, a wooden figure bangs a drum, and after 10 li, another figure strikes a bell.

In the west, the machines were lost until Leonardo DaVinci reconstructed Vitruvius' account in diagram, but never himself made it. A modern implementation of DaVinci's diagram was constructed by Andre Wegener Sleeswyk in 1979, diagrammed below:

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Etymology of Algorithm

Muhammad ibn Musa al-Khwarizmi (ca.780 - ca.850 AD) contributed heavily to Arab mathematics and astronomy, substantiated algebra as a method of calculation, and introduced Indian numerals to the west. The second of his two landmark works on mathematics was lost in its original Arabic form, but Latin translations (~12th century) still survive. The image depicts how the document begins with Dixit algorizmi, Latin for "so says Al-Khwarizmi". Algorism and algorithm began to refer to the technique of performing arithmetic using the Hindi-Arabic numeral system.

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Etymology of Algebra

By the 5th century AD, algebraic techniques from Indian astronomy that were used to handle calculations with variables garnered a name, Bijaganitam- “the other math” (bija- another or second, ganitam- mathematics). Bija also has etymological ties to “seed”, the implication being algebra was recognized as the original method of computation.

The 13th century saw the Islamic Renaissance and India was a frequent target of invasion. The Arabs assimilated this Indian form of calculation, and named it Al-Jabr- “the reunion of broken parts” (Al- the, Jabr- reunion), from which we get "algebra".

Image credit: Agathe Keller, Making diagrams speak, in Bhaskara I’s commentary on the Aryabhatıya

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The success of Indian mathematics [in the Gupta period] was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension.

A.L. Basham, The Wonder That was India

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We need some categorization of this [much needed] research in order to draw from several projects going on all over the world, under different names, but which satisfy our conceptualization of Ethnomathematics I. Research in culturally diversified environments. II. Curriculum development projects and classroom applications. III. Out-of-school applications. IV. Conceptual and theoretical foundations.

Ubiratan D'Ambrosio, A Research Program in the History of Ideas and in Cognition

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History of zero as a placeholder

like in the number 103

Babylonian tablets show Babylonian math from as far back as 1700 B.C.E., but it wasn't until about 400 B.C.E that a placeholder showed up, looking like two marks ".  103 would have appeared like 1"3 (granted, their number system appeared entirely different from ours).  In Babylon before 400 B.C.E., 103 was expressed as 13, and its difference from the actual 13 was intended to be derived through context.  (we still do this: "three-fifty bus fare" and "three-fifty airplane fare")

Meanwhile, the subject was of no concern to the Greeks because their math was geometry based and consisted of pictures.

In India, a similar system of a dot as a placeholder slowly transformed into a small circle up top º eventually becoming 0.  Western culture assimilated their system and here we are.

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The Algorithmic Collection Project coordinated by Professor Daniel C. Orey at California State University, collects and studies basic written arithmetic operations performed by students from around the world.  The examples shown are from two recently immigrated students at Encina High School in Sacramento, California.

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Mariana Ferreira: You have one pen, and you obtain a second. How many pens do you now have?

Indigenous woman: I have one

Ferreira: How did you obtain this answer?

Indigenous woman: I have one pen, and I do not need the other

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Traditionally in mathematics classrooms, the relevance of culture has been strangely absent from the content and instruction. [...] In many classrooms, students are not permitted to construct a personal understanding of the mathematics that is presented. The values, traditions, beliefs, language, and habits reflective of the culture of the students are ignored. In such situations, the ways that children might invent personally meaningful conceptualizations are not respected. Children are expected to assimilate prescribed procedures by rote without necessarily gaining a deeper and conceptually significant understanding of the mathematics that they are studying. This style of instruction unfortunately restricts learning to the length of time that students accurately remember the procedures. An application of the learning is also often context specific and poorly generalized because it is limited to the types of problems practiced when the procedures were taught. Students should be encouraged to construct personal mathematical understandings and be able to explain their work.

Ubiratan D'Ambrosio. "What is ethnomathematics, and how can it help children in schools?", Teaching Children Mathematics, Volume 7 Issue 6

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We call Ethnomathematics the art or technique of understanding, explaining, learning about, coping with and managing the natural, social and political environment, relying on processes like counting, measuring, sorting, ordering and inferring which result from well-identified cultural groups.

Ubiratan D'Ambrosio, A Research Program in the History of Ideas and in Cognition