The term ‘ethnomathematics’ was first used in the late 1960s by a Brazilian mathematician, Ubiratan D’Ambrosio, to describe the mathematical practices of identifiable cultural groups. Some see it as the study of mathematics in different cultures, others as a way of making mathematics more relevant to different cultural or ethnic groups, yet others as a way of understanding the differences between cultures. But perhaps the most powerful claim for the new discipline has been made by D’Ambrosio himself (quoted in The Chronicle of Higher Education, 6 October 2000): This makes ethnomathematics a rather unusual discipline, because it attempts to meld science and social justice. This isn't something that sits comfortably with many scientists: science, they argue, is science, and trying to make it politically correct will only impede its progress. Some educators fret that teaching mathematics using an ethnomathematical approach reduces it to a social-studies subject that teaches students little about ‘real’ mathematics. Others simply ridicule the whole notion: according to one disparaging journalist, 'Unless you wish to balance your checkbook the ancient Navajo way, it’s probably safe to ignore the whole thing'. But there are also many scientists, educators and commentators who see ethnomathematics – in all its definitions – as a legitimate discipline with plenty to offer the modern world. Examples of non-Western mathematics Many non-Western cultures have developed complex mathematical systems. One often-cited example is that of the ancient Mayans. This civilisation, which emerged more than 3000 years ago, recognised many patterns in their observations of the universe, and developed mathematical relationships and symbolic systems to describe these patterns. The Mayans were keen astronomers who developed a complex system of calendars to keep track of the solar and lunar cycle and other planetary events. The counting system used to support the calendars was based on cycles of 5 (quinary) and 20 (vigesimal). (The number system predominantly used today is decimal, based on composite units of ten.) Some historians suggest that the vigesimal system derived from the total number of digits on a person’s hands and feet (just as the decimal system probably arose from finger-counting). The Mayan counting system is elegant in its simplicity. It uses only three symbols: dots, which represent single units; lines, which represent units of five; and a shell-like symbol, representing a place that could hold a number. The three symbols could be combined to represent any number. When the Spanish conquered central America in the 1500s they destroyed many artefacts of the Mayan civilisation, including religious icons and texts. One of the few surviving texts is called the Dresden Codex – after the city in which it now resides. This document reveals the sophistication of the Mayan's knowledge of mathematics and astronomy (Box 1: The Dresden Codex). Counting systems in Papua New Guinea Counting systems based on composite units of 5 and 20 are also common in Papua New Guinea. The 800 different language groups have their own counting systems with a variety of basic number words. Commonly used number words are hand as 5, and person (10 fingers and 10 toes) as 20. A few groups have a hand as 4 (without the thumb) or as 6 (with the thumb as two knuckles). The counting systems in Papua New Guinea are best described in terms of the cycles (rather than the base) that they use. For example, if pairs were important to a language group, then the counting system might feature a 2-cycle, with six objects being thought of as three groups of two. Many systems would probably have a second cycle combining number words. The second cycles are commonly cycles of five so that, for example, the number 14 might be two fives and two twos. Several language groups use parts of their bodies – up one arm, across the head and down the other arm – to mark off different numbers. These counting systems are known as body tally systems. However, there are not always clear links to physical or cultural aspects in the Papua New Guinean counting systems. Many groups use words that are not related to parts of the body, and combinations of counting words may not be linked to specific cultural activities. Australian Aborigines Mathematics is used by different cultures for activities in addition to counting. For example, the complex interpretation of natural data using (among other things) some fundamental mathematics has ensured the survival of the Aboriginal people in the often-hostile Australian environment for at least 50,000 years.
The navigators The Polynesians, Micronesians and Melanesians, who populated thousands of islands scattered across the western, central and southern Pacific Ocean, needed a different kind of mathematics. Without a reliable navigation system, sailing and paddling between islands perhaps thousands of kilometres apart would be highly dangerous. The Marshall Islanders, for example, used a combination of techniques when they ventured onto the high seas, complementing celestial navigation – using the moon, the sun and the rising and setting of specific stars – with a detailed understanding of wave and current patterns.
Learning to read wave patterns in the ocean swell was not an easy skill to master. The Marshallese knew that the swell, which was created by the reliable trade winds and could run in a straight line for great distances, was reflected, refracted and diffracted when it met islands, setting up patterns in the waves that could be ‘read’ to help determine position. Over the centuries, the Marshallese acquired a great deal of knowledge about wave patterns and how they were affected by islands and island groups and by changes in wind direction, but without a written language it was difficult to pass such knowledge to young navigators. They solved this problem with stick charts, lashing sticks together to form geometric shapes illustrating the patterns of currents and waves that would be encountered on a given voyage. The stick charts were used as a visual aid for budding navigators, who would spend years learning these and other patterns, forming the mental maps of the ocean that they would need as they made their way from one tiny atoll to another. What is the point? Some people maintain that ancient mathematics systems are irrelevant today. This is unfortunate. Many non-Western mathematics systems remain ‘alive’; some Mayans, for example, still use traditional calendars for religious purposes and to help determine the agricultural cycle. Moreover, Western mathematics does not meet the needs of all people and is not always easily understood outside the ‘mainstream’ culture. For years, Australian educators have noted that Western mathematics often has little meaning in remote Aboriginal communities and is therefore difficult to communicate. Approaches that take into account the cultural context and the mathematical systems in use within the community are likely to be much more effective. Box
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