Ethnomathematics – a rich cultural diversity

Key text

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.

Related site: The lost seasons Looks at the validity of indigenous weather knowledge. (Australian Broadcasting Corporation)

Some examples of the traditional mathematics of Australian Aborigines include counting in non-decimal systems, recognition of the patterns in relationships between clans, and calendars based on natural changes in the environment. For example, a 'season' may be defined by the sets of natural phenomena that occur at a given time, such as the flowering of particular plants, the activities of bees and the direction of the wind. Other mathematics of the Australian Aborigines is also based on the relationships between things. A spear may be 'too long' for a particular person, too short or just right: the length of the spear is thus measured relative to the user.

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.

Related site: Traditional navigation in the western Pacific – a search for pattern Using animated diagrams, explains how Micronesian navigators were able to sail between islands. (University of Pennsylvania Museum of Archaeology and Anthropology, USA)

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 1. The Dresden Codex

The Dresden Codex ('codex' simply means 'ancient manuscript') is thought to have been produced about 800 years ago, probably based on a document written 500 years before that. It was written on a sheet of what is called Amatl paper, which is made of flattened bark covered with lime-paste.

The codex is about 3.5 metres long (folded to make double-sided pages) and painted in colour with hieroglyphics, illustrations and mathematical symbols.

Mayan astronomy

The deciphering of the Mayan number and hieroglyphics systems found in the Dresden Codex and other similar documents (as well as those etched in stone in the temples and palaces found in the ruins of ancient Mayan cities) has taken scholars well over a hundred years; in the case of the hieroglyphics system, the deciphering is still not complete.

The information contained in the codex demonstrates the mathematical sophistication of the Mayans. It includes, for example, a table predicting the dates of eclipses and the astronomical fortunes – such as the dates of conjunctions with Mars and Saturn – of the planet Venus.

These predictions were surprisingly accurate for more than a century into the future. But because the Mayans had such crude instruments for measuring time, the errors become noticeable thereafter. Even so, the level of accuracy they achieved required a good basis in mathematics, because only by understanding the patterns of planetary, lunar and solar events could such robust predictions of future events be made. The ancient Mayans also developed a 'calculator' (something like the modern-day times tables) to assist in tasks of addition, subtraction, multiplication and division.

Related sites

• The Dresden Codex (Dresden University of Technology, Germany)

• Calendars – keeping track of time (Nova: Science in the news, Australian Academy of Science)

Activities

• Department of Education, Training and Employment, South Australia
  • The development of counting and numeration systems – provides structured lessons for students to find out about different ways of counting and representing numbers.
  • Establish shared knowledge of the context – describes a unit of work in which students develop a local tourist guide.

• Peace Corps (USA)
  • Marshall Islands activities grade 6-9: Stick chart – students learn about stick charts and make one of their own.

• The Math Forum (Drexel University, USA)
  • Mayan arithmetic – shows students how to add and subtract using Mayan symbols.

• Michiel Berger (Netherlands)
  • Maya mathematics – an explanation of the Mayan system of mathematical symbols with an interactive number converter (requires Java).
  • The Maya calendar – introduces students to the Mayan calendar and provides an interactive calendar converter (requires Java).

• Mark Millmore's Ancient Egypt (UK)
  • Egyptian math: Numbers – describes the basic symbols in the Egyptian decimal system and provides problems for students to solve.

• Science and Mathematics Initiative for Learning Enhancement (Illinois Institute of Technology, USA)
  • How to use an abacus
  • Boomerang – students make an X-shaped boomerang using two rulers and show that the distance it travels is determined by the angle at which it is thrown.
  • Geometric designs in the game of 'Life' – students predict and generate patterns as they play a board game with counters.

• The abacus: The art of calculating with beads (Ryerson University, Canada)
  • A brief introduction to the abacus – students learn the basics of using an abacus and how it represents numbers. They also compare the Chinese, Japanese and Aztec abacuses.
  • Performing addition on the abacus and Performing subtraction on the abacus – explains how to do these calculations using an abacus.
  • Build a LEGO abacus – instructions and plans for building an abacus.

• The Math Forum (Drexel University, USA)
  • Symmetry and pattern: The art of oriental carpets – introduces the idea of studying symmetry by analysing patterns in carpets. Click on 'Educational resources' for suggestions about student activities (eg, pattern-making, observing symmetry and making tessellations).

• MEGA Mathematics (Los Alamos National Laboratory, USA)
  • Activities with knots – provides a series of activities for students to explore the mathematical connections associated with knots.

• Simon Fraser University (Canada)
  • Quipu: A modern mystery – describes the quipu (knot) used by Incas for accounting and census taking. Click on 'To learn how to make your own quipu' for instructions.

• Pacific Northwest National Laboratory (USA)
  • River crossing – students solve a problem of how best to cross a river by breaking the problem down into a series of mathematical steps.

Some games to develop reasoning skills:

• Lawrence Hall of Science (University of California at Berkeley, USA)
  • Shongo networks – students draw a simple pattern without lifting their pencil.
  • Tower of Hanoi – students transfer different size disks from one of three poles to another. Note: Students can play online or build their own game.

• British Go Association (UK)
  • Introduction to the game of Go – gives the background and history of Go, an ancient board game for two players in which stones are moved on a grid with the aim of claiming territory. Click on 'How to play Go' for the rules and an example of a game.

• John Miller Crawford (Auckland University of Technology, New Zealand)
  • A Maori game from Aotearoa New Zealand – students play an interactive game against the computer.

• Scientific Computing and Visualization Group (Boston University, USA)
  • Peg game – online jumping game where students use their mathematical reasoning skills to outwit the computer. (Instructions appear after your first move.)

Further reading

Useful sites

Base valued numbers (Poseidon Software and Invention, USA)

Presents some historical aspects of numbers (eg, 'Trading and numbers') and gives examples of different number base systems (eg, 'Base 2 (binary)' and 'Base 5 (hand)').
http://www.psinvention.com/zoetic/basenumb.htm

Indigenous mathematics – a rich diversity (Glen Lean Ethnomathematics Centre, Australia)

An overview of the different counting systems and other mathematical concepts that are used in Papua New Guinea and Oceania.
http://www.uog.ac.pg/glec/Key/Kay/diversity.htm

Maths 'needs to listen' to other cultures (News Online, Australian Broadcasting Corporation)

Explores the connections between mathematics and different cultures.
http://www.abc.net.au/news/newsitems/200602/s1572327.htm

Ethnomathematics in Australia (The Australian Institute of Aboriginal and Torres Straight Islander Studies)

Provides a series of articles on Aboriginal mathematics.
http://www1.aiatsis.gov.au/exhibitions/ethnomathmatics/ethno_contents.htm

Links to Pacific Islander ethnomathematics (International Study Group for Ethnomathematics, Rensselaer Polytechnic Institute, USA)

Included in the available links are 'Oceania – mathematics and the liberal arts' and 'Glen Lean Ethnomathematics Centre in Papua New Guinea'.
http://rpi.edu/~eglash/isgem.dir/links.dir/islands.htm

Ethnomathematics Digital Library (EDL) (Pacific Resources for Education and Learning, USA)

A new (and still growing) online database of ethnomathematics materials, with links to relevant websites.
http://www.ethnomath.org

University of St Andrews (Scotland)

• Mayan mathematics
  An overview of the mathematical achievements of the Mayan civilisation.
  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Mayan_mathematics.html

• Mathematics of the Incas
  Describes how the ancient Incas used knots in strings (quipu) as a method of recording numerical information.
  http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Inca_mathematics.html

Glossary

hieroglyphics. A form of writing in which pictures and symbols are used to represent objects, concepts or sounds.

reflected, refracted, diffracted. All three terms refer to a change in the direction of a wave. A wave has long crests and valleys called wavefronts. The distance between successive crests is called the wavelength. The wave always travels in a direction that is at right angles (90°) to the wavefronts. A wave is reflected when it bounces off a smooth obstacle that is long compared to the wavelength. A wave is refracted when it travels from one medium to another in which its speed is different (eg, from deep water to shallow water); and it is diffracted when it passes through a small opening in, or around the edge of, a large obstacle. A wave is scattered when it bounces off a small or rough obstacle. For more information see Behaviour of waves (The Physics Classroom, USA) and Wave behavior (Alaska Tsunami, USA).

stick chart. A three-dimensional map of ocean patterns that was used by Marshall Islanders to teach and preserve their navigational knowledge. Each map was created by tying together midribs of coconut-palm leaves or pieces of split bamboo in patterns that represented wave and current patterns. Shells were used to indicate the position of islands in relation to the ocean currents. For more information see Traditional Marshallese Stickchart Navigation (Dirk Spennemann, Charles Sturt University, Australia).