Building on the Rich Diversity of Ethnomathematics in Papua New Guinea

Kay Owens and Rex Matang

Charles Sturt University and University of Goroka

kowens@csu.edu.au matangr@uog.ac.

Melanesian cultures in Papua New Guinea. Each has its own language. Each has its own culture. Each has its own mathematics and use of mathematics. These languages were oral and first written down by linguists among the various visitors (e.g. colonial government officers, missionaries, and researchers). There may be similarities between the different groups especially if they have a related language or live in the same geographic region. Some of the diversity of mathematics can be found in the counting systems that are used and in their measurement methods (Owens, 2001a). Mathematics is used in many different activities:

• weaving, decorating,
• hunting,
• canoeing and sailing,
• exchange ceremonies,
• carving,
• design making and using (Figure 1),
• building houses and bridges (Figure 2),
• playing,

A B C

Figure 1: A. Part of a carved column decorating the University library representing the highlands region carved by a Sepik river carver.

B. A Sepik river mask of clay on tortoise shell with cassowary feathers.

C. Kwila columns carved with different provincial designs ready for installation.

Figure 1. Examples of carvings and decorative masks.

Figure 2. House designs illustrating the use of eye, sticks and diagonals to make rectangular houses and sticks for round houses.

Experiences like these influenced Alan Bishop (1988) to suggest mathematics be considered as a part of groups of activities like those above found in all societies. We might ask

• What is the mathematics in each culture?
• What are the differences and similarities between the groups?
• How is it related to school mathematics?

For example in carving and mask-making traditional methods can assist students in studying symmetries, similarities, negative spaces, repetition, and measurement.

Figure 1 about here

A piece of grass or a stick is used to mark off points that are the same distance away from the centre line when marks or carvings are made (Figure 1). This physical and concrete marking of the distance helps students understand that points are equidistant from the axis of symmetry.

The diversity of patterns can be seen in the carved logs seen in Figure 1. New patterns, particularly in woven mats and bilums (string bags,Figure 2) are created and passed onto other people. The bilum makers study the designs, imitate each other if possible, and use numbers to recreate a design. Their visual image of the design is studied in their minds. They work on the number of colours they will need and how they will work these together.

Insert Figure 2 about here

In the various decorated items, there is a great deal of repetition of lines and small shapes as illustrated by the carvings. These practical examples could be discussed in terms of their cultural and artistic use as well as their mathematical relationships.

Insert Figure 3 about here

House designs also contain mathematical ideas. There is considerable estimation of size and older men who are regarded as good builders will know just what affect on quantities of rope and logs and matting and grass will be needed if a house plan is enlarged. Men use their eye to get the sticks in a straight line as shown in Figure 3A. However, the builder knows that both diagonals marked by a stick or rope will be the same length if the house is “square” (Figure 3B). If not, the house will end up being less safe and at a “slope” (a parallelogram) (Figure 3C). If a round house is built a long stick or rope is used to mark off points that are equal distant from the centre point (Figure 3D) and then people developed an improved house design as shown in Figure 3E.The learnt that having the additional cross-rafters, they could cut off the central pole (personal communication Henry Atete Kapao, Enga, 1997).

New Educational Reforms Encouraging Early Education in the First Language

In different parts of the country, pre-elementary classes have been held in the vernacular for many years. About 20 years ago, before the Bougeainville separatist crisis, this island had Tok Ples (village language) preschools while many Church schools had minimal schooling in the vernacular 50 years or more ago. Recently, the government began a reform to encourage Pre-elementary, Grade 1 and Grade 2 in Tok Ples. The lingua franca (Tok Pisin) is used where children come from many places as we find in towns. The elementary school buildings have to be built or supported by the community. The teachers receive minimal training in an apprenticeship model. The children learn to read in their own language. It seems that in most places, they are also learning to count and to develop other school mathematics in the vernacular. The syllabus encourages many basic concepts like ordering, counting, classifying, and finding relationships to be adjusted to the community. In a similar way, stories for reading can be community stories which abound. Story tellers and actors are identified by most communities and telling stories about everyday events and myths is common practice.

However, some teachers are reluctant to count in mathematics as a system of classifiers or non-decimal cycles makes the teachers unsure of what and how to teach the mathematics. Other areas of mathematics like shape and measurement are included in the Cultural Mathematics syllabus for elementary schools.

Towards the end of elementary school and as a transition to English in early primary schools (Grades 3-8), the children will learn to count and describe mathematical situations in English which is the language of instruction for the following educational years.

Glen Lean Ethnomathematics Centre

Glendon Lean was an Australian by birth, initiated into a Tolai family. He taught mathematics at the PNG University of Technology while the first author (Kay Owens) also lectured there over a 15 year period interspersed with teaching at a community (now called primary) teachers college. For 22 years, Lean collected data from the world’s historic resources (e.g., first contact language data) and by questionnaire given to university students and teachers and by field work. He analysed this data on around 900 PNG and Oceania counting systems and subsequently wrote a doctoral thesis on the diversity of systems and about the spread of counting systems in the region. The Centre will make his thesis available on their website and summaries can be found in Owens (2000, 2001b).

Glen’s work was a mammoth task given the fluidity of the languages and the multitude of names and in some cases dialects for different languages. Alan Bishop, as academic executor, sent Glen’s materials back to PNG with his two students Wilfred Kaleva (1998) and Francis Kari. These men wrote theses on attitudes to ethnomathematics held by secondary teachers and students. Kaleva helped to establish the Centre at the University of Goroka. It was opened by Richard Saxe in 2000. (Two papers are referenced below: Saxe, 1985, in press.) The second author (Rex Matang) is the Director of the Centre (see Matang, 1997). The Centre now houses much of Glen’s photocopied materials along with other, usually more recent materials on ethnomathematics. Much of this material comes from linguists and anthropologists. The Centre is establishing a website and database of materials, raising awareness and valuing ethnomathematics, preserving it, and fostering it in schools. The database has mainly been the work of the first author during two visits to the Centre supported by the University of Western Sydney and the Hawaii Center for Pacific Resources for Education and Learning. Several papers, lesson ideas (several collated by Richard Zepp during a visit), and parts of the material that Glen had collected will be available on the web.

A glimpse at the diversity of the counting systems can be seen in the next section describing possible advantages of learning counting in the vernacular. There are systems having main cycles of 2, 3, 4, 5, 6, 8, 10, and 20 while the body tally systems can have any number like 27 depending on the number of upper body parts identified for tallying.

Impact of Counting in their own Language on the Learning of Arithmetic Strategies

The Director of the Centre will be researching the impact of learning to count in the vernacular on students learning. Some of the expected advantages are given for the types of languages listed below.

• Digit tally systems are common. For example, in Kåte (Rex’s language and a common common Church language), counting equals 1, 2, 2+1, 2+2, 5=hand, 5+1, 5+2, 5+3, 5+4, 2 hands, 2 hands+1,…2 hands + 1 foot, …2 hands +2 feet or man. The system has a main cycle of 20. However, it also has a two and five cycle. Students incidentally learn that groupings of numbers can vary and that English has selected a consistent decimal grouping for its place value system. The counting is likely to reinforce the use of five in combinations to 10 which makes adding across the decade more readily accessed.
• Manus-type systems, for example Buin, have patterns like 7 is “3 before”, 8 is “2 before” and 10 is “complete”. The system encourages a strong sense that 7 and 3 is 10 and 10 is a unit-group.
• A few languages use combinations of numbers 1 and 2. For example, 7 is “2+2+2+1” in Aruamu. This is a basis for even and odd numbers. Some of these languages are fluid and will use variations like 3x2+1.
• Languages using a 5 cycle reinforce products of 5 (e.g. Goroka) while others like Wiru and Hagen have four cycles with products of four as well as other cycles or systems.
• Body part tally systems provide a number line up the arm. For example, Fasu using 27 parts up the arm and across the head and down the other arm. For a sum like 17 add 2, students visualise two parts along from 17 and give its name.

Figure 3 Counting in Hagen and English.

An example of two visual aids developed from conversations between Hedwig Aspro (Mt. Hagen, 2003) and the first author will illustrate how the Hagen language (which has two dialects with variations in both) may use its 8 cycle system effectively. There are many new counting words as compensation claims are now up to ten thousand kina (PNG currency like a Euro). As they count, the people bend down their small finger on the left hand followed by the other fingers excluding the thumb and then they start on the right hand. Traditionally the people built some numbers on the embedded 4 cycle e.g. 12 is 3x4 while the main cycle is an 8, marked by two fists. People count many items, for example pigs in an exchange for a bride or compensation, and pairs are common. They might also count in tens with the fists followed by the two thumbs and so 8+2=10 is supported by the counting words and hand actions.

Building on the Diversity in Upper Primary School – Grades 7 and 8

Knowing different counting system patterns besides their own language and base 10 gives a richer understanding of the place value system and number systems. To illustrate just how widespread and diverse these area, Figure 4 shows where 2-cycle systems can be found. Most of these systems have 5 and 20 cycles as well.

Figure 4. Two cycle systems with supercycles noted.

Attitudes of secondary teachers and students is positive towards ethnomathematics (Kaleva and Kari). This encourages ownership of mathematics and hence increased pedagogy. A parallel can be drawn with architecture students who feel that their cultural heritage has encouraged them to be good architects because of the encouragement of traditional culture in design (Owens, 1999). There is a revised high school curriculum to encourage the recognition of ethnomathematics due to the University of Goroka researchers being on the curriculum committees.

Primary and secondary teacher education courses include work on investigating number and pattern using a study of their own counting system. However, more work is needed on this aspect so that teachers have general principles for planning for maintenance of the traditional culture and an enriched bridge from the Tok Ples where they happen to be teaching and mathematics in English.

References

Bishop, A. (1988). Mathematical enculturation: A cultural perspective on mathematics education. Dordrecht, Holland: Kluwer.

Kaleva, W. (1998). The cultural dimensions of mathematics curriculum in Papua New Guinea. Teachers beliefs and practices. Doctoral thesis, Monash University.

Matang, R. (1997). The role of ethnomathematics and reflective learning in mathematics education in Papua New Guinea. Papua New Guinea Journal of Teacher Education, 4 (1), 7-10.

Owens, K. (1999). The role of culture and mathematics in a creative design activity in Papua New Guinea. In E. Ogena & E. Golla (Eds.), 8^th South-East Asia Conference On Mathematics Education: Technical papers, (pp. 289-302). Manila: Southeast Asian Mathematical Society.

Owens, K. (2000) Traditional counting systems and their relevance for elementary schools in Papua New Guinea, Papua New Guinea Journal of Education 36 (1 & 2), 62-72.

Owens, K. (2001). Indigenous mathematics: A rich diversity. In Mathematics: Shaping Australia. Proceedings of Australian Association of Mathematics Teachers, pp. 157-167. Available on CD. Adelaide: AAMT and through www.science.edu.au/nova/.

Owens, K. (2001). The work of Glendon Lean on the counting systems of Papua New Guinea and Oceania. Mathematics Education Research Journal 13(1), 47-71.

Saxe, G. (1985). Effects of schooling on arithmetical understandings: Studies with Oksapmin children, Papua New Guinea Journal of Educational Psychology, 77(5), 503-513.

Saxe, G. (in press). Making change in Oksapmin trade stores: A study of shifting practices of quantification under conditions of rapid shift towards a a cash economy.