Mathematics: Shaping
157
Indigenous Mathematics — A Rich Diversity
Kay Owens
Introduction
I once taught, and learnt, extensively in Health Education. One thing I learnt about Indigenous nutrition issues is that food taboos are very common. In order to cope with their impact on nutrition, it was common practice to encourage people to consider whether the food taboos fitted into the category of (a) good for nutrition, (b) bad for nutrition, or (c) it did not matter whether the food taboos were followed or not as far as good nutrition was concerned.
For example, I had a close teaching friend. She was pregnant and she looked awful. She was tired and very anaemic. We discussed her food taboos as I knew she came from an area in which eating fish and dark green leafy vegetables were taboo during pregnancy (although these provided the most common protein and iron and B12 needed to avoid anaemia). I suggested that she could try eating other protein foods like peanuts. Unfortunately, over the years, it seemed that the taboo on fish protein had spread, with a little nutrition knowledge, to all protein. We discussed why this taboo had developed. It was probably related to the problem of large babies and difficult village births for mothers who were well-nourished as adults but not so well-nourished and hence small bodied as children. This in fact was not a problem for this teacher and she had access to a hospital. The taboo on fish also related to totems.
I think we can view Indigenous mathematics in much the same way. Does the
Indigenous mathematics reinforce the school mathematics, does it lead to conflicts (which may need resolving) with school mathematics, does it not matter. My personal view is that in the cases with which I am familiar, whether enhancing or different, it is most likely to be beneficial to mathematical learning. It does, in fact, mean both teacher and students' mathematics is developing.
I wish to present some aspects of Indigenous mathematics that is different to school mathematics but which can be used to enhance all teachers, and students of both the Indigenous culture and others.
Like food taboos, mathematics is not free of other, very significant aspects of culture. In recognition of this, I wish to say that I hope I do not offend or misrepresent the Indigenous mathematics.
Mathematics: Shaping
158
I am drawing extensively on the work of the late Glendon Lean whose thesis The Counting Systems of
I was also drawn into considering the comments of Robin Williams at the 1999 MANSW conference on the Yupno body-part tally system (original source, Wassman & Dasen, 1997). I found no confirmation of this data (Lean, 1991, 1993; Smith, 1988; Wurm & Hattori, 1981) but my check took me through a number of issues such as the likelihood of counting systems changing, especially dying out. The system did, however, prove to be unusually unique in that it was unlike, in fact unsupported by other body-part tally systems (Owens, 2000). It is therefore worth having Glen's records and his summary of common features although systems are unique. It is also worth knowing about other ways of determining quantity and the cyclic nature of counting systems.
The languages of the region are classified into the Austronesian languages, Oceanic languages, and the non-Austronesian or Papuan languages. The Austronesian languages are generally around the coast and represent a second-wave of languages. The non-Austronesian languages are considered to be older and have a huge variety of structures unlike the Austronesian languages.
Different cycles
A 5-cycle system will have the basic number words of 1, 2, 3, 4, 5 with other counting words being made up of combinations of these words. (5, 20) digit-tally systems use hands and feet with 15 being two hands and one foot and 20 being one man. Additional basic number words are now included, for example the word for 20 which is also man. Many counting systems have several cycles before settling down into one supercycle like the 20 cycle.
Table 1 illustrates the large diversity of counting systems that Glen had collated and analysed. He selected to group all systems with the lowest cycle. The table shows what is frequent.
Mathematics: Shaping
159
Table 1
Summary
of counting systems in
|
Type of System |
Alternatives |
Distribution |
|
10-cycle |
189 out of 217 AN; NAN influenced by AN neighbours |
|
|
‘pure’ type 10, 100, 1000; |
||
|
‘Manus’ type 7=10–3, 8=10–2, 9=10–1; |
||
|
‘Motu’ type 6=2x3, 8=2x4 (some with 7=2x3+1, 9=2x4+1); |
||
|
(10, 20); |
||
|
(10, 60) |
||
|
5-cycle |
(5, 10) or (5, 10, 100) |
Second most common AN |
|
(5, 10, 20) |
All systems in New Caledonia, some AN and NAN in PNG and Irian Jaya |
|
|
(5, 20) |
AN and NAN |
|
|
2-cycle |
201 NAN, 30 AN |
|
|
‘pure’ with 1 and 2 |
37 NAN, 2 AN; usually associated with body-part tally systems |
|
|
(2, 5) or (2, 5, 20) |
Most common, 18 AN; digit-tally system with 2 subordinate |
|
|
(2', 4, 8) |
||
|
2 more types |
||
|
4-cycle |
Highlands of PNG |
|
|
Cycle from first 4 |
||
|
Cycle from second 4 |
||
|
Superordinate cycles 28; 48; or 60 |
||
|
Special name for each cycle |
Enga dialects |
|
|
3-cycle |
Restricted areas |
|
|
6-cycle |
Restricted areas |
|
|
Body-part tally system |
Diversity |
Highlands of PNG, possibly once among Australian languages and other provinces |
Note. AN are Austronesian languages, NAN are non-Austronesian or Papuan languages. 2 ’
are modified 2-cycle systems. Not all languages are recorded.
Body-part tally systems
These systems can have a range of different cycles depending on which body parts
are included in the cycle the most common is 27 but they range from 18 to 74.
They occur now in PNG and Irian Jaya but seem to have occurred also in Torres
Strait and Australian language groups. Tallying usually begins on the small finger of
the left hand, to the wrist and then along the arm, shoulder, left ear and eye, nose or
Mathematics: Shaping
160
central part, and then down the other side of the body. If vocalised they tend to use the body part (see Figure 1).
Figure 1. Body-part tally system of the Fasu, Southern Highlands Province.
Classifiers in counting systems
In some areas, particularly on Bougainville but also in New Ireland, New Britain, and Milne Bay and to a lesser extent in other Provinces, a morpheme may be used to distinguish different classes or groups of objects. They occur in both Austronesian and non-Austronesian languages. This morpheme is associated with number words providing a different set of counting words for different classes. In some cases, the counting cycle size and words change for the different classes of objects but this is likely to be a ‘borrowed’ idea from a neighbouring language or trading partner.
Non-counting systems
In a few different language groups, the larger amount of objects is compared by the amount of space taken up rather than by counting objects precisely. This is not an area or volume idea per se but a recognition that approximation and spatial abundance can be sufficient for a transaction.
Mathematics: Shaping
161
Bases and cycles
There are few counting systems that have a regular base in which numerals are used for powers of the base. For example, only some island languages have a true base 10 system with numerals for 100 = 10×10, 1 000 = 10×10×10 etc.
For this reason, the recognition of cycles and patterns within the counting system was more beneficial than bases for Lean to describe, collate, and analyse the data.
The patterns of the counting system
Lean (1991) has recorded and documented the patterns of the counting systems in his collection. He uses the term operative pattern for regular patterns. Operative patterns may include how the numbers between 6 and 9 are formed, the regular use of decades, (i.e., 20=2x10, 30=3x10), digit tally and body-part tally.
The digit-tally systems with (2, 5, 20) cycles have the following operative pattern which combines the frame words 1 and 2, 3 = 2+1, 4=2+2, then there is a frame word for 5, then 6 to 9 are combinations of the word for 5 (or another morpheme) and the words for 1 to 4. In a system like this, the counting frame is 1, 2, 5, and 20.
Geographical distribution of the counting systems from both Papuan and Austronesian language groups
The linguistic systems in Polynesia and
Some uses of numbers
Case studies show that some non-Austronesian societies make extensive use of numbers in ceremonial contexts (e.g. Melpa). Others count a wide variety of objects using a 2-cycle variant system (e.g. Mountain Arapesh) while others adopted a 10- cycle system (e.g. Ekagi). Some societies place little importance on counting despite their 10-cycle systems and place an emphasis on the indivisible mass of a visual display. In these circumstances, some societies like the Loboda retained their 10-cycle system while others like the Adzera have numeral systems modified to a 2-cycle with digit-tallying like their non-Austronesian neighbours. The ability to amass large quantities of wealth items accords status and prestige to clans and individuals but the judgment of quantity, however, varies in each society from the visual impression to the precise counting. Lean notes that such diversity is also evident when other questions are considered, for example, (a) what are countable objects, (b) when are they counted, (c) which economies and exchanges use counting, (d) are different types of objects counted in different ways, (e) are some objects not counted, and (f) how are totals recorded. ‘Counting does not exist in isolation. it quantifies and qualifies relations between people, objects and other entities’ (Bowers & Lepi, 1975).
Mathematics: Shaping
162
It is often heard that some of these systems are too ‘primitive’ to count large numbers. It is not possible to generalise about the use of large numbers in these languages although the systems can be used to generate large numbers and continue forever. Some further points are made by Lean: (a) some of the words for large numbers may mean countless or indefinitely large; (b) the same word is used in different languages for 1 000 or for 10 000; and (c) non-Austronesian languages tend not to have single terms for large numbers, like a million, unless borrowed from Austronesian influence.
Fractions are generally not used except a half. The Chuave make use of half to refer to a hand, that is half of the hands of a man. This word is repeated for the two hands. Usually man is used in the (2, 5, 20) systems for 20.
Multiples of two are common. Melpa and other groups seem to like twos and tend to count in twos and to give in twos, especially making 8 or 10 items (Strathern, 1977). Completing a pair seems important.
Enga is an unusually large language group with over 170 000 resident speakers (1982 census data) with 11 dialects and apparently some recent influence of 10-cycles from the English or Tok-Pisin systems. Some young informants thought that the nondecimal counting system was only used for large ceremonies. Some of the words for one dialect given in Table 4 was recorded in 1938. The 60-cycle system consists mainly of a sequence of 4-cycles beginning at 9 and each cycle typically has the construction: cycle unit+1, cycle unit+2, cycle unit+3, cycle unit+end where the cycle units, 13 in all, are not numerals as such but may be words or phrases with some non-numerical meaning like dog.
The origins of counting systems
Glen's thesis emphasised that counting systems did not spread around the world from the Middle East. The Indigenous counting systems belong to cultures that are much older than those of the Middle East. While the cycle of 2 is strong in remote places, nevertheless there is evidence to suggest that the parts of the body could influence the development of number. A key example is that of (5, 20) cycles and the 10 cycles. It is also obvious in the systems that have a 4 or 6 cycle in which the hand is then considered as having 4 or 6 parts.
Other mathematical concepts
Other mathematical concepts are also embedded in cultural activity. For example, all groups measure. In particular, comparisons are made by using existing lengths like a length of string. Equal lengths are particularly important in building in which shapes like rectangles are used with equal length sides and diagonals. Floor spaces are frequently divided up into equal parts and again equal lengths are used. A length of rope or a long stick is used to mark the points of a circle when circular houses are built.
When carving, men mark symmetrical points by marking off with equal-length sticks or string.
Mathematics: Shaping
163
Figure 2. Symmetry, rope becomes diagonals of rectangle, bamboo volume units.
Water is regularly carried in containers, especially bamboo lengths. The volumes of thick and thin bamboos are associated with the amount of garden that can be watered. These proportional relationships are intuitively used. If the garden is three times the basic area that can be watered by one length, then the three length container will be chosen.
Spatial thinking is extensively employed in making decisions. For example, if a standard house is enlarged the increases in materials is known by the master builder. A good mental image seems to be held of the size of a house when a floor plan is enlarged.
Designs are regularly shared and modified. Shared patterns are usually illustrated by example but the number of strips under which a leave is tucked while weaving or the number of stitches made when weaving a string bag is noted. The overall design is dominant but how to get the design has known tactical procedures.
Interesting ideas are used by teachers naturally in primary school classrooms. For example, in joining papers together two techniques have been used, neither needing glue or paper fasters. One is to make a hole in each piece of paper and join with a narrow cylindrical paper. The other is to slit two pieces so they can slide together.
Many activities require organisation. The order in which things are carried out is significant and emphasised. The collaboration of people's effort is also recognised. This takes the form of deciding on time for an activity and then allocating work and amounts of materials to be used to different people.
Time is also well developed as a concept. While marking in hours is not, until recently, well delineated, nevertheless the amount of time needed to undertake an activity was well developed. For this reason, people could rise in the middle of the night to get ready to set sail for a distant airport or to walk to the road head to catch transport to market to sell their garden produce. Lengths of time were intuitively compared. Time to walk to different places was also well established by experience and could be, to some extent used for deciding on other walking lengths.
Balance is another key idea that is well
established. For example, many bridges are counter-balanced. Heavy rocks are
used to counter the weight of a bridge swinging out across a fast-flowing
stream. The number and thickness of posts needed for different types of houses
and different parts of the house (walls or ceiling) is also
Mathematics: Shaping
164
established. In a round house, the use of cross-beams in the ceiling and care with circularity has led to a recognition that the central pole can be cut away and the equal force of the ceiling on the top of the central pole is sufficient to keep it in place.
Figure 3. Partly counter-balanced bridge, circle formed by marking off points with a stick.
The seasons are a particularly important feature of
hunter-gatherer societies that exist in
Maps are also used. These generally feature spiritual connections to the land and to relationships between people. The connectedness of places frequently dominates the map with direction and distance being secondary.
Mathematics: Shaping
165
Indigenous mathematics and teaching
There are several points to raise. Where the Indigenous culture is either strong or in need of preserving, students need to learn the mathematics of the culture. These conceptual, contextual mathematics have intuitive meaning for children. They form the foundation of learning.
Just as it was important to know when teaching nutrition that food taboos exist, so it is important for teachers to know about the existence of forms of mathematics other than the school mathematics with which we grew up.
It may be that some of our students come from cultural groups
in which the mathematics has significant differences to those we are teaching.
For example, many Indigenous cultures of the Pacific and the
But the classifiers actually enhance another form of mathematics. Classification is a key mathematical process. It varies from culture to culture. Take for example the Greek and hence the Western form of classifying shapes. There are other ways of classifying, equally as valid, in other cultural groups. Doesn't this knowledge give us, as teachers, a bigger picture of mathematics itself. It should be shared with children at some stage.
Similarly, knowledge of different bases or rather different cycles, which is a much more useful term for describing different systems, enriches our understanding of mathematics. It certainly helps students to recognise a key feature of the base 10 place value system and this too should be shared with students at some stage, as it is currently in Year 7.
Of course, this brings us to another issue. Are there key processes in all mathematics. The closest we can probably come to that is the six mathematical activities suggested by Bishop (1988). These are measuring, designing, classifying, playing, making, and counting. All require mathematics of some form but they are activities not processes per se.
Finally, Indigenous mathematics indicates that some histories
of number and some comments about Indigenous mathematics need to be questioned.
The Middle East is not the only centre for the
development of numerical mathematics nor is
Furthermore, our teaching of geometry and shape often misses
some critical issues. What do you think of when I mention the properties of rectangles. Many people cannot remember anything about
diagonals but young men involved in rectangular house-building in
Mathematics: Shaping
166
Now what of the issue of whether any Indigenous mathematics may be harmful. Just as it is frequently an issue that can be side-stepped with nutrition by substituting one food for another, so it is with mathematics. For example, village garden lands are frequently compared by using the sum of length and width. Now many gardens are not rectangular and even when they are, two different areas can have the same sum of length and width (or semi-perimeter). You do not have to be in a traditional culture to confuse perimeters and areas of rectangles etc. Try out how many students think that all rectangles with perimeters of 12 cm have equal areas. However, reference to traditional mathematics is a neat way in which confusions can be addressed. There is much more to deciding land issues than mere size.
Despite the number of times, that I have used my non-Indigenous mathematics to explain what I regard as Indigenous mathematics, I am deeply aware that I do not have the language or the experience to really have that ‘sense’ of Indigenous mathematics or that real mathematical understanding that comes with language and culture. However, that does not mean that we should not make links. It does mean that learning mathematics in a first language is very important. It also means that mathematics must be seen as socially constructed and where differences seem to appear, these must be addressed.
Traditional mathematics may remain the providence of that society, it may have links with Western society mathematics, and it may be basically equivalent.
References
Bowers, N., & Lepi, P. (1975). Kaugel valley systems of reckoning. Journal of the Polynesian Society, 84(3), 309–324.
Lean,
G. (1991). Counting
systems of
Lean,
G. (1993). The counting
systems of
Owens,
K. (1999). The role of culture and mathematics in a creative design activity in
Papua
Mathematical Society.
Owens, K. (2000). Creating space. University of Western Sydney, Bankstown.
Owens, K. (2000) Unravelling the mysteries of mathematics. University of Western Sydney.
Owens, K. (in press, 2001). The Work of Glendon Lean on the counting systems of Papua
Siegel,
J. (1982). Traditional bridges of Papua New Guinea.
Lae,
Appropriate Technology Development Institute, PNG University of Technology.
Smith,
G. (1988). Morobe counting systems. Papers
in
1–132). Canberra: Pacific Linguistics.
Strathern,
A. (1977). Mathematics in the Moka,
16–20.
Wassmann, J., & Dasen, P. (1994) Yupno number system and counting. Journal of Cross-
Cultural Psychology, 25 (1), 78–94.