The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania

Kay Owens

University of Western Sydney

Glendon Lean collated data on nearly 900 counting systems of Papua New Guinea, Oceania, and Irian Jaya (West Papua). Lean’s data came from a questionnaire completed by students and talks with village elders. He read old documents written in English, German, and Dutch. He made comparisons between older and new accounts of the counting systems and compared neighbouring counting systems from both Austronesian and non-Austronesian languages. His work drew attention to the rich diversity of the systems and suggested that systems based on body parts and cyclic systems developed spontaneously. Digit tally systems were also relatively common. Lean’s thesis on spontaneous developments of these ancient cultures challenged traditional theories describing the spread of number systems from Middle East cultures.

If my life is richer for the people I have met and the books I have read, then let me share it with others. If the ethnomathematics that I have read, heard, seen, and explored has influenced me, then I am pragmatic enough to think that it will influence others. Some argue that ethnomathematics should only be available to those whose culture and mathematics it is. Sound ethical reasons suggest that consideration of ownership and context has to be taken into account. Cultural roots, however, should be recognised by students in all disciplines. I recently undertook an interview study of some Papua New Guinea (PNG) architecture students (Owens, 1999). The response that impacted on me most was the affective and somewhat intangible response by students that their PNG heritage was significant in their designing. It was as if their pride in the PNG design tradition meant they were good designers. Design was part of their culture; their culture was part of their designing. In some cases, this was evident in specific aspects of their sculptures, but at other times it was not obvious or explainable by them. A fishing tool, a house roof, spirals, weaving, a symmetrical face, sea devils, and abstract design were represented, but many students did not express their culture in a specific design feature.

I also asked them about their use of mathematics in their designing, but there was little impact of mathematics, even though some did use some simple measuring and symmetry copying. A few even rediscovered such mathematical shape formulae as the relationship between the curved surface of a cone and the diameter of the base circle, and how a rhomboid prism could be varied. I wished that mathematics was seen as part of the culture and valued in the same way as traditional design. Some students could also explain how mathematics was used traditionally and they were pleased to explain it, but it meant little in its links to school mathematics. Interestingly, although they had done school mathematics, it was primary school activities¾and to a lesser extent traditional mathematical approaches¾that were more commonly used, though without their recognition until interviewed.

 Should ethnomathematics be used just to “unfreeze” the mathematics in it in order to teach Western mathematics which some consider more advanced? This has been an argument and approach used by Gerdes (1999). He has consistently made links between traditional mathematics and that of Western culture. This may be a good thing, but it is likely to miss the point of the traditional mathematics. Others say it is misappropriating the traditional mathematics. The view I take is that it may be viewed as making links between two mathematical systems. It is not always the Western mathematics that must dominate. For example, in using the diagonals of a rectangular floor plan to build a house, the diagonals take on new prominence in thinking about rectangles. It is possible to link it to theorems on congruent triangles and proofs of properties of rectangles but it is more traditionally linked with the use of sticks and ropes to show equality in a great variety of situations. The use of the diagonals to “square” a house may remain a form of measurement and a way of deciding equality. Equal lengths are also used to gain equal space for two parties, as a means of checking or a useful approximation, and as an easy length to change if alternative dimensions are needed.

Whenever I am introduced to a new aspect of ethnomathematics, I re-evaluate my limited view of mathematics. In particular, I had thought about different bases in arithmetic and their strict power structure. But when I read Glendon Lean’s work, I was confronted by cycles. What is the difference? How does that change my view of bases? Why is that significant?

I write this article aware of current criticisms of ethnomathematics and its uses and abuses. I write it aware that I am neither Papua New Guinean nor the author of the research I am about to describe. I bring to bear on this research the opinions of others gleaned from my own literature search and from personal contacts. I do, however, think my life and my mathematics is richer for this knowledge, and I therefore wish to make Glendon’s work available to others.

Glendon Lean, the Languages, and His Thesis

For over 20 years, Lean meticulously gathered, recorded, and analysed data. He would question and search for answers using the world’s resources of documents on the languages of PNG, and young and old informants in the various societies. He also collated from written records data from other parts of Oceania. He collected into one document the natural language numerals of 883 of the 1200 languages spoken in the region encompassing PNG, Irian Jaya or West Papua, the Solomon Islands, Vanuatu, Fiji, New Caledonia, Polynesia, and Micronesia (Lean, 1992).

In the Melanesian region, there are three groups of languages. Twelve of the languages are known as Polynesian Outliers. There are at least 707 Papuan (non-Austronesian) languages which are considered to be daughter languages of proto-languages spoken by groups of people who migrated into the island of New Guinea at least 10,000, probably 40,000 years ago. The third group consists of 500 or more Austronesian languages thought to be descendants of proto-Austronesian spoken by people migrating into the Indonesian area about 5,000 years ago and proto-Oceanic which Ross (1988) suggested spread from near the north-east coast of PNG.

To collate so much data and to systematise it geographically and mathematically was a magnificent feat in itself. However, Lean went further and used his data to test diffusionist theories of the spread of counting systems. Standard histories of numeration begin at the time when written numerals were first used. These histories suggest that the majority of systems have a base of 10, although systems or vestiges of them are recorded with 12, 20, and 60¾as in the English, French, and ancient Sumerian systems. They further suggest that enumeration is fairly consistent, and that it spread from the ancient city states of the Middle East. This view is untenable if the indigenous cultures of Oceania are considered. Lean has provided evidence that their counting systems exhibit a fascinating diversity and he provides an alternative view of the diffusionist theory suggested by Seidenberg (cited in Lean, 1992). Seidenberg suggested 2-cycle systems spread first from the Middle East and were later replaced by successive waves of counting systems, except on the verges of civilisation where the 2-cycle system survived. Neo-2-10 cycles, claimed Seidenberg, came before the 10-cycle system. However, Lean showed that 2-10 cycles do not exist in Oceania as Seidenberg’s theory would suggest. He also points out that Melanesian counting systems are older than the Middle East city states, that changes to the counting systems occurred in some members of both the earlier and later Melanesian migrations, and that systems could develop more spontaneously due to the use of body parts, especially digits, for counting.

Data Sources

Lean collected counting-system data for 550 languages from students (mainly from the PNG University of Technology) and teachers. He acquired others from informants, from unpublished linguistic sources, and from published sources that often contained first- and early-contact data. The Oceania data was almost entirely from published sources. Whenever possible, first- and early-contact material was sought.

Many of the counting systems that Lean has collected were derived from linguistic surveys; but these gathered just the words for the numbers 1 to 5, 10, and perhaps 20. This is often insufficient data for deciding the mathematical cycles of the language. It is also difficult to see how some morphemes are linked to each other when there are modifications (as in, for example, twenty and two tens).

At other times, Lean’s information was from the Counting System Questionnaire. His report indicates the difficulties of describing the counting system definitively when it was completed by only one or two students from particular villages or where two sources of data differed. Differences were mainly due to the lack of use of the counting system in the traditional culture, or its replacement by counting in Tok Pisin (a PNG lingua franca, often called pigeon English) or English. Fortunately, his questionnaire asked for lists of body-part words that are used in counting, and other pertinent comments from his informants. These data were very important for his analysis and for his description of different counting systems. They were often richer than the linguistic surveys.

Analysis of Counting Systems by Cyclic Structure

Bases and Cycles

In PNG and Oceania, there are few counting systems with 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, 1000 = 10 ´ 10 ´ 10, and so on. For this reason, it was more beneficial for Lean to describe, collate, and analyse the data in terms of cycles and patterns within the counting system rather than simply in terms of bases.

As well as recording bases and cycles, Lean documented what he called the frame of number words and the operative pattern of the system, that is, the pattern in the ways the number words were constructed.

Each counting system has a set of number words from which all other number words in the system can be generated by combination. Thus, a complex number word may be analysed as having a syntactic combination of simpler number words. We can infer the arithmetical operation(s) carried out on the simpler numerals to obtain the complex numeral. For example, the number word for 10 might have the number morphs for 2 and 5. This system might have the operative pattern of 2 ´ 5 or 5 ´ 2. For another system, we may be able to infer that ten is 5 + 5, or 6 + 4, or a distinct number word in itself.

Cyclic structures were introduced by Salzmann (1950, cited in Lean, 1990). It is important to note that a base 20 counting system is better understood in terms of its cycles. For example, it may have both a 5-cycle and a 20-cycle. A (5, 20) cycle system has number words represented as 1, 2, 3, 4, 5, 5+1, 5+2, 5+3, 5+4, 2´5, (2x5)+1, (2x5)+2, ... , (2´5)+5, (2x5)+5+1, ... , 20, ... , 2´20, ... . Such a system can be distinguished from (2, 5, 20), (5, 10, 20), or (4, 20) cycle systems.

A 5-cycle system will have the basic number words of 1, 2, 3, 4, and 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. Non-Austronesian systems generally do not have a cycle of 10 unless influenced by Austronesian neighbours. One possible exception is the Ekagi in Irian Jaya, who have a (10, 60) system.

Table 1 illustrates the large diversity of cyclic counting systems. Lean has selected to group together all systems with the same lowest cycle. In my opinion, this may distort the numbers placed in a particular cyclic group. For example, the number of languages in the 2-cycle category seems large but the majority of these languages have a (2, 5, 20) cycle system forming a digit-tally system. In addition, there are other (5, 10, 20) and (5, 20) digit-tally systems.

Body-Part Tally Systems

Body-part tally systems can have a range of different cycles depending on which body parts are included in the cycle¾the 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, moves to the wrist and then along the arm, shoulder, left ear and eye, nose or central part, and then down the other side of the body. If vocalised the words reflect the body part (see Figure 1).

Table 1

Summary of Cyclic Counting Systems in Papua New Guinea and Oceania

System type






“Pure” type: 10, 100, 1000

“Manus” type: 7=10-3, 8=10-2, 9=10-1

“Motu” type: 6=2´3, 8=2´4 (some with 7=2´3+1, 9=2´4+1)

(10, 20)

(10, 60)

189 out of 217 AN; NAN influenced by AN neighbours


(5, 10) or (5, 10, 100)

(5, 10, 20)



(5, 20)

Second most common AN

All systems in New Caledonia, some AN and NAN in PNG and Irian Jaya

AN and NAN



“Pure” with 1 and 2



(2, 5) or (2, 5, 20)


(2’, 4,  8)

2 more types

201 NAN, 30 AN

37 NAN, 2 AN; usually associated with body-part tally systems

Most common, 18 AN; digit-tally system with 2 subordinate


Cycle from first 4

Cycle from second 4

Superordinate cycles 28, 48, or 60

Special name for each cycle



Highlands of PNG

Enga dialects



Restricted areas



Restricted areas

Note. AN = Austronesian languages, NAN = non-Austronesian (Papuan) languages, 2' = modified 2- cycle system.

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, Milne Bay, and to a lesser extent in other Provinces (East Sepik), a morpheme may be used to distinguish different classes or groups of objects. They occur in both Austronesian and non-Austronesian languages. In some cases, the counting cycle size and words change for the different classes of objects, but this is likely to be an idea borrowed from a neighbouring language or trading partner.

Non-Counting Systems

In a few different language groups, large numbers of objects are 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.

Geographical Distribution of the Counting Systems

The maps in the Appendix provide an idea of the distribution of the systems in PNG. Austronesian languages are mostly found around the coast. The most remarkable aspect of these maps is the extent of 2-cycle systems. However, many of these systems also have (5, 10, 20) or (5, 20) cycles. In practice, these supercycles are more akin to the base of a system.

The linguistic systems in Polynesia and Micronesia suggest that the 10-cycle systems of the Austronesian languages in the region retained the essential cyclic nature of their counting systems over a long period partly due to their relative isolation in Oceania.

Two Non-Decimal Systems

In order to illustrate the way in which cycles can be recognised in counting systems, two non-decimal systems are described below.


This large language group in the Southern Highlands Province, south-east of Mendi, was estimated by Franklin (1971) to be spoken by over 50,000 people in 1968. It is a non-Austronesian language. Beside the 4-cycle system there is also a body-part tally system with a 47-cycle. Counting begins on the little finger. The morpheme for hand is ki indicating that fingers but not thumbs are tallied. Table 2 gives only one of the recorded Kewa systems.

Table 2

A Kewa Counting System





ki laapo na kode repo

two hands, three thumbs



ring finger


ki repo

three hands



middle finger


ki repona kode (pameda)






ki repona kode laapo



kode or kina kode

the thumb


ki repona kode repo



kode laapo

two thumbs, i.e., one hand and two thumbs


ki maala



kode repo

three thumbs, i.e., one hand and three thumbs


ki maaloma kode (pameda)



ki laapo

two hands


ki maaloma kode laapo



ki laapo na kode

two hands, one thumb


ki maaloma kode repo



ki laapo kode laapo

two hands, two thumbs


ki su


Sources: Franklin & Franklin (1962, p. 403), Franklin (1971, p. 403), Pumuge (1975), Lean (1991, Vol. 10, pp. 37–38; 1992, pp. 107–108)


This non-Austronesian language is one of three 6-cycle languages spoken on Kolopom Island adjacent to the south coast of Irian Jaya, just west of the Irian Jaya-PNG border. There are distinct terms for 18 and 36. Both 72 and 108 are compounds of 36. Ndom has a (6, 18, 36) cyclic pattern.

Table 3

The Ndom Counting System





mer abo meregh






mer an thef






mer an thef abo sas












tondor abo thef






tondor abo mer



mer abo sas






mer abo thef






mer abo ithin



nif thef



mer abo thonith



nif ithin


Sources: Galis (1960, p. 148), Drabbe (1949, pp. 6–7), Lean (1992, Vol. 5, p. 122; 1992, pp. 113–114)

Classification of the Counting Systems

Lean used five major methods for analysis. First, he listed the languages by families. He then listed the frames, cycles, and operative patterns for each of the languages in a particular province. This helped to highlight similarities between counting systems in the region. He also listed the first five numeral words and the word for hand. This again highlighted similarities and possible spreading of one counting system to another. A hand-related morpheme is used in most digit-tally systems and Austronesian languages.

Other writers have selected to analyse the data in other ways. Lancy (1978) summarised the counting systems as four types. This included body-part tallying, use of objects or bundles of fixed size, systems using tallying on feet and hands, and those employing a base 10 system. Smith (1984) and Lean (1992) point out that the second type is found as a supporting system in other types of systems. Both refute the vaguer classification by Ray (1926, cited in Lean, 1992) of quinary and vigesimal systems as often overlapping. Smith (1984, 1988) used eight types in order to evaluate the diversity of counting systems within a language Family. He broke up systems that used two, three, or four numerals besides a numeral for five, and those that use numerals for 10 and 20. Lean selected not to use this kind of typing because there were too many possibilities. Rather he referred to cycles, frames, and operative pattern or type. The operative patterns cover (a) body-part tallying, (b) digit-tallying, (c) regular decades, (d) an unexpected construction for 4, and (e) patterns for numbers between 6 and 9. The patterns for 6 to 9 included (i) a morpheme plus the morphemes for 1 to 4, (ii) 10 subtract 4 to 1, and (iii) 1 to 4 towards 10. Similarly, Lean classified counting systems according to their base cycle (often 2, or a modified 2 cycle) which was frequently subordinated by a supercycle such as 5, 10, or 20. Few counting systems had a regular base with powers of the base for later cycles. Frames were the basic numerals used in developing the remaining counting words. Two recent publications (Butterworth, 1999; Wassmann & Dasen, 1994) only refer to the more restricted types of systems given by Lancy (1978). It is important that future researchers make use of the classifications set up by Lean (1991, 1992) or Smith (1984).

The Difficulties of Making Decisions About the Counting Systems

Linguistic records or informants’ knowledge frequently provided insufficient data for decisions to be made on the system’s operative patterns. Another decision for Lean was the possible lack of knowledge of an informant. For example, in the case of the informant’s data on Mebu, another Yupna language, the language words were similar to a neighbouring language of a different family. Lean decided that the data source was inconsistent with Yupna languages and so was unreliable.

Although the use of words for hand and other body-parts assisted in decisions regarding the type of system, the meaning of the various morphemes and their variants sometimes confounded these decisions.

The difficulty of collecting and analysing so much data will be demonstrated further using the Yupno language example. This commentary on the Yupno counting system highlights the kinds of debates that Lean had with himself as he tried to understand the counting system data. Initially, he needed to match languages or dialects when different names were given for the same one. This was assisted by questionnaire informants being encouraged to give alternative names for the language and to name their village and district. It was particularly difficult when Lean had older recorded data.

Many of Lean’s counting system tables give multiple names and multiple dialects. Some of these he had gleaned from the older records of languages in the area. The Wurm and Hattori (1981) language maps and geographical maps, despite the movement of villages since they were made, were essential for establishing language classifications and names.

I wished to follow up comments made by Butterworth (1999) on the Yupno counting system because he claimed that it was a type of body-part tally system which I had not read about in Lean’s thesis or appendices. First I went to Butterworth’s source, Wassmann and Dasen (1997). Wassmann had first undertaken anthropological studies in PNG in the 1970s. His recent work was on the numerical counting and basic operations of the Yupno. In my comments, I refer to his paper and to the work of Lean (1991, Volume 15) and Smith (1984) who give the counting systems for several of the Yupna languages. Both these men had been working in PNG and collecting counting systems for more than 20 years.

Smith (1984) and Lean (1991) refer to Isan, one of the Yupna languages, which, according to the Wurm and Hattori language atlases, seemed to be the language referred to by Wassman and Dasen. Smith’s and Lean’s informants gave essentially the same words for the counting system, although Smith’s was more complete. Lean’s informant came from the village of Isan. Wassmann and Dasen’s informants came from Gua. Both villages are near the Teptep station in the Finisterre ranges, and informants say that both villages speak the same language, Yupno (Matang, 2000, personal communication). The Wassman and Dasen reference does not give the language counting words, so direct comparison with Lean’s data is not possible. Wassmann and Dasen’s (1994) informants were ten older and middle-aged men who had not attended schooland, it is claimed, had no knowledge of Tok Pisin; four middle-aged men who had been to school; and six school students. They asked the people to count without referents and then to do some additions, subtractions, and multiplications. They found division could not be explained adequately. Only the older men gave the body-part tally system, and the data varied between them.

Classifying the Counting System

Wassman and Dasen describe the Yupno counting system as a body-part counting system. Both Lean (1991) and Smith (1984) regarded the counting system as a digit-tally system. Their informants stopped counting at 20 as if it were the last new counting word. Wassman and Dasen say that 5 was the finger with which one peels bamboo shoots, that is, the thumb; also that the sum or addition of numbers is usually indicated by one hand, showing the closed fist. Ten is two hands, also called mother. According to Smith (1984) and Lean (1991), the words used for counting are initially 1, 2, 3, 4 = 2+2; 5 has a hand morpheme, then the other hand is counted, followed by the feet¾giving a man similar to other digit-tally systems. However, Wassman and Dasen claim that the older men continue counting with body-part tallying. It should be noted that a hand morpheme and not a thumb morpheme is used by Lean (1991) and Smith (1984). This would suggest that the body-tally system, if it existed, was added to the digit-tally system rather than the reverse, although big toe was used by one of Smith’s informants as well as some of Wassmann and Dasen’s informants, as expected of body-part tally system using leg parts.


Lean watched for systems that differed too much from the expected. For example, discrepancies in the same language family that had characteristics of a neighbouring non-family counting system indicated a change or diffusion of a counting system. His analysis in terms of families, frames, operative patterns, cycles, and numerals for 1 to 5 and hand assisted his decision-making. However, he also built on the linguistic knowledge of the spread of languages and a possible proto-Austronesian language.

Again the Yupno case illustrates the issues. The Wassman and Dasen description can be compared with other body-part tally systems.

First, the Yupno counting system is much further east than other recorded body-part tally systems in PNG. It does not have any neighbours using the system, as far as current records show. The other Madang Province body-part tallying systems are Gende, Kobon, and Kalam (in the far west of the Province near Enga and Western Highlands Provinces), and Murupi, which is only a few kilometres inland from Alexishafen near the coast and “the easternmost location of a body-part tally-system within Papua New Guninea” according to Lean (1991, Volume 15, p. 115).

Second, according to Wassman and Dasen, Yupno begins as a digit-tally system with cycles of 5 and 20 but then changes to a body-part tally system. Lean gives a number of modified counting systems. For example, several body-part tally systems have grafted onto them a 2-cycle system (usually as far as 4 = 2 + 2) and thereafter the body-part systems remain unchanged. The school-age Gende speakers indicated their 31-cycle body-part system had been modified to a 10-cycle system by truncating the original body-part system to a 10-cycle system. However, no mention is made of combinations of body-tallies being grafted onto a digit-tally system in any systems recorded by Lean (1991). He does suggest that digit-tally systems may have existed before neighbouring body-tally systems were introduced in some limited areas. For example, Hewa (Lean, 1991, Volume 9) in the Southern Highlands and Alamblak (Lean, 1991, Volume 14) in Sandaun Province both have body-part tally systems like their neighbours. Both digit-tally and body-part tally systems are used in Alamblak. They are not grafted together, and the body-part tally does not switch sides. The closest account of a body-part tally and digit-tally system merging is provided by the limited data available for Kwomtari in the Sandaun Province, which appears to have been influenced by neighbouring languages. The informants suggested that there is a 10-cycle system with counting on two hands (but not a digit-tally system using a hand morpheme), as well as a body-part tally system.

Lean found no evidence of languages possessing body-part tallies among the Finisterre-Huon Stock to which Yupna languages belongs or other neighbouring stocks in the Trans New Guinea Phylum (Lean, Ch. 8, 1992). Lean’s general conclusion was that 2-cycle and body-part tally systems were present in proto-systems with the digit-tally system being introduced into the Trans New Guinea Phylum subsequently (Lean, Ch. 8, 1992).

Third, the system recorded by Wassman and Dasen (1997) differs from other body-part tallying systems in that the people move from one side of the body to the other (which may follow from the side switching for digit tallying) and include the testicles and penis. The latter word was usually, but not always, used as the last counting word by Wassman and Dasen’s informants. Side switching may occur in one other system and the use of genitals is unknown in any other recorded body-part counting system.

 Some informants have noted that lower parts of the abdomen are not used in tallying systems. Biersack (1982, p. 813, cited in Lean, Volume 13) observes that, with the Paiela, the lower part of the abdomen is associated with “abominations such as intercourse, menstruation, childbearing, defecation, and the like”. “Tallying, thus, is an abomination-free process” (Lean, Volume 9, on the Ipili). Nevertheless, Lean (1991) does report that the recorded data for Nagatman of the Sandaun Province appear to have a 74 body-part tally system requiring lower parts of the body to the toes but only the sides of the body. Tallying proceeds up the left arm and then down the left side. The toes are tallied and upon reaching the big toe the tally stands at 36. Then the side of the left foot is tallied and the symmetrical points on the right side of the body are tallied in reverse order.

Fourth, Wassmann and Dasen report that Yupno women do not count in public. Lean refers to only one report of a different counting system for women. Bruce (1984, cited in Lean, 1992) reported that Alamblak use three types of systems: One is a digit-tally system with a (2, 5, 20) cyclic pattern. Another, used exclusively by men, is a body-part tally system with a 29-cycle. A third, used exclusively by women, is similar to the men’s system except that the breasts are used as tally-points instead of points on the face.

Fifth, the fact that different men had different tallying systems, using different body parts or a different order, is known in other counting systems (e.g., Kewa: Lean, 1991, Volume 10). But it is unusual that they do not have the same final number.

The many unusual features of the Yupno tallying system as recorded by Wassman and Dasen (1997) may caution a researcher familiar with as many counting systems as Lean. Nevertheless, they make several interesting comments. First, the idea of counting large numbers of pigs for a bride price was considered preposterous, so much so that the researchers did not give a planned subtraction involving bride price to the older men. The researchers do not go on to indicate what objects are counted. Second, when Wassman and Dasen questioned the informants about the differences in their different counting systems, the informants said that the one given by the more important informant was correct. Wassman and Dasen note that this may have been deference to the more important person. Third, the fact that counting was a man’s activity might suggest that many of the communities from which data has been analysed as digit-tally systems may also have rituals that develop body-tally systems. No data have been collected on this, although we do know that in some places traditional counting is used in ceremonies and not necessarily on other occasions.

Seidenberg, Crump, and Wurm (all cited in Lean 1992), as well as Lean himself, comment on how systems might influence each other¾especially how a few of the neighbouring Austronesian, Tok Pisin, and English base-10 counting systems have influenced existing systems. Lean (1992) suggests a few counting systems have spontaneously developed from tallying with body parts—a point also raised by Butterworth (1999)—and that there is some change resulting from contact with other systems. What might be interesting in the case of the Yupno is whether the earlier contacts with Lean and Smith had started them talking and developing their interest in counting systems. Lean’s questionnaire had a diagram of a body for marking matching numbers. Smith also checked for body-tally systems. There is always the possibility of counting systems developing in an unexpected way.

Lean was aware of his own biases in collecting data, the various cultural contexts in which the data was collected or used, possible modifications and the reasons for these. Clearly, this task was problematic and, in my opinion, only a person like Glendon Lean could achieve such a mammoth task in a responsible way.

Using Numbers

Case studies show that some Papuan societies (e.g., Melpa) make extensive use of numbers in ceremonial contexts  and some count a wide variety of objects. It does not necessarily depend on whether they have a 10-cycle system or not. Some Papuan groups (e.g., Mountain Arapesh) who emphasised counting used a 2-cycle variant system, while others (e.g., Ekagi) adopted a 10-cycle system. Some societies place little importance on counting despite their 10-cycle systems, emphasising instead the indivisible mass of a visual display. In these circumstances, some societies like the Loboda have retained their 10-cycle system, while others (e.g., the Markham Valley 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 varies in each society from visual impression to 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) whether different types of objects are counted in different ways, (e) whether some objects are not counted, and (f) how totals are recorded. “Counting does not exist in isolation. It quantifies and qualifies relations between people, objects and other entities” (Bowers & Lepi, 1975).

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 arbitrarily large numbers. 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 they are subject to Austronesian influence.

Except for a half, fractions are generally not used. 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 for 20 in the (2, 5, 20) systems.

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. Relatives of the Melpa dialect given in Table 4 below used a (4, 28) system. Certain numbers such as 8 or 10 are particularly significant in ceremonial situations, but it would be wrong to invest them with sacred connotations.

The Melpa Counting System

Melpa is a dialect of Hagen. The people count up to four while simultaneously bending their fingers, starting with the little finger of the left hand, and then continuing similarly on the right hand. At this point, a further 8-cycle can be used or a set of 10 can be formed by tallying one and then the other thumb, giving hands of one man. With the two thumbs down, they finish with slapping the knuckles together at ten. The anthropologist Strathern (1977) provides extensive information of the use of the number 10 in counting pigs or other items during exchanges, maybe explaining some of the variations in the two recordings. He explains that another man tallies the number of tens. The 8-cycle system has vestiges of 2-, 4-, and 5-cycles. The Melpa data have several variations and show the complexity with which Lean worked to classify the counting systems. Table 4 gives some early recorded data which differ from more recent data such as that collected by Lean in a study in 1978 or given by some of the informants on the Counting System Questionnaire.

Table 4

The Melpa Counting System




tembokakapoket pombin ti ngkutl




tembokakapoket pombin rakl ngkutl (12+2)




tembokakapoket pombin rakltika ngkutl (12+3)




ki rakl




ki rakl tende ngkutl


pombingkutl dende or ngkutl rakl

(5+1 or 4+2)


ki rakl rakl ngkutl


pombingkutl rakl or kotrakltika


ki rakl rakltika ngkutl


engkaka or ki dende


ki rakl tembokaka


pombi ti ngkutl


ki rakl pombingkutl




ki rakl pombingkutl dende




ki rakl pombingkutl rakl




ki rakltika

Source: Vicedom and Tischner, 1948, Vol. 1, pp. 237-8 (cited in Lean, 1988, Vol. 9, p. 27).

The Enga Data

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. A decimal system is also reported by Engans. The 60-cycle system consists of a sequence of 4-cycles beginning at 9, each cycle having the same construction. For example,

17   yupun(ya) me(n)dai   dog, 1

18   yupun(y) lapo   dog, 2

19   yupun(ya) tepo   dog, 3

20   yupun(ya)gato   dog, end

The cycle units, 13 in all, are not numerals as such but may be words or phrases with some non-numerical meaning such as dog, sweet potato, flat object, or what can I say? According to Lean, there has been no given explanation for the choice of cycle names. Only the first, fourth, and seventh counting words are numerals. The last cycle is kaeapalu (I stop) and reaching 60 is described as exciting like a house full (burning) (Yaru, personal communication; Lean, 1988, Vol. 9, pp. 12-14).

Lean meticulously checked data where possible, but diversity and change made this task difficult. For example, on finding discrepancies in the Enga data (including some recorded in 1938), Lean visited a village to check. The above data came from Mai dialect informants and was confirmed, with minor suffix variation, by students and teachers completing the Counting System Questionnaire. Some young informants thought that the non-decimal counting system was only used for large ceremonies. The ceremonial counting system may have been known by informants’ old fathers who had seen several large pig exchange feasts, but their infrequent occurrence led some students to think it was based on powers of 2 with cycles like 4, 8, 16, 32 (Yaru, personal communication).

Despite differences and difficulties, the importance of Lean’s work has been its immensity of detail and analysis, the links between systems, and the employment of other research from linguistic and other non-mathematical fields to restructure the prehistory of counting.

Glendon Lean’s Model of the Prehistory of Number

The 850 different languages that have sufficient data provide a wide range of counting and tallying systems with diverse cyclic structures. From this complex picture, Lean analysed the systems and linguistic family connections to establish what might have happened in the development of counting systems in New Guinea. The Papuan (NAN) languages, spoken by the descendants of groups who moved into the island of New Guinea prior to the Austronesian (AN) migrations, possess the greater diversity of counting systems. The majority of the groups begin with 2-cycle systems, with a smaller number having 5-cycle systems. Decimal (10-cycle) systems are relatively rare and seem to occur in some languages as a result of influence by Austronesian groups. The body-part tally systems, found largely in the isolated interior of New Guinea, are used only by non-Austronesian groups. While it is difficult to reconstruct the situation regarding numeration within New Guinea prior to the Austronesian intrusion, it seems likely that 2-cycle systems predominated and that body-part systems existed together with 2-cycle and some of the less common types, for example, 4-cycle and 6-cycle systems. The digit-tally system with a (5, 20) cyclic structure also existed before the Austronesian intrusions. On this point, Lean agrees with Seidenberg. These groups were there before 8,000 B.P. (Before the Present) when the Torres Strait was formed, and earlier than Seidenberg’s claim.

Lean disagreed with Seidenberg on several other accounts. He did not think that the Motu 2-cycle system is a development of the pure 2-cycles, but he did think that the complex body-part tallies may not be as old as the 2-cycles or digit-tally systems. Lean stated that his data disagreed also with Seidenberg’s theory (1962, cited in Lean, 1992) on the cultural origins of counting¾in particular tabu on counting certain things, and the link of body-part tallies to customs such as the distribution of bodies to various gods. The anthropological evidence available to Lean was not supportive of this theory.

With the advent of the Austronesian migration, the 10-cycle counting system was introduced. Those AN groups which remained relatively isolated from the Papuan groups, or which left their new homeland for islands further east, generally retained their original 10-cycle systems. Those groups that interacted with the non-Austronesian speakers, generally had their decimal system modified in some way. For example, in coming into contact with the (5, 20) digit-tally system some groups incorporated a 5-cycle into their existing system, resulting in a (5, 10, 20) system. Other Austronesian groups, particularly those in the Markham Valley region of PNG, entirely abandoned their 10-cycle system for their neighbour’s 2-cycle system. Others incorporated 4-cycles. Certain NAN groups acquired systems with (5, 10), (5, 10, 20), (10), and (10, 20) cyclic patterns. There is no evidence that neo-2 systems (so named by Seidenberg, who linked them to 2-10 systems) had primary status. Where these systems (that is, the Motu type) do occur, they are present in Austronesian languages which spread from proto-Austronesian systems that developed in the region. Indeed, the pure 10-cycle system introduced by the Austronesian immigrants has primary status. This result casts doubt on Seidenberg’s suggestion that 10-cycle systems were first diffused as neo-2-10 systems. Lean is quick to point out that even later Austronesian systems were present at least 2,000 years before the Sumerian civilisation, thereby disputing that they spread from the Middle East.

Figures 2 and 3 give Lean’s (1992) genealogy of non-Austronesian and Austronesian counting systems and tally methods. The suggested dates are the latest dates for the development of these languages; it is expected that the systems developed much further back in time.

The 10-cycle nature of the proto-Oceanic system was already established prior to its introduction into New Guinea, as the reconstruction of the proto- Austronesian numerals showed (Ross, 1988, cited in Lean, 1992). Although the introduction of a 5-cycle into Austronesian systems may have developed from

(Trans New Guinea Phylum)

              body-part tally                  pure 2-cycle                                    (5, 20) digit tally

                   Ù                                         Ù                                                            Ù

                   Ù                                         Ù                                                            Ù

                            Ù                                                                  Ù                                                                                           Ù             

                                       |                                     |                  |               |                     |

                   hybridised body-part             (2,5)          (2’,5)     (2’’, 5)        Ù


                                                                                                              |                          |

                                                                                                         4-cycle               6-cycle

Figure 2. Genealogy of the non-Austronesian counting systems and tally methods prior to the AN migrations (from 15,000 to 5,000 B.P.).




                                            Proto-Austronesian (~7 000 B.P.)


                                                  Proto-Oceanic (~5 000 B.P.)


       NAN influence 2-cycle                        Ù                        NAN influence 5-cycle

                           Ù                                            Ù                                            Ù

                |                        |                                  |                     |                  |

            (2,5)               (2,10)                            Ù               (5,10)        (5,20)     (5,10,20)



                                         |                                                |

                                     Motu                                      Manus

                                      Type                                         type

                           (post-2 000 B.P.)                   (post-4 500 B.P.)

Figure 3. Genealogy of the Austronesian counting systems.

contact with non-Austronesian systems, it is also likely that it developed through the habit of accompanying serial counting with finger tallying, a mechanism which induces change without the requirement of an external stimulus such as the sort of diffusion suggested by Seidenberg (Lean, 1992). As a result, proto-Oceanic systems use the word for hand for the number 5 (lima).

Lean’s (1992) investigation of the variants of the numbers 6 to 10 by distribution shows that neighbouring and related languages can have all three different constructions. This development would not be likely if diffusion, as suggested by Seidenberg’s theory, were strong. Further, the existence of the 4-cycle and 6-cycle systems cannot be explained by the diffusionist view. These two kinds of systems are not widely distributed and are sporadic. They seem instead to be localised inventions, variants of digit tallying. Displacement of systems with new waves of migration or economic power does not appear to have occurred to any great extent until colonial times, although there is evidence to suggest that the borrowing of part of a numeral lexis to increase the primary cycle of a counting system has occurred in a number of instances.

Seidenberg’s thesis is that the pure 2 system spread first from a Middle East culture and is now only found on the verges of its spread in South America, Africa, and perhaps Australia, having been displaced by successive waves of counting systems from the Middle East. Lean views this as both Eurocentric and dismissive of other ancient civilisations such as those of India and China. Indeed, evidence of engraved bones and tallying devices from at least 35,000 BC (Ifrah, 1998) and the above discussions of the prehistory of New Guinea and Oceania languages suggest that counting systems predate the ancient civilisations by many thousands of years and have their origins in so-called primitive societies. Nevertheless, there is considerable speculation that the European and Indian languages developed from a proto-Indo-European language. Deakin (1995) expresses the similarities of these languages and suggests that Seidenberg’s thesis is somewhat simplistic; therefore, Lean has perhaps been a little too dismissive of the possible spreading of language and counting systems around the world. Indeed, Lean does rely on a theory of proto-languages to suggest a possible spread of languages. Lean refers to the archaeological evidence of a piece of pottery from the west appearing in Fiji and linguistic evidence that suggest languages spread east from the New Guinea islands region to Fiji.

Implications for Research and Teaching

The education reforms in PNG have included the establishment of elementary schools at village level. These schools are to teach in the mother tongue of the children. The children begin school at age 5 and attend elementary school for three years. During this time, basic arithmetic and other mathematical concepts will be established. As the teaching is in Tok Ples (village or place language), materials have to be prepared and teachers need to be trained.

The resource of counting systems and an understanding of how they are best understood from a mathematical perspective is very useful. It means that school mathematics can make use of the linguistic formation of counting words, community uses of mathematics for realistic examples, the natural groupings (e.g., fives and 20s in digit-tally systems), and classifications (where classifiers are used). In some cases, writers of materials may not be mathematics educators, and Lean’s analysis can assist them in understanding and unravelling their own complex systems when preparing materials (D. Ope, personal communication). These elementary schools are not only providing an education in the mother tongue but they are also providing maintenance of language and culture, including the counting systems. Students will codeswitch (Clarkson, 1996) as they learn the mathematics in English, but this will only strengthen their understanding of the more advanced concepts if they have a solid understanding from their mother tongue.

In teacher education in PNG, at both the primary and secondary level, students are encouraged to prepare lessons that centre on community activities and ethnomathematics. For example, a collation of lessons prepared by student teachers based on traditional mathematics has been made at the University of Goroka (Zepp, 1993). So many lessons have been prepared over the years that it is doubtful that such an emphasis will easily be ignored in future. In other words, ethnomathematics makes sense for establishing mathematical concepts.

Students in high school can be made aware of the rich diversity of counting systems in their country. In addition, they can look for patterns and commonalities, and they can understand grouping and place value at a deeper level. Its value for this older age level is to embed mathematical ideas in their cultural background, and to establish a sense of mathematics belonging to their cultural heritage rather than being alien. Lean’s study also provides a relevant historical summary of the history of number. Even in schools outside of PNG, this data provides new insights into number systems and into the different systems used by different peoples. No longer do we just study ancient systems of the Babylonians, Egyptians, Romans, and Mayans, we can consider the wealth found in counting systems still in existence today. The idea of counting systems not being a rigid base system is also important. For example, the notion of cycles could be introduced into our mathematics curriculum in this way.

Perhaps the most important aspect of learning about different counting systems is that it highlights that people invent mathematics, that it is a culturally developed phenomenon. In addition, knowing that counting could be done in different groupings or composite units makes the base 10 place value system have more meaning.

From a research perspective, Lean’s study has expanded our understanding of counting systems and the history of number. He has illustrated procedures for analysis of linguistic materials for the study of mathematics. His meticulous and multi-disciplinary work is to be commended. However, the difficulties that he encountered should not be dismissed. Although much of his recording is tentative, it may already be too late for further work as the counting systems become supplanted by English or Tok Pisin. On the other hand, further research may reveal yet more intriguing systems and applications.

Lean’s work covered only the counting systems. Other areas of research have included work on measurement, design, and location. However, none of this work has been systematically explored in the same way as the counting systems. Each of these areas, and operations on numbers, could provide extensive areas of further research. Just as the counting system data is not just for elementary schools, it is likely that different approaches to number operations will be found that will enhance our understanding of how children construct numbers. Work on other aspects could also richly expand our understandings of mathematics. Both Wassmann and Dasen’s (1997) emphasis on operations and Saxe’s (1981) exploration of children developing number concepts could be repeated in other places.

If I return to the architecture students who were set the problem of designing a sculpture out of paper and cardboard without using glue, several solved the problem by putting rolled paper through holes punched in the papers to be joined (Owens, 1999). It is no coincidence that floor joists of some traditional houses pass through holes in the posts on which the house stands. Like the architects solving a problem, mathematics students might use traditional ideas to solve mathematical problems in future. Like the architects who felt they were good designers because of their PNG cultural heritage, mathematics students may feel they are good mathematicians because of their PNG mathematical heritage.


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Clarkson, P. (1996). NESB migrant students studying mathematics: Vietnamese and Italian students in Melbourne. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 225-232). Valencia, Spain: Program Committee.

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Map 1. Distribution of 2-cycle systems with supercycles noted.



Map2. Distribution of 4- and 6- cycle systems.



Map 3. Distribution of 5-cycle systems of Austronesian languages in New Guinea


Map 4. Distribution of 5-cycle systems of Papuan languages in New Guinea


Map 5. Distribution of 10-cycle systems



If a security officer at the PNG University of Technology had run out of cigarettes at 3 am in the morning, he could be sure to find some with his wantok, Glen, sitting at his desk sillouhetted by the study light of his campus house. The purr of the engine and a soft cough would bring Glen to the door with a quick greeting, “Moning nau, yu olsem wanem” followed by a brief chat in Tok Pisin or a local language. Such an idiomatic greeting could only be used by a respected friend. Like others, the security officer would say, “Glen was my friend.” Glendon was a person gifted in many ways; the diversity and depth of his interests we could not know. One always felt he had the interest and time to spend with you. He shared jokes or stories or literature or academic discussions as he chose. Glendon warmly remembered his early days at Monash University where he made life-long friends from the drama society. “He played out the dramas and loves of his life in Papua New Guinea”, remembers a close friend from PNG. Glendon quoted Auden in the abstract of his thesis, saying:

To bless this region, its vendages, and those

Who call it home; though one cannot always

Remember exactly why one has been happy,

There is no forgetting that one was.

Glendon had expertise in literature, art, mathematics, mathematics education, history, anthropology, and linguistics. He spoke Tok Pisin and Tolai fluently and conversed in several other PNG languages. A couple of weeks after his special doctoral graduation ceremony held in Australia by the Vice-Chancellor and Registrar of the Papua New Guinea University of Technology, Glendon died, a week before his 52nd birthday, March 1995.

Glendon was also assisted and honoured by Deakin University, where he worked after leaving PNG.

In recognition of his work, the University of Goroka has set up the Glen Lean Ethnomathematics Centre under the leadership of Wilfred Kaleva and Rex Matang, encouraged by Alan Bishop. The papers which Glendon had collected over the years have been catalogued and stored. These include German, Dutch, English, Australian, and United States records and papers dating from the mid-1800s, from first contact and colonisation, and from researchers interested in the languages of the East Indies and the Pacific. The Centre has also collected and catalogued numerous other papers on ethnomathematics, especially from PNG. A computer database of the languages has been set up using Glendon’s appendices. This is a huge task as there are nearly 900 languages. When this is finished, it will be made available on a website.

The work of the Centre is mainly to encourage continuing work into ethnomathematics. While ethnomathematics is regarded as an important aspect of school mathematics in PNG, there is a real need for it to be recorded. There is, in one sense, an urgency to preserve the cultural heritage of the people of PNG in their mathematics as well as their language. Individual student teachers are encouraged to explore their own and others’ mathematics. The Centre also wishes to develop research projects that draw together the diversity of measurement systems and other mathematical activities  used by different language groups (Bishop, 1988). Already Wilfred Kaleva and Francis Kari have considered the attitudes of teachers and students in PNG to ethnomathematics (doctoral theses, Monash University). The reader may be able to support a national researcher or have data that may be of interest to the Centre. The Glen Lean Ethnomathematics Centre can be contacted on <>.




Kay Owens, School of Education and Early Childhood Studies, Bankstown Campus, University of Western Sydney, Locked Bag 1797, Penrith South DC 1797, Australia. E-mail: <>.