Kay Owens

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

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.

The impact of a teacher education culture-based project on identity as a mathematically thinking teacher

Identity as a mathematics teacher is enhanced when a teacher explores the cultural setting of their mathematics. The reports of projects that link culture and mathematics were analysed to explore the impact of sociocultural situations together with affective and cognitive aspects of self-regulation on identity. The reports were written by pre-service and in-service, mainly secondary, teachers at the University of Goroka, Papua New Guinea. While all 239 reports were read and considered in terms of the thesis of this article, 60 (25%) were analysed in detail developing the argument. The results indicate the strengths of such projects to take account of cultural knowledge when colonised education systems are further modified through reforms that emphasise culture. The significance for teacher education is the role that an activity which links culture and school mathematics plays in building values and identities.

Covering shapes with tiles: Primary students’ visualisation and drawing

Students’ early area concepts were investigated by an analysis of responses to a worksheet of items that involved visualising the tiling of given figures. Students in Years 2 and 4 in four schools attempted the items on three occasions and some of the students completed ten classroom spatial activities. Half the students had difficulty visualising the tiling of shapes, but students who participated in spatial activities were generally more successful in determining the number of tiles that would cover a shape. Students’ drawings indicated a varying awareness of structural features such as alignment and tile size. Students who drew the tilings were more likely to be successful on the items involving trapezia. The tiling items were part of a test of spatial thinking, Thinking About 2D Shapes, and scores on the overall test were very highly correlated with results for the tiling items.

Visuospatial Reasoning in Contexts with Digital Technology

The ecocultural perspective on visuospatial reasoning was established by considering Indigenous communities and their practices and appropriate schooling and other diverse ecocultural practices illustrating visuospatial reasoning. However, today is the digital age so does research on visuospatial reasoning support this ecocultural perspective. Since most of the research on visuospatial reasoning has been focussed on dynamic computer-generated images, it is important to consider digital technological facilities as an ecological context. How can an ecocultural perspective of visuospatial reasoning enhance our understanding and valuing of visuospatial reasoning? In this chapter we consider how a computer-facilitated learning age influences an ecocultural identity and both self-regulation and visuospatial reasoning. It is then important to consider how these personal dispositions impact on mathematical identity.

Body-Part Tally Systems

Unique systems of counting found in Papua New Guinea (PNG), West Papua, Torres Strait Islands Australia and south-east Australia involving tallying against body parts. These differ from the digit-tally systems that only involve fingers and toes. In body-part tally systems, words for 1-4 are likely to be specific finger names, the word for 5 or thereabouts is thumb rather than hand and then other parts of the arm are used. Most of these systems reach a mid-point, usually on the head and then go down the other side in a symmetrical fashion. However, these systems vary considerably. Some are truncated, and the number in the cycles can vary considerably from 13 to 59. The last counting word for the cycle could be the last tally point or a word to indicate total. These words are believed to only involve the upper body parts with one system involving the outside of the legs. To count beyond the final tally point would require going to a second person, at least abstractly if not as a person. These systems often occur in 2-cycle system areas and there are some occasions where other systems exist. In terms of spread of counting systems, these appear to only occur in pockets in certain areas and not to be widespread.

An Overview of the Studies, Papua New Guinea, Oceania, Languages and Migrations

This multidisciplinary study draws on archaeological information, linguistics, and an understanding of mathematics. A summary of the background archaeology for New Guinea and Oceania provides some evidence of the longevity of these cultures and the archaeological evidence for the spread of languages in New Guinea and Oceania. The diversity of language groups is a result of movement, colonisation, influence and innovation over time. The overview presented in this chapter permits the reader to link the pursuing discussion in a time and place. The chapter finishes with an overview of the book that sets out the diversity of counting system cycles, where they are established, and how they may have developed.

Number and Counting in Context, Classifications and Large Numbers

This chapter begins the development of the importance of number in Indigenous societies. It takes a number of case studies to illustrate the role of counting in these societies and the influences of neighbouring tribes with different, e.g. Papuan language contexts on an Austronesian language. Counting is part of mathematical activities and is used for decision-making, comparison, and relationships. One aspect that the collation and analysis of counting systems in this region has highlighted is the notion of qualitative decisions about number. Another is the link with classifiers used in several areas and having considerable variation across systems. Other issues such as large numbers, alternate representations, and changes in language and practice over the years are addressed.

Testing the Diffusion Theory

This chapter addresses the issue of how different counting systems occurred and in particular the theory of counting systems spreading from a centre. The most comprehensive theory of this kind before 1990 was that of Seidenberg. This theory is expounded and then several queries are raised. In general, the argument is put that the counting systems of Papua New Guinea and Oceania did not spread from the Middle East and the prominence of so-called neo-2 cycles and 10 cycles cannot be supported.

10-Cycle Systems

There are among the Austronesian (AN) and Non-Austronesian (NAN) languages of Papua New Guinea and Oceania, examples of 10-cycle systems that do not have a 5-cycle or 20-cycle system. Of these, there are a number that have (10, 20) or (10, 20, 60) cycle and given the ancient origins of these cultures, it is evident that the Babylonians were not the only people to have 60-cycle systems nor that they were diffused from Babylon but rather were unusual, unique local developments. In addition, some systems have a replacement for 10 in higher numbers and others regard the specific group of 10 as more important than the number 10 itself leading to a range of words for 10 as in Motu. Lean (1992) also indicated that pairs may dominate and there are some systems in which 10 is likely to refer to 10 pairs. In addition, a number of systems refer to numbers such as 8 as 4 × 2, 9 = 4 × 2 + 1 while others in the second pentad refer to the number required to reach the complete group of 10 such as 8 = 10-2. Furthermore, a 10-cycle system is by no means the norm for AN languages. Interestingly, classifiers for number groups are also found in a range of languages from those in Manus (AN), to NAN languages in Bougainville to Polynesian languages of Micronesia as well as in the Solomons (Chapter 8 has more details.). Comparative data indicate the probability of Proto Polynesian language having this kind of classifier rather than it occurring discretely.

Towards a Prehistory of Number

The previous chapters focused on the cycles of the languages, but this chapter summarises the similarities and differences of the various phyla to indicate how different systems influence others. This provides evidence of much earlier dates for people in this region and a possible timeline with the known movements for counting systems to have been modified especially from the Austronesian 10-cycle. As a result, it is possible to dispute Seidenberg’s timeline and theory of the diffusion of number from civilisations in the Middle East. An alternate timeline is provided based on known events such as the end of the ice-age, uplifts and volcanic activity in island regions, the spread of Lapita pottery, and cultural developments. The variation in counting systems also provides evidence when considered in terms of the other evidence from linguistics, archaeology, geography, genetics and biogeography. Thus a possible timeline for the history of number is suggested indicating the probable antiquity of these counting systems from around 40 000 to 5 000 years ago in Melanesia.