Chapter 5

 

Ethnomathematical Practices in PNG

 

 

Papua New Guinea is a living manifestation of Well’s time machine. In half a day one can witness the application of the most advanced Space Age technology (in telecommunications, for example) and the application of a Neolithic technology that has remained unaltered for centuries. It is a nation in transition. The direction is clear, the pace uncertain. The motivating forces are partly international and partly local. Papua New Guineans are in no sense ashamed or rejecting of their roots; yet they are also enthusiastic about the products of the industrialised world. [Lancy, 1983, p. 11]

 

 

There is probably no better way to describe the cultural, political and educational context in PNG than the above description provided by Lancy (1983) in his introductory remarks to chapter 2 of his book, Cross-Cultural Studies in Cognition and Mathematics. It is a description of PNG which is somewhat typical of a developing country which is both enthusiastic and determined to play its part within the international community of nations. On the other hand however, it is also a country of immense cultural diversity, comprising well over 700 distinct languages, traditions, rich cultural values and belief systems. Moreover, when coupled with many of its small indigenous rural communities separated by rivers and rugged terrain, these physical factors together have simultaneously provided both the colonial government and now the independent government of PNG with the cultural richness that is very unique, and an educational experience that is perhaps the most challenging one when compared with similar situations elsewhere.

 

5.1        Background

For more than 100 years, PNG has served as a natural laboratory with many of its exotic flora and fauna contributing much in stimulating comparative biology. In medicine, Carelton Gajdusek (cited in Lancy, 1983) won the Nobel Prize mainly because of the strength of his discovery of kuru, a disease found only in PNG. Part of any introductory textbook in anthropology will be devoted to studies done in PNG. The country is therefore very much alive today particularly in terms of basic research in many fields with this situation showing no signs of abating (Lancy, 1983). Comparative research in both education in general and mathematics are no exception in this regard, particularly in terms of providing the relevant data in analysing the cultural influences on the teaching-learning process not only in PNG, but also in educational systems elsewhere. This is particularly true for mathematics education in the light of studies conducted in PNG analysing the impact of language in the learning of mathematics (Clarkson, 1992; Lean, Clements & del Campo, 1990; Souviney, 1983) and on mathematics and cognition (Bishop, 1979; Lancy, 1983; Saxe, 1981, 1982). The question of where these studies lead then arises.

It is in partly answering the above question and in the light of the implications of the current practices in mathematics education noted in the preceding chapters that this chapter further explores the notion of ethnomathematics with specific reference to its practices in the cultural context of PNG. As defined in section 4.2.4, ethnomathematics is the study, in its cultural context, of knowledge as cultural adaptation to the world in which people live. Therefore, if the cultural diversity in the country is taken as any indication, the Papua New Guinean ethnomathematical models of reality are many and varied, and continue to form an integral component of its people’s view. An overview of ethnomathematical practices in PNG reveals that it deals with reality as an integrated whole rather than focusing on its parts (that is, it is holistic in nature). It classifies and explains cultural phenomena with reference to their interaction with human beings (it is anthropocentric), and is concerned with practical applications and results more than with abstract theorising (Vlaardingerbroek, 1991).

 

5.2        Cultural Tools and Mathematical Thinking

Before proceeding to looking at specific ethnomathematical practices in PNG, it is necessary that a background on the notion of cultural tools be provided basically for two reasons. The first reason is to reveal their role in the development of mathematical thinking among members of any culture group, and secondly to use such a role as a theoretical base for the teaching-learning process with respect to mathematics education in PNG.

The beginning of tool use has been seen as one of the significant events in human evolutionary history. As tools became more complex, more and more effort was required to transmit to each new generation the skills for using these tools, a responsibility that has fallen with culture. This is noted by Bruner (cited in Stigler & Baranes, 1988) who argues that the only means whereby man could fill his evolutionary niche was through the cultural transmission of the skills necessary for the use of priorly invented techniques, implements and devices. Thus, one of the most powerful ways in which culture can influence mathematical thinking is by providing tools for solving problems and representing mathematical content. Taken in this context, Stigler and Baranes (1988) therefore suggest that tools can be general purpose like that of language, which is the prime example of a general tool which can also be applied to the domain of mathematics, or they can be specialised artefacts, concrete representations, or procedures passed down through education. A good example of a specialised tool found in nearly all parts of PNG is the wooden digging stick (or dibber), an ancient and very simple yet very effective instrument of agriculture (Lancy, 1983). Once the forest has been cut and burned, the digging stick is used in every phase of gardening from soil preparation through weeding to harvesting.

While there is still controversy surrounding the research on the acquisition and use of cultural tools in terms of their relevance in understanding cognitive development in general, Stigler and Baranes (1988) argue that “much, though perhaps not all, of what moves cognitive development along is the acquisition of skills in using language and other tools” (p. 262). Likewise in their view, within the domain of mathematics, much of what students come to know about mathematics results from their learning to skilfully use specific representations and algorithms. This view of cognition is well-expressed by Brown, Collins and Duguid (1989) who argued that:

Tools share several significant features with knowledge: They can only be fully understood through use, and using them entails both changing the user’s view of the world and adopting the belief system of the culture in which they are used. (p. 32)

 

It has to be pointed out that, if the background information provided in chapter 2 is taken as any indication of the staggering cultural diversity in PNG, it logically suggests that it is beyond the scope of this study to describe in detail all the possible ethnomathematical practices that could be identified in each of the 700 cultural groups identified. Furthermore, it has to be also noted that cultural differences or for that matter similarities of many ethnomathematical practices in PNG do not only exist across different cultures, but  within a culture these practices also vary depending on their context and purpose (Nunes, 1992).

In the light of the above background and on the basis of  the six ‘universal’ mathematical activities (Bishop, 1991b) described in chapter 4, the next two sections will discuss traditional or cultural activities of mathematical practices namely: (1) counting and measuring; and (2) geometrical activities of basket and blind weaving. These cultural activities, though not representative of all types of mathematical activities within the cultural context of PNG, were chosen not only because of their illustrative role, but most importantly their significant role in generating further mathematical ideas and inferences which could be subsequently incorporated into the mathematics classrooms in PNG.

 

5.3        Cultural Practices of Counting and Measuring

Within many of the past and present individual traditional societies in PNG, the cultural practices of ‘counting’ and ‘measuring’ have featured so prominently that they have become part and parcel of many of the day to day societal routines and activities as well as forming an integral part of its people’s views. Moreover it has to be pointed out that traditionally these two practices are concurrently used particularly in situations where one of them is unable to account for certain aspects of objects the are to be quantified, i.e. determining an amount of water. This aspect of the two activities is reflected by this section in the use of both activities simultaneously. Based on the six universal activities (counting, locating, measuring, designing, playing, explaining) described in section 4.2.2, the activity of counting refers to a systematic way to compare and order discrete objects (Bishop, 1991b). It may involve body or finger counting, tallying or using objects or strings to record, or special number names. Measuring activities involve quantifying qualities such as length and weight for the purpose of comparing and ordering objects (Saxe, 1981; 1982).

In PNG some specific cultural situations in which these two mathematical practices have become more apparent include the important traditional exchange ceremonies such as bride price payments, pig killing festivals, compensation payments and the traditional barter system that has existed for centuries before European contact and influence. It has to be pointed out that because of the significant role of counting and measuring practices in the day to day routine activities in many societies in PNG, it is uncommon to find one of them without at least some basic means or ways of performing these two practices. In this respect, they are therefore seen to be culturally fundamental practices confirming Bishop’s (1991a) analyses of universal mathematical activities noted in section 4.2.2. Moreover, these two activities are also used in determining a person’s wealth and status in society, mainly during ceremonies of bride price and traditional compensation payments. Such determination of status is based on how much one is able to offer in terms of materials and goods such as pigs, cassowaries, baskets of food items and artefacts, just to name a few. In other words, the more a person is able to offer, the higher the status of the person within the society.

Such determination of wealth and status in society undoubtedly involved, at least in this author’s view, some basic form of mental representation, for without them this would not be possible. In this context such activities can be considered as ways of representing selected aspects of objects and situations (Nunes, 1992). In order to measure, one has to choose what dimension will be quantified, for example, a class of objects when counting or a quality like length or weight when measuring. Like any form of representation, it is argued by researchers (e.g. Julie, 1993; Lancy, 1983; Nunes, 1992; Saxe, 1981) that the initial choice of what to represent and what to ignore involves an abstraction. This simply means that everything else will be ignored except the aspect that is to be quantified.

As noted in the background section of this chapter, because of the ‘holistic’ characteristics of many of the ethnomathematical practices in PNG, the activities of counting or measuring are usually carried out for some larger purposes. For example, during special ceremonies or festivals, the contributions made by different individuals from a culture group is taken as a contribution representing that particular culture group as a whole. Examples of such practices can also be found in the present-day activities involving money. For example, when making a particular purchase, one needs to count the money to determine if there is enough money to make the purchase. Also, a table is measured to determine the size of the tablecloth that will cover it properly. Carrying out such activities is what  makes counting and measuring very meaningful (Nunes, 1992).

There are two areas in mathematics namely, the notions of logic and a base systems of numeration where the cultural activities of counting and measuring are seen to have contributed significantly. Despite some variations in counting and measuring activities as determined by specific contexts and purposes, basic underlying logic seem to be present in all situations (Nunes, 1992). In thoroughly analysing the development of counting skills among children, Gelman and Galistel (cited in Nunes, 1992) specified four basic logical principles that must be satisfied if an activity is to be classified as counting. These principles are summarised as (1) establishing a one-to-one correspondence between the things to be counted and the counting labels, (2) maintaining counting labels in a fixed order, (3) recognising the irrelevance of the order in which the objects are counted, and (4) applying the cardinality principle, that is using the last label to represent the number of objects in the set .

While these four principles have a strong logical appeal, it is argued by Nunes (1992) that a system based only on these logical principles has limited application because of the difficulty in obeying these principles in the absence of a culturally organised numeration system. For example, with regard to principle 2, it is questionable just how many labels one can remember in a fixed order if the labels are unrelated and are not part of a system that makes their production easy. Because of the limited capacity of the human mind to memorise ordered lists, this therefore necessitates the use of some structural support (Nunes). Thus, with some form of numeration system, a counting system can go indefinitely otherwise it will be restricted to only low numbers.

This then leads to the question of how different cultures have addressed the problem of memory load in counting. In particular, the research studies conducted by Lancy (1983) and Saxe (1981) among the Kewa and Oksapmin people of PNG indicated how these two cultural groups have developed their numeration systems to help them maintain fixed order through the use of names of body parts as labels in counting. This is a cultural and conventional solution to the problem of memory load since the body parts to be named and the order in which they are to be used must be agreed upon (Nunes, 1992). Interestingly, the two studies have also indicated that there are some body parts chosen which do not have clearly identifiable labels in many Western cultures, notably the three locations on the forearm and six locations between the shoulder and the neck. The use of these body parts has enabled the Kewa to count up to 68 (Lancy, 1983).

Though closely related to memory load in counting, the second area where counting and measuring activities have contributed relate to the concept of a ‘base’ system in counting. A base in a numeration system refers to a grouping scheme used to reorganise counting. To be defined as a base in a numeration system is to choose a conventional unit to be used in counting. Thus a base-numeration system involves counting natural objects, organising them in conventional groups that become new counting units, and grasping the semantically complex structure underlying the numeration system (Lancy, 1983; Nunes, 1992). The notion of a base system is illustrated by the three examples of traditional base-numeration systems in PNG which are described below in Table 5.3 with their corresponding equivalent numerals in the Hindu-Arabic system. It has to be pointed out that many of the counting systems in PNG are characterised by lower number labels as well as a base system. The three numeration systems are namely, (1) the Kote numeration system from the Morobe Province (this author’s traditional counting system), (2) the Kuanua from East New Britain Province (Amo, Podarua & Polikran, 1993), and (3) the Enga counting system from Enga Province (Leme & Jikap, 1993).

Looking at Table 5.3, it is obvious that ‘Kote’ is a ‘base-five’ numeration system while ‘Kuanua’ and ‘Enga’ are ‘base-ten’ numeration systems. Thus, the notion of a ‘base’ in a numeration system not only enables the learners to capture the meaning and generate number labels indefinitely, but also enables them to think of the natural objects as they are counted as well as understanding the structure of meaning in the numeration system (Nunes, 1992).

 

Table 5.3          Three traditional base-numeration systems of PNG

 

Numeral

Kote

Kuanua

Enga

   1

moc

tikai

mendai

   2

jajahec

aurua

lapo

   3

jahecomoc

autul

tepo

   4

jahecojahec

aivat

kituma

   5

memoc           

ailima

kondape

   6

memoc o moc

laptikai

yangisimange

   7

memoc o jajahec

lavurua

sakatia

   8

memoc o jahecomoc

lavutul

mangelap

   9

memoc o jahecojahec

lavuvat

tukulap

   10

mejajahec

avinum

akalita

   11

mejajahec o moc

avinum ma tikai

akalita kisa mendai

   12

mejajahec o jajahec

avinum ma  autul

akalita kisa lapo

 

 

One of the notable features concerning all three numeration systems described in Table 5.3 is that they do not have a word label or even a symbol representing the notion of ‘zero’, a feature which is almost universal for all traditional counting systems in PNG. This lack of representation of zero further signifies the fact that nearly all the counting systems in PNG, or for that matter other cultural activities, are based on the contextual reality of providing the appropriate meaning rather than on abstract theorising which somewhat characterises many of the ‘Western’ thoughts. It is therefore no surprise to find many Papua New Guinean students having difficulties understanding the presentation of abstract mathematical knowledge because of the absence of contextual meaning which characterise many of the mathematics classrooms in PNG.

 

5.4        Cultural Activities of Geometry in PNG

As highlighted in section 5.2, the notion of ‘tool use’ has been seen as a most significant event in any human evolutionary and cultural history. This is particularly apparent from a recent cognitive point of view where it is considered as a significant factor in the development of mathematical thinking among members of specific cultural group (Stigler & Baranes, 1988). This idea is further highlighted by White (cited in Bishop, 1991a) in his classification of the cultural components described in section 4.2.3, in which he argued that the technological factor is the most basic one with all other factors namely, ideological, sociological and sentimental all dependent upon it. That is to say that because the role of technology is primarily on shaping the environment, it therefore at least in a general way determines the form and content of the other three factors. The connections between technology and the cultural activities of geometry is fundamental because the widespread cultural activities of geometry somewhat epitomises both the notion and the role of technology in society.

Based on the six universal activities (Bishop, 1991b) described in section 4.2.2, the cultural activities of geometry can be classified under the ‘designing’ activities which are concerned with  the ‘manufactured’ objects, artefacts and technology which all cultures create for their home life, for trade, for adornment, for warfare, for games and even for religious purposes (Bishop, 1991b). To some extent, the activities of ‘designing’ can also apply to the spatial environment like in the case of constructing houses, villages, gardens, fields, roads and the two cultural activities of ‘basket’ and ‘blind’ weaving considered in this section. There are many and varied examples of geometrical activities found within the cultural context of PNG. For reasons of space limitations, only two activities namely, ‘basket weaving’ used for various purposes and ‘blind weaving’ for constructing the walls of the house have been chosen.

These two activities, though very similar in nature, are chosen mainly because of the significant involvement of geometric patterns in the process of weaving. Moreover, they also involve, on the part of the weavers, the conceptualisation and visual imagination of the ‘spatial’ environment described in section 4.2.2 under one of the six universal activities namely, ‘locating’ (Bishop, 1991a; 1991b). It is perhaps also true to say that there is no better way to prove the weaver’s ability to conceptualise the spatial environment than to actually see such representation reproduced through the geometrical patterns such as those found on the baskets and wall blinds. As Bishop (1991b, p. 39) clearly put it:

Designing involves imagining nature without the ‘unnecessary’ parts and perhaps even emphasising some aspects more than others. To a great extent, ..., design concerns abstracting a shape from the natural environment.

 

The underlying idea of designing is that it involves transforming a part of nature, that is, taking some natural phenomenon, be it wood, clay or ground, and fashioning it into something else, perhaps a carved artefact, a clay pot or a garden (Bishop, 1991b). In other words, it involves imposing a particular structure on nature.

The above suggestion is particularly true in the case of basket and blind weaving activities in PNG. The weaver’s ability to conceptualise the spatial environment is particularly crucial at the earlier stage of the weaving process because it involves mentally ‘locating’ the points of intersections of the strips of coconut leaves (in the case of basket weaving) and bamboo or sago stalks (in the case of blind weaving) in forming the required pattern of various geometrical shapes. An example of one of these geometrical patterns commonly found on the blind walls of houses in many parts of the coastal and highlands regions of PNG is illustrated below in Figure 5.4 and made from strips of paper woven by the author.


 


Figure 5.4         PNG traditional patterns of geometry

 

A number of mathematical ideas can be derived from such patterns. While the actual finished product is not mathematically important, it is the planning, the structure, the imagined shape, the perceived spatial relationship between object and purpose, the abstracted form and the abstracting process that are of significant importance to mathematics education (Ascher & D’Ambrosio, 1994; Bishop, 1991a; 1991b). Furthermore, these geometrical patterns of shapes can also be investigated further for their specific properties such as lines of symmetry, angles, number of sides and so on. Further analysis of these patterns can also lead into the idea of number patterns as well as arithmetic sequences and series. For example, in the above pattern in Figure 5.4, one can find that the number of strips to be covered or lifted by the weaver in forming the pattern is based on the set of odd numbers 1, 3, 5, 7, . . . which further illustrates the rich contextual meaning such cultural activities can offer.

As in the case of the activities of counting and measuring noted in section 5.3, there are slight variations across different cultural groups particularly in terms of the patterns or designs used. These designs or patterns are of significant importance to individual cultural groups because they represent specific ideas, histories and cultural phenomena or stories concerning specific cultural group. Traditionally, it is considered unethical to use patterns of other cultural groups without first obtaining the permission from the owners of such patterns, a feature that is somewhat equivalent to the notion of ‘copy right’ found within ‘Western’ societies.

 

5.5        Conclusion

The discussion of two ethnomathematical practices considered in this chapter further demonstrates the notion that such cultural practices may be useful in the teaching of many of the current abstract mathematical ideas that are found to be difficult by many learners in PNG. This proposition is supported by the recent cognitive view which suggests that much of what students come to know about mathematics is a result of their learning to skilfully use specific representations and algorithms as embedded within the appropriate cultural activities found within their social and cultural environment.

Although the two cultural practices examined in this chapter are not representative of all the cultural practices of mathematics in PNG, they however illustrate the type of cultural practices that could be incorporated into the actual teaching practices of mathematics in the classroom. For this to happen, it is therefore necessary that a meaningful dialogue be established between the teacher and learners. This can be achieved via the teacher-learner interactions which will in turn provide a learning environment characterised by discussion and reflection, thus developing learners’ critical thinking in mathematics, one of the key features that is lacking in current practices in PNG. Thus the role of a teacher in such an approach will be seen as a facilitator of the learning process rather than an authority in the classroom. Moreover, such an approach will also increase the self-esteem on the part of the learners. The proposition of such an approach will become clearer during the discussion in the next chapter which will focus on the implications of an ethnomathematical approach to mathematics education in terms of the curriculum, teaching process and teacher education.

Finally, it has to be pointed out that while the two cultural practices used in this chapter are illustrative of what can be incorporated into the classroom environment, it is also difficult to objectify many of the cultural practices in PNG in terms of curriculum content because of their ‘holistic’ and ‘religious’ characteristics involving mental imaginations. Moreover, they also vary from one cultural group to another based on purpose and context. Some of these practices include among others, the traditional coastal navigation, many cultural phenomena of explaining the environmental situations such as the thunder and lightning, practices of spiritual worship, and practices of traditional dancing just to name a few. Educationally, the holistic and religious aspects of many cultural activities in PNG can be both an advantage and a disadvantage in that they can become a barrier as well as providing contrasting situations for comparing different mathematical practices found in various cultural groups.