Thesis Review

Ethnomathematics: Mid-Wahgi Counting Practices in Papua New Guinea

Charly Muke

Lecturer in Mathematics

1.0 Introduction

This paper aims to describe briefly the research report written for my Masters thesis. The study was done in line with the current education reform that the government of PNG endorsed and started to implement in 1992. The particular policy this study aimed to comply with, was the emphasis on the inclusion of local language and cultural knowledge from students’ background in the schooling system. The particular cultural knowledge this study targeted was ‘counting practices’ from Mid-Wahgi culture (the culture to which I belong). This culture is found in the Western Highlands Province of Papua New Guinea.

2.0 Literature review

The way teachers view school (academic) mathematics affects their attitude towards the subject and in turn affects their teaching. The dominant perspective of many teachers is categorised as an absolutist view (Ernest, 1996). This view is that school mathematics is not part of humanity and is generally culture-free. Teachers with such a view promote rote learning and assume that children come to school with no mathematical ideas.

However the philosophy of mathematics as a fallibillist phenomenon challenged this perspective and through its challenge lent support for ethnomathematics. From a fallibilist perspective, mathematics was no longer seen as a body of pure and abstract knowledge which existed in a superhuman, objective realm (Tymoczko, 1986). Instead, mathematics was associated with sets of social practices, each with its history, persons, institutions and social locations, symbolic forms, purposes and power relations (Ernest, 1996).

According to history, any form of knowledge, including mathematical knowledge, arose from necessity as experienced by humans (Eves & Newsom, 1966). The need to count, locate, measure, design, play and explain (Bishop, 1988) in their immediate social and physical environment, stimulated humans to establish valuable knowledge that could be considered mathematical.

For instance, the cultural origin of the content we know as geometry in academic or school mathematics, can be traced to the Egyptians. The need that stimulated this knowledge was the annual inundation of the Nile valley which forced the Egyptians to develop some knowledge for determining land marks and this is where the word geometry originated, which means ‘measurement of the earth’ (Eves & Newsom, 1966).

Similarly, the need to convert the land along the Tigris and Euphrates rivers into a rich agricultural region, encouraged the Babylonians to put together the valuable mathematical knowledge of engineering which helped them put up structures of drainage, irrigation and different ways of flood control (Eves & Newsom, 1966).

From such cultural bases of knowledge that were established to satisfy the immediate needs of ancient people, mathematics developed into an abstract form of knowledge, that created an image of absolute truth or of being a culture-free subject.

From history, it seemed clear that the early Greeks might have promoted this form of knowledge and view (Eves & Newsom, 1966). The Greeks, through their motivation and interest, that was not shared by all cultures of the world, converted the real life mathematics of Egyptians and Babylonians into an abstract form, by insisting that mathematical knowledge should be based on deductive reasoning, and by discouraging and even destroying other forms of establishing knowledge.

As Kline (1964, 1972) confirmed:

The Greeks insisted that all mathematics conclusions be established only by deductive reasoning. By their insistence on this method, the Greeks were discarding all rules, formulas, and procedures that had been obtained by experience, induction, or any other non-deductive method and that had been accepted in the body of mathematics for thousands of years preceding their civilization. It seemed then, that the Greeks were destroying rather than building …. (p.28)

From the beginning of what might be termed ‘formal education’, educationalists such as Plato, were faced with the question of what mathematics should be taught when mass education was beginning to spread rapidly (D’Ambrosio, 1997). According to D’Ambrosio (1997), the answer advocated by Plato was that it should be mathematics that maintains the economic and social structure, reminiscent of that given to artistocracy when a good training in mathematics was essential for preparing the elite. Of course, at the same time this allowed this elite to assume effective management of and power over the productive sector (D’Ambrosio, 1997). The best option available at that time to satisfy such interest, was seen to include mathematics that the Greeks promoted and given a place as ‘academic or school mathematics’. This mathematics was adopted throughout the world as one of the schooling subjects and also became entrenched in non-Western cultures that accepted a formal schooling system.

This meant that for many hundreds of years, or as long as the schooling existed in a society, the absolutist view dominated (Ernest, 1991). Such a view had many connotations, but the one that penalized a lot of non-western cultures was the impression that mathematics was far removed from the affairs and values of humanity and could be found in the head of a selected few: a person, a culture or a selected textbook.

Mathematical knowledge may be established through deductive reasoning that accomplished a degree of high certainty, but it did not supersede mathematical knowledge that was established through experience, induction, or reasoning by analogy (Kilne, 1953). For centuries the Egyptians used mathematical formulas drawn from experience. Had they waited for deductive proof, the pyramids at Gaza would not be squatting in the desert (Kline, 1953).

Similarly, the living mathematical knowledge of weaving in Africa studied by Gedes (1999), the different counting systems in Oceania and Papua New Guinea with counting cycles (bases) of 2, 4, 5, 6 or 10 by Lean (1992) and the logic of kin relations of Warlpiri in Australia studied by Ascher (1991), did not wait for such reasoning to arrive from Greece in order to answer questions arising in their daily life, and help them survive and make sense of their environment.

All forms of mathematical knowledge were important for the cultures in which they were embedded, despite the difference in their philosophical foundations. Therefore, mathematical ideas of all cultures including Mid-Wahgi, are equally important as school mathematics and must be included in schooling processes.

3.0 Methodology

This section begins by locating the study area, identifying the people and the language that was studied. The next section identifies the participants of the study, which were mostly school children and teachers. In addition, it reveals that this research used an ethnographic study design. The last two sections reveal the tools used to collect data and the research questions that guided this study.

3.1 Study location, people and language

The term Wahgi refers to the biggest and the most fertile valleys and also the most quiet, slow moving and meandering river, the mighty Wahgi, in the Western Highlands of Papua New Guinea. The people speak the Wahgi language, classified as South Wahgi and North Wahgi dialects (Muke, 1993). The speakers of South Wahgi dialect were the primary focus of this study. The locals call their language Yu Wooi. There are more than 30000 speakers of Yu Wooi (Burton, 1988). These people are organised into 18 contemporary sociopolitical and territorial groups. This study had participants from 9 tribes.

3.2 Participants in the study

The participants in the study were mostly students, teachers and a few local people. There were forty (40) students, thirty (30) teachers and three (3) local participants. From this number of participants, thirty-seven (37) students, twenty-one (21) teachers and three (3) locals were speakers of Yu Wooi, living in nine tribes. The others spoke the Kumai, Yu Nimbang and North Wahgi dialects.

3.3 Research design

The knowledge of counting in Wahgi was subsumed within a cultural context. The way this knowledge was to be studied required a research design that could consider the technique of long term observation and partial or complete integration, in order to gain a better understanding of the overall culture before any attempt was made to isolate and study any part of the knowledge for its own sake. The need to follow such a design of study was summarized by researchers as ‘Ethnographic Design’ (Banister, Burman, Parker, Taylor & Tindal, 1994).

In this research project, the researcher was from the culture and had a good understanding of the culture before he initiated the study. Therefore, he was an appropriate person to conduct ethnographic research in the chosen context.

3.4 Data collecting tools

The study used two types of tools to collect required data. They were by

· questionnaire

· interview.

Out of the 73 participants, 72 filled in questionnaires and one (researcher’s father) was interviewed.

3.5 Research questions

There were four research questions that guided this study.

· What are the Waghi counting systems of both the past and present in terms of:

verbal – counting names

symbols – both written & on artefacts

practice – other practices of counting?

· What are the social relationships (e.g. conceptual, understanding, values, beliefs etc.) in the culture associated with counting practices?

· In what contexts are Wahgi counting systems found?

· How can this information be used to contribute to the learning of mathematics in the school system?

4.0 Results

Only two areas of the findings will be reported in this section. The first will report on number names and expressions identified for selected numbers such as 1 to 10, 27, 250 and 1376. The second part will examine whether these number names and expressions were used within common events found in this culture. The events that were examined in this study were bride price ceremony (amb kolme), compensation (kong hi) and pig kill festival (kong gar).

Before presenting the most common number names and expressions, it is important to present the amount of variation identified.

4.1 Variation in number expression

A great deal of variation in expressing numbers was identified. It is categorised into two groups, by difference in pronunciation and in concept.

[Pronunciation]

Most variations were caused by different ways of pronunciation. To give an idea of such a variation, the following table outlines the number of responses found for each number.

Table 1. Number of responses for each number

No.	1	2	3	4	5	6	7	8	9	10
Tchrs	2	1	4	6	5	10	13	16	11	8
Stds	1	1	3	2	4	5	6	8	7	5

[Variation in concept]

When number expressions were grouped into the concept they were expressing, again there were variations. The following table shows how many different concepts were expressed for each number by the participants.

Table 2. Variation in number concepts

Number	1	2	3	4	5	6	7	8	9	10
Concepts	2	1	1	4	2	4	4	4	5	3

However, the most common number expressions are reported in the following section.

4.2 Number names for 1 to 10

Results revealed that the participants used a lot of varying names and expressions for each number. The most common expressions for numbers 1-10 are given on the following table.

Table 3. Number names in Yu Wooi

No Yu Wooi Concept

1 endi 1

2 tak 2

3 takendeka 2 + 1

4 taksi taksi 2 + 2

5 angek yem other hand

6 angek yemsi endi other hand + 1

7 angek yemsi tak other hand + 2

8 angek yemsi takendeka other hand + 3

9 angek yemsi taksi taksi other hand + 4

10 angek yem yem both hands

4.2 Number expressions for 27, 250 & 1376

For number 27, again there were variations, but the most common (52%) expression used for the concept was both hands and legs (20) + other hand (5) + tak (2).

There were also varying responses for numbers 250 and 1376. However, the participants commonly responded that they did not know how to use vernacular expression for big numbers like 250 (44%) and 1376 (62%).

4.3 Counting in cultural festivals

Wahgi cultural festivals such as pig kills, bride price or compensation could involve numbers in hundreds or thousands. However, the study discovered that the people did not use the number expressions or names to count a large number of items. Instead, they used a tally or matching system.

The items used for tally included body-parts tally and accessible materials found in their environment. The body parts tally included digit (finger)-tally, other common body parts such as head, nose, ears, hands etc, and whole person tally. The object tally included banana fruit (kong tau tom), food parcel (ango boki), tying knots (amb mongi kan gope), stakes (kong ond), money poles (ku ond) and matching pigs (kong tumdi tom).

5.0 Discussion

This section gives the definition of words and expression for numbers 1-10, variation found within the expressions, fractions discovered to be used in this culture, and the common practice of counting in some cultural festivals.

5.1 Definition of number names and expressions

Distinct number names can be identified from table 3. They are endi for one, tak for two and angek yem for number five. The expression used for five is the concept of five fingers that one or the other (yem) hand has.

Using these distinct numbers and the use of additive and multiplicative systems, other numbers are generated. A feature of the number expression that implicates this system is the suffix – si attached to them, except for numbers three and ten. In Yu Wooi, the suffix si means ‘take, combine or add’. This means that whenever it is used between two number names, they will be added to generate or express the concept of the targeted number. For example, taksi taksi means two and two (2 + 2) to express the concept of four. A similar feature is observed for numbers six, seven, eight and nine.

For number three (tak-ende-ka), the word -ende may be different but it gives a similar meaning as this suffix. This word means ‘to add another one to’. Therefore, when it is used in this case, it means add another one to two and this expresses number three. The structure of the expression also has another suffix- ka. It means ‘good’. It might mean that number three might bring good fortune. However, this suffix is commonly used in the names of certain sociopolitical groups or tribes.

For instance, in the Tangilka (researcher’s tribe), Kamblika, Koponka and Konombka, tribes that speak Yu Wooi language, it may be a polite way of naming.

The expression for ten (angek yem yem) does not have a suffix. The expression means both hands. It is understood to mean the same as two lots of five (2 x 5 = 10).

5.2 Combination of modern number names

An interesting feature was identified from the participants’ responses while trying to express bigger numbers, such as 250 and 1376. In many responses, there was a combination of modern words and Yu Wooi counting names. The common modern words used were ten, hundred and thousand from the English language. People used this combination to express numbers 250 and 1376. For instance, 250 was expressed as hundred taksi (2 x 100) and (ala) ten angek yemsi.

Similarly, 1376 was expressed as thousand endi si (1000), hundred takendeka si (300), ten angek yems (50) si tak (20), angek yemsi (5), endi (1).

The researcher believes that such adaptation by the culture and its language is a sign that the culture is responding to changes, in this case linguistically. Using such bases of understanding, further recommendations may be made in terms of the role of language in teaching and learning mathematics.

5.3 Tallying system

The purpose of studying bride price (amb kolme), compensation (kong hi) and pig kill (kong gar) ceremonies was to investigate whether or not number expressions in local language were used to count large number of items used within these important cultural events. This study discovered that number names/expressions in the local language were not used very much, but instead a tallying system was used. In other words, tally systems were the dominant practice used by the Wahgi people to quantify items in their hundreds or thousands during the three cultural festivals.

Quantities in hundreds and thousands were quantified using two mediums of tallying, namely body parts tallying and object tallying. To illustrate, the following section describes one of the systems of body tallying. It begins with digit/toe tally and ends with a whole person tally.

[Digit tally for units]

Digit-tally is obviously used in the early stages of Mid-Wahgi counting system. The distinct number name for five is a tally name for the hand (representing five fingers). In addition, a second method of digit tally was used. This system was only used when there was a lot of items to be quantified. This system involved tallying by two fingers at a time. The spoken word accompanied this action was ‘iraksi, iraksi, iraksi,…..’. The word comprises of two terms, irak-si, where irak means ‘this two’ and si means take. So, it means take this two. As this system used all fingers, this led to further digit and toe tally for tens.

[Digit and toe tally for tens]

When digit-tallying in two reached ten, which involved both hands, the people would tally each of their fingers and toes for a ten. As they physically point at the fingers and toes, they would say angek yem yem elsi, elsi, elsi,…. This means, both hands (angek yem yem), equal this one (el-si), this one, this one, …. When they tallied a ten for each of their fingers and toes, it helped them express the quantity of two hundred (200). The Mid-Wahgi people went further from here by tallying whole person.

[Whole person tally for 200]

When fingers and toes of both hands and legs were tallied for two hundred, the way forward was tallying a whole person for this quantity. In other words, the concept for two hundred (200) was expressed by tallying a whole person. This meant that to quantify 1000 items, the Mid-Wahgi people would tally five people, 2000- ten people and so on.

5.4 Fractions

While studying the use of counting systems in the three cultural festivals, it became evident that Mid-Wahgi people had known and used two common fractions. They were quarters (¼) and half (½). The respective names given were kekep – ¼ and arhka – ½. They also understood the sum of fractions.

They seem to know that two quarters (¼ + ¼ = ½) gave half, four quarters gave two halves or one whole pig.

Discovery of the use of fractions was identified within the sharing of pig meat. The fraction names would refer to fractions or pork meat or any four-legged animal.

6.0 Conclusion

This section presents a summary of the paper and a recommendation based on the findings of this study.

6.1 Summary

It was obvious through this study that Wahgi people’s desire to count was aided by their well-established tallying system. This study did not identify traditional number names for big numbers such as 250 and 1376. Yet, within the three cultural festivals, items such as pigs were used in hundreds and thousands. It was an honest response when the participants expressed that they did not know how to express these numbers using their local language, because, the normal practice to quantify items in bigger numbers was tallying. For instance, this study identified six tally systems that were used during the three cultural festivals that assisted in quantifying items in hundreds and thousands.

The tendency to tally is indicated early in the counting system. Within the expressions for numbers 1-10, three distinct number expressions were identified. However, expression for number five could not be one of them, because it is a phrase expressing the act to tally one hand for the concept for five. This tally system continued to assist in generating numbers 6, 7, 8, 9, 10 and 27.

There may be many reasons why Mid-Wahgi language had distinct words for only numbers one and two, and did not progress to establishing names for other numbers. It may be hypothesised that they had no motivation to establish further number names, because the tallying system already satisfied the need to quantify.

6.2 Recommendation

From this study, it is recommended that the following number names and expressions be used for schooling purposes in situations involving Wahgi children. The pattern could be used to generate other numbers.

Number Expression in Yu Wooi

¼ kekep

½ arka

1 endi

2 tak

3 takendeka

4 taksi taksi

5 angek yem

6 angek yemsi endi

7 angek yemsi tak

8 angek yemsi takendeka

9 angek yemsi taksi taksi

10 angek yem yem

20 simnb angek yem yem

100 hundred endi

200 hundred tak

300 hundred takendeka

400 hundred taksi taksi

500 hundred angek yem

600 hundred angek yemsi endi

700 hundred angek yemsi tak

800 hundred angek yemsi takendeka

900 hundred angek yemsi taksi taksi

1000 hundred angek yem yem

or thousand endi

Source document

Muke, C. (2000). Ethnomathematics: Mid-Wahgi Counting Practices in Papua New Guinea. (Unpublished Masters thesis). University of Waikato, New Zealand.