RATIO AND PROPORTION: BARTER OF SAGO FOR FISH IN TAU DIALECT OF KWANGA LANGUAGE, WEST SEPIK

Tom Kaviambu, Antonia Kachau, Patrick Nembeli.

Introduction

Just as in many traditional ways of applying mathematics to counting things or the addition and subtraction of anything, this lesson attempts to introduce the ways which our people use. In the Sepik provinces we generally have the same usage of our number systems or the counting systems. Mainly they are used in the following areas: addition, subtraction and ratio and proportion. Not much is known about the operation of multiplication and division. In this ethnomathematics lesson we will concentrate on ratio and proportion used in the barter system and the money system. The main objective is that students would be able to apply the traditional ideas of ratio and proportion into the modern mathematical operation of ratio and proportion. They would also appreciate the relationship between the traditional method and the modern method.

In addition we would like to introduce our counting system in the Tau language (spoken along the coastal area of Wewak). The Tau counting system is a base 5 system. All the other numbers can be made out of the basic base 5 numerals. Therefore with the understanding of the Tau number system students can also translate the traditional numerals to the modern ones and apply them in any of the operations except in multiplication and in division. However it is possible to improvise on the Tau system to include multiplication and division. Indeed, it is often claimed that the absence of terms for multiplication and division in the Tau language indicates that Tau culture is unaware of multiplication and division. However, the barter of numbers of bags of sago for numbers of fish indicates an appreciation for ratio and proportion which presupposes the concepts of multiplication and division. For the time being we will only include the numbers in the ratio and proportion by comparing the same quantity and one quantity with the other in terms of size, quality and quantity of mainly food materials used in trade and traditional money.

Description of Tau Traditional Mathematics.

The traditional method of ratio and proportion can be seen in the barter system and the traditional money system. The barter system as a means of trade was practiced all through out the Melanesian society and it has a prominent value in the many cultures and groups of people. In the Sepik provinces mainly food materials and the traditional money for pigs and for bride price are exchanged proportionally. That means that the people do not only give and take, they base their judgment on these three forms of standards; size for quantity, size for quality, and quality for quality.

In the pictorial language the proportion of size for quality means, for instance, that a cylindrical shape of prepared sago called samau (approximately two feet in length or primitively taken to be the length from the sole of the feet to the knee) would be exchanged for either ten small fish or five large fish. The samau is said to be the standard length and it is also traded with the inland people for yams and mami (a form of yam but is shorter than the real yam). It would be one samau for say five or more mami or so depending on their best judgment and the standards they set.

The other areas where ratio and proportion are used most is the money system. That also is common throughout the Sepik provinces. The traditional shell money comes in various sizes and values. The shell money is used for bride price and pig payment. The values for these shells correspond to the size of the shells. Therefore the values range from the smallest to the biggest (kalem wuru, kalem, kalem wusu, kalem wasung wasung, wurul tai). Refer to the number system for the values. Note that there are no smaller units. Te shell money can be exchanged for shell money too, or instance exchanging two little shells for a large one.

The Tau counting system

Hindu-Arabic system Tau system

1 tai
2 wuru
3 tual
4 viat
5 viliri
6 viliri tai
7 viliri wuru
8 viliri tual
9 viliri viat
10 kalem tai
11 kalem tai tai
12 kalem tai wuru.

Note that the other numbers are taken from the base 5 system.

The payment of pigs is based on standard said to be the height of a man (it is not sure exactly what the height of a man means) But usually the girth of a pig is taken and then it is used as a standard for the next purchase. Depending on that girth length, the value of the pig is determined.

In the case of the bride price usually the parents of the lady value the lady according to the sort of character their daughter possesses. If she is a very hard working lady, careful and would be productive in bearing children then she would be worth more. That is an example of money for quality.

LESSON PLAN

Grade 7

Objective: To use traditional idea of ratio and proportion in the modern mathematical operation of ratio and proportion.

Materials: Cards shapes of varies sizes of fish, a cylindrical shape to represent samau.

Teaching procedure

Introduction

Ask the students if they had seen their people exchanging one thing for another. Did they just give and take, or did they carefully look at the other person’s item before the exchange? What conclusions should be reached before they can exchange their goods? What should the agreement be based upon< or what criteria are agreed upon for exchange?

Tell students of an interesting mathematical idea that is used between villages in the past and almost every day today. That is exchanging one thing for another in trade according to ratio and proportion either money for goods, or goods for goods.

Tell the students of the Tau system and give some related real life examples.

Body

We will work only with samau and fish or yams and mami. Note the standard for the exchange exercise is
1 samau is equal to 5 fish (1 samau = 5 fish)
1 samau is equal to 5 mami (1 samau = 5 taro)
(Tai samau raniak fish viliri, mami viliri.)

Note that the number of fish, samau and mami can vary depending on the quality, size and quantity.

Based upon the above standards we will try some exercises.

Exercises

1. a) 3 samau = _____ fish. c) 20 samau = ____ fish.
b) 6 samau = _____ fish. d) 11 samau = ____ fish.

2. a) ___ samau = 10 fish. c) ___ samau = 25 fish.
b) ___ samau = 15 fish. d) ___ samau = 100 fish.

3. Word problems

a) I have 3 samau and my friend has 10 fish. But we want only samau. How many samau would we have if we traded our samau for fish?
b) We all combine our fish together and counted it to be 300 Early in the morning we went to the mountain people and traded with them for samau. How many samau would we return with?
c) One of the trading partners complained that he wants 10 fish for his quality samau so he went to everyone asking and finally he met someone. His friend has 30 well smoked fish so they traded. How many samau would his friend got for his 30 fish? How many fish would he get for 2 samau.

Conclusion

To understand that topic of ratio and proportion the students should have to apply what they have known about their own exchange system. The idea of 1 samau is to 5 fish in their exchange system or barter system uses the same principle of ratio and proportion in the modern operation of ratio and proportion. Therefore after a short reintroduction to the idea the students can appreciate the relationship between their usage of ratio and proportion its usage in modern math.

Thus with the stated reason above we think the goal of our lesson would be achieved. Not only that but the students will then be geared and equipped to do more of the harder ratio and proportion questions that will come in the lesson.

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