Counting on Your Body in Papua New Guinea

James Rauff
Milliken University

 

1. Counting.

Counting is usually the first thing that comes to mind when people think of mathematics.  Indeed, counting is the first bit of school mathematics learned by children. The NCTM's Principles and Standards for School Mathematics (2000) asserts that counting is "the foundation for students' early work with number" (p.79).  Counting is also regarded as one of the earliest manifestations of mathematics, but is usually passed by quickly in standard histories in order to get on to the meatier mathematics.  Katz (1998) devotes a mere four pages (out of  850) to counting while Burton (1999) ups the coverage to 6 pages (out of 671).  Neither of these authors discuss body-part counting.  Standard histories have a lot of mathematics to cover. Consequently, speculations on how counting began or how some non-Western cultures go about tallying  may seem less important than solving a cubic equation.  Nevertheless, there is value in examining the mathematical notions of people outside of the main currents of Western mathematical history.   Mathematical thinking is universal and the number sense is innate (Butterworth, 1999).  Mathematics is a part of culture and its manifestations across cultures are part of its story (D'Ambrosio, 1989, 1992).

 In that spirit of ethnomathematics, this paper is about a particular kind of counting that developed among many of the people now living in the country of Papua New Guinea (PNG).  Body-part counting is unique and fascinating.  It opens a window to other cultures and provides new narratives to the story of mathematics.   Too often, however, the mathematical notions of non-technological or non-literate people are deemed "primitive", not mathematics, or simple.  It is important at the onset to emphasize that the use of body-part tally-systems does not imply "primitive" mathematical ability.  Indeed, some of the people who use these tally systems have algebraically complex kinship systems (See Tjon Sie Fat, 1990).  Furthermore, despite its apparent  simplicity, counting requires a wide range of cognitive abilities.  Lakoff and Nunez  (2000, p.51) list nine "cognitive capacities needed in order to count on our fingers." Counting is a complex cognitive activity and is intricately interwoven with language and culture (See Tyler, 1969 for a discussion of culture and cognition). 

A detailed examination of the social, cultural, psychological, and mathematical dimensions of body-part tally-systems is well beyond the goal of this essay.  Instead, I hope to raise the awareness of body-part tally-systems within the mathematics community, exhibit some of the wonders and mysteries of the practice as a humanistic-mathematical phenomenon, and to offer avenues of investigation for those who are drawn to its magic.

2.  Papua New Guinea. 

Papua New Guinea (PNG) is a constitutional, parliamentary democracy of over 5 million people.   PNG encompasses a group of islands and  the eastern half of the island of New Guinea  between the Coral Sea and the South Pacific Ocean (See Map 1.)  PNG has over 5000 kilometers of coastline and includes more than 1400 islands.  The topography of Papua New Guinea varies dramatically from beaches and mangrove forests and swamps to rain forest to high mountains that can receive snow.  The rugged interior of PNG has enabled its indigenous populations to develop without contact with European culture until well into the twentieth century (in some cases as late as 1970).  There is some disagreement among linguists as to the number of languages spoken in PNG, but the estimates generally hover around 800 (e.g., Ethnologue, 2002; Wurm, 1982).

Map 1. Papua New Guinea (CIA, 2002)

 

In a collection of over 800 distinct languages one would expect to find wide variation in numerals and counting words.  (See Lancy (1983) for a classification scheme of 225 PNG languages.)  Indeed, Papua New Guinea has enough variety to keep students of numeral systems busy for decades  A couple of examples will serve to illustrate the complexity and variety of PNG numeral systems.

Consider Nasioi, a Papuan  language spoken on the island of Bougainville.  Nasioi possesses an extensive system of numerical classifiers.  Different numerals are used depending upon what is being counted.  Table 1 shows the variations in Nasioi counting.  The basic linguistic pattern of Nasioi numerals is clear.  The number morpheme is a prefix on the noun class. The number morphemes are  (1) na-,  (2) ke-, (3) be-  and  (4) kare-.  Why the noun classes are divided the way they are is an interesting question of ethnoscience that awaits a future investigator.

 

Table 1. Nasioi (Solomon Province) (Lean, 2002, Vol. 3, 23ff).

 

1

2

3

4

People

narung(m) nani(f)

kenankara

benaura

karenaura

Quadrapeds (mammals)

navoro

kevoroka

bevoroi

karevoroi

Birds

nau

keura

bekuri

karekuri

Fruit

nau

keuka

bekupi

karekupi

Hollow objects

naro

keroka

beropi

kareropi

Tools

nari

kerika

bevintu

karevintu

Bag-like objects

nara

keraka

berapi

karerapi

Ropes

navin

kevinta

bevintu

karevintu

Leaf-shaped objects

nane

keneka

berapi

karerapi

Years & months

navera

keveraka

beverapi

kareverapi

Days

namung

kemunta

bemuntu

karemutu

Trees

nave

kevera

beveru

kareveru

Flat objects

namo

kemoka

bemopi

karemopi


 

The case of Enga, spoken in the Western Highlands Province on the island of New Guinea, is also intriguing.  Table 2  shows the Enga counting words from 1 to 32.

 

Table 2:  Enga (Western Highlands) (Lean, 2002, Vol.9, pp.5ff)

Number

Word

Analysis

1

me(n)dai

1

2

lapo

2

3

tepo

3

4

kitome(n)de

4

5

yungi

5

6

tokage

6

7

kalage

7

8

tukulapo

two arrows

9

tukutepon(ya) me(n)dai

two arrows + 1

10

tukutepon(ya) lapo

two arrows + 2

11

tukutepon(ya) tepo

two arrows + 3

12

tukutepon(ya) gato

two arrows + end

13

mapun (ya) me(n)dai

sweet potato + 1

14

mapun (ya) lapo

sweet potato + 2

15

mapun (ya) tepo

sweet potato + 3

16

mapun (ya) gato

sweet potato + end

17

yupun (ya) me(n)dai

ground + 1

18

yupun (ya) lapo

ground + 2

19

yupun (ya) tepo

ground + 3

20

yupun (ya) gato

ground + end

21

watakapun (ya) me(n)dai

wild + 1

22

watakapun (ya) lapo

wild + 2

23

watakapun (ya) tepo

wild + 3

24

watakapun (ya) gato

wild + end

25

paipan (ya) me(n)dai

come and go + 1

26

paipan (ya) lapo

come and go + 2

27

paipan (ya) tepo

come and go + 3

28

paipan (ya) gato

come and go + end

29

yanapun(ya) me(n)dai

dog + 1

30

yanapun(ya) lapo

dog + 2

31

yanapun(ya) tepo

dog + 3

32

yanapun(ya) gato

dog + end

 

The Enga counting system consists of a cycle of 60 subdivided into 15 cycles of four.  The first few counting words are numerals, but after that the counting words follow the pattern base+n where n takes on the values 1, 2, 3, and "end", and the base is a word or phrase with no numerical meaning.   The Enga count continues in like fashion until it ends at 60 with the appropriate kaeapalun(ya) gato ("I stop + end").  This is a fascinating pattern and begs for an ethnomathematical analysis.  Nevertheless, it is not my intention to offer such an analysis here, nor to provide a summary or analysis of PNG counting systems.  Instead, I focus solely on body-part tally-systems.

3.  Body-part tally-systems.

My discussion of PNG body-part tally-system relies heavily on the data compiled by Glendon Lean (2002).  Lean's extensive notes on counting systems survey 740 languages in Papuan New Guinea. I cross-checked Lean's data with other published work  (Biersack, 1982; Carrier, 1981; Cheetham, 1978;  Conant, 1896;  Franklin and Franklin, 1962; Hurford,1987;  Lancy, 1983; Levy-Bruhl, 1926; Panoff, 1970; Pumuye, 1978; Saxe, 1981; Smith, 1978). I included theYupno counting system as described by Wassmann & Dasen (1994) which is not included in Lean's notes.   Of these 741 languages, 38 have explicit  body-part tally-systems. 

Nearly all of languages investigated  begin counting on the little finger of one hand.  In some cases, the left hand is specified as the starting hand, but in most the count may start on either hand.  The count then proceeds through the fingers of the starting hand, then to the thumb and then up the adjoining arm to either the head or chest.  The count next proceeds down the other side of the body until it ends at the little finger of the other hand.  Figure 1 shows the count for Anggor, a language spoken in the West Sepik Province.

Anggor includes the wrist (6 and 18), forearm (7 and 17), elbow (8 and 16), upper arm (9 and 15), shoulder (10 and 14), breast (11 and 13) and the sternum (12) in its tally-system.  The count ends at 23 and it is customary to call such a system a 23-cycle system.  For counts above 23, such systems would usually continue on with a phrase like "one man and " begin the count anew.  (Although see Saxe, 1981 for a count that proceeds back up the arm in the reverse direction.) 

The Faiwol body-part tally-system is an example of one that includes counting locations on the head.  This 27-cycle system is shown in Figure 2.  The inside of the elbow is counted 8 and 20 in Faiwol in contrast to the outside elbow in Anggor.  Eleven and  seventeen in Faiwol are counted at the collar bone.

There is great variation in the cycle length of PNG body-part tally-systems.  Cycles of length  12, 14, 18, 19, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 47, 68, and 74 have been reported.  When the cycle length is odd there is usually a unique midpoint to the count that is located along the person's line of symmetry (e.g., the sternum in Anggor and the nose in Faiwol).  Table  3 summarizes the cycle-lengths of PNG body-part tally-systems and gives the midpoint location for cycles of odd length.

Miriam is unique in having three count locations on the body's line of symmetry (Lean ,2002, Vol. 12, p.36).  Yupno differs from all other languages in several ways.  First, the count in Yupno proceeds back and forth across the body with 6 -10 being counted on the right hand after 1 -5 is counted on the left. Second, the toes are included in the count with 11-15 on the left foot and 16-20 on the right foot.  Finally, Yupno includes locations below the waist with 31 counted at the left testicle, 32 at the right testicle, and 33 at the penis. (Wassmann and Dasen, 1994, p.84).

 

Figure 1.  Anggor (West Sepik) (Lean, 2002, Vol. 13, p. 14)

 

 

Figure 2.  Faiwol (West Province) (Lean, 2002, Vol. 12, p. 1)

 

 

Table 3.  Cycles of PNG Body-Part Tally-Systems

Cycle Length

People

Province

Midpoint

12

Ama

ES

 

14

Duna

SH

 

18

Dabu

WP

 

 

Anggor

WS

 

 

Agob

WP

 

 

Mabiuag

WP

 

19

Gizra

WP

sternum(10)

 

Gidra

WP

sternum (10)

22

Baruga

OP

nose(12), mouth(13)

 

Sakam

MO

 

23

Awin

WP

sternum(12)

 

Anggor

WS

sternum (12)

 

Yuri

WS

sternum (12)

 

Purari

GP

chest (12)

 

Kabon

MA

hole above breast bone (12)

 

Kalam

MA

bone of neck (12)

 

Murupi

MA

sternum (12)

25

Enga

WH

nose(13)

 

Amanab

WS

heart(13)

26

Giri

MA

 

27

Faiwol

WP

nose(14)

 

Mianmin

WS

nose (14)

 

Telefol

WS

nose(14)

 

Tifal

WS

nose(14)

 

Oksapmin

WS

nose(14)

 

Hewa

SH

nose(14)

 

Orokola

GP

nose(14)

 

Sanio

ES

nose(14)

28

Paiela

WH

nose(14)

29

Alambak

ES

nose(15)

 

Miriam

WP

navel(14), top of chest(15), front of throat (16)

 

Huli

SH

nose(15)

30

Ninggirum

WP

 

31

Gende

MA

nose(16)

32

S.Mende

SH

 

33

Yupno

MA

none-alternates sides

35

Fasu

SH

nose(18)

37

Fol

SH

centre of nose (19)

47

W. Kewa

SH

bridge of nose (24)

 

E. Kewa

SH

between eyes (24)

68

Kewa

SH

 

74

Nagutman

WS

 

Guide to Provinces in Table 3:  ES (East Sepik), GP (Gulf Province),  MA (Madang), MO (Morobe),  OP (Oro Province), SH (Southern Highlands),  WH (Western Highlands),  WP (Western Province), WS (West Sepik),

 

4. Language.

The relationship between the words spoken while counting, and the body part referenced, is complex.  Lean (2002, Vol. 12, p.2)  observes that "body-part tally-systems tend to fall into two types: 1) those in which all tally-words have a body-part referent, and 2) those in which the first few number words are numerals with no body-part referent, the remaining tally-words all referring to body-parts."  

Baruga, spoken in Oro Province, is of particular interest here because of the generality of the spoken words.  The Baruga body-part tally-system is a 22-cycle system that begins with the little finger of the right hand .  Baruga counting is shown in Table 4.   Notice that the counting must be visual since the word "doro" stands for 3,4,5,19,20, and 21.  Baruga counting also is unique in including the left eye before the central nose and mouth, thus breaking the vertical axis symmetry pattern.

 

Table 4.  Baruga (Oro Province) (Lean, 2002, Vol. 5, p.9)

Number

Word

Meaning

1

angusi

little finger-right hand

2

doro 

ring finger - right hand

3

doro

middle finger-right hand

4

doro

index finger-right hand

5

ubei

thumb-right hand

6

tama

wrist- right

7

ungubo

elbow-right

8

visa

shoulder-right

9

dengoro

ear-right

10

diti

eye-right

11

diti

eye-left!

12

mendo 

nose

13

bee

mouth

14

dengoro

ear-left

15

visa

shoulder-left

16

ungubo

elbow-left

17

tama

wrist- left

18

ubei

thumb-left hand

19

doro

index finger-left hand

20

doro

middle finger-left hand

21

doro

ring finger- left hand

22

angusi

little finger - keft hand    

 

Fasu  (Table 5) and  Kewa (Table 6) are examples of counting systems that use a specific body-part word for each number.   In each of these languages the counting words for the numbers on the other side of the midpoint (after 18 in Fasu and after 24 in Kewa) are marked lexically with the word other or another.   This is typical of body-part tally-systems with or without a midpoint on the line of vertical symmetry.  Fasu is representative of several langauges that give the last number in the cycle a designation of closure.

 

Table 5.  Fasu (Southern Highlands) (Lean, 2002, Vol. 10, p.8)

Number

Word

Translation

1

meno

little finger

2

teta

ring finger

3

isia

middle finger

4

kitafa

index finger

5

kakorea

thumb

6

nama

palm of hand

7

yatipinu

inside wrist

8

kari

forearm

9

tokona

inside elbow

10

kaeyako

upper arm

11

kinu

shoulder

12

keno

collar bone

13

fufu

neck

14

senaki

ear

15

pare

cheek bone

16

hi

eye

17

no

side of nostril

18

terayia

ridge of nose

19

taku no

other side of nostril

20

taku hi

other eye

 

 

 

33

taku isia

other middle finger

34

taku teta

other ring finger

35

kenake urutae

last one counted

 

Table 6.  East Kewa (Southern Highlands) (Franklin & Franklin, 1962)

Number

Word

Translation

1

kegali

little finger

2

kegali yame

ring finger

3

ada ki

middle finger

4

mala

index finger

5

su

thumb

6

su midi

heel of palm

7

wara

palm

8

kerepo

wrist

9

pala ki

forearm

10

noe

large arm bone

11

noe luabu

small arm bone

12

koma

above elbow

13

winya ropa

lower upper arm

14

ali ropa

upper upper arm

15

pasa

shoulder

16

pasa kuli

shoulder bone

17

pasa midi

neck muscle

18

ma

neck

19

yaga

jaw

20

kale

ear

21

pae

cheek

22

le

eye

23

paki

inside corner of eye

24

riga

between the eyes

25

paki meda

inside corner of another eye

26

le meda

another eye

 

 

 

47

kegali meda

another little finger

 

Table 7 gives the counting system for Amanab, a language spoken in West Sepik Province.  Amanab is an example of a language in which the body-part tally-system coexists with a pure numeral system.   In a pure numeral system, the words designating a number have no other semantic content. Amanab counters may choose to name body parts for the numbers  1 to 4 or they may use number words.

 

Table 7.  Amanab (West Sepik) (Lean, 2002,, Vol. 13, p.13)

Number

Word

Translation

1

angik   or  munggo

little finger  or one

2

angangik or sambaga

ring finger or two

3

nibwari or sambaga munggo

middle finger or two-one

4

figinik or sambaga sambaga

index finger or two-two

5

afa wa goho-k

thumb

6

onuweso

wrist

7

enemsakak

mid-forearm

8

erengeg

inner elbow

9

enembuguk

mid-upper arm

10

guk

shoulder

11

hwanifuk

armpit

12

tut

breast

13

oruk

heart

14

mink tut

other breast

15

mink hwanifuk

other armpit

16

mink guk

other shoulder

17

mink enembuguk

other mid-upper arm

 

 

 

25

mink angik

other little finger

 

In Mianmin  (Table 8), the option for body-part words is not available.  Mianmin counters use number words to count up to six and then shift to body-part words for the numbers thereafter.  Neither Lean nor his sources give information for Mianmin counting beyond 23.  However, we would expect to see body-part words for these numerals.  It is also notable that Mianmin begins counting on the left thumb.

 

Table 8. Mianmin  (West Sepik) (Lean, 2002,, Vol. 13, p.35)

Number

Word

Translation

1

ele-yem

this-alone

2

asu

two

3

asu-matna

two - one more

4

asu-ke asu-ke

two-and two-and

5

asu-ke asu-ke mak-e

two-and two-and other

6

asu-ke asu-ke asu-ke

two-and two-and two-and

7

ban-lim

forearm on

8

hetlefab

inner elbow

9

tumin

shoulder joint

10

nakal

shoulder

11

tam-lim

side of face-on

12

klon-lim

ear-on

13

kin-lim

eye-on

14

munung-lim

nose-on

15

kin-milim

eye-other side

16

klon-milim

ear-other side

17

tam-milim

side of face-other side

18

nakal-milim

shoulder-other side

19

tum-milim

shoulder-joint-other side

20

hetlefab-milim

inner-elbow-other side

21

ban-milim

forearm-other side

22

gong-milim

wrist-other side

23

kweit-awok-milim

hand-thumb-other side

 

Although not a body-part tally-system, Yahang counting  (see Table 9) illustrates the blend of pure numerals, number words with no other semantic value, and body-part referents.

Yahang counting uses pure numerals for one, two, and perhaps threeFour is an additive combination of two two'sFive has the same meaning as hand.  The semantic correlation of five with hand  is widespread among PNG languages that do not use body-part tallies.  This correlation is present in many of the languages of the world  (Butterworth, 1999; Conant, 1896; Hurford, 1987).

The patterns of counting seen in these examples provide evidence for Hurford's (1987, p.82) assertion that  "The normal process of language-change over a long period would lead to words which originally had body part associations losing those associations and becoming pure numeral, or at least counting, words."

 

Table 9. Yahang (West Sepik) (Lean, 2002,, Vol. 13, p.32)

Number

Word

Translation

1

numusuku

one

2

kolou

two

3

kololomu

three (?? 1 + 2)

4

kolikolou

2 + 2

5

wom nateng

hand

6

nor nateng numu

hand and one

7

nor nateng kolou

hand and two

8

nor nateng kololomu

hand and three

9

nor nateng kolikolou

hand and four

10

kolou nateng

two hands

 

 

 

20

kolou nateng kolou inding

two hand; two feet

 

5.  Cultural context.

The semioticist Umberto Eco asserts that "usually a single sign-vehicle conveys many intertwined contents and therefore what is commonly called a 'message' is in fact a text whose content is  a multileveled discourse." (1976, p.57)   Viewed as semiotic systems, body-part tally-systems do more than count.  They reflect culture and communicate.

Consider the case of the Kewa pig-kill calendar (Pumuye, 1978).  The body count terms correspond to the months of preparation for the pig killing festivals that occur every five to seven years.

Table 10 was compiled from the information given by Pumuye (1978) and should be compared to the Kewa counting system shown in Table 6 which is slightly different.  Pumuye (1978) reports a 68-cycle body-part tally-system for East Kewa while Franklin and Franklin (1962) report a 47-cycle system.  This kind of variation is not uncommon in reports. It exhibits either the dynamic nature of the systems or differences in memory and understanding among informants. 

 

Table 10. Kewa Pig-Killing Calendar (Pumuye, 1978)

Month Number

Month Name with Number Word Italicized

Gesture

Activity

1

Ki-kegali eke mere

Touch or bend little finger.

Prepare raw materials to build long rows of houses for the festival.

2

Ki-kegali-yame eke mere

Touch or bend ring finger.

Assign festival participants a room in row of houses.

3

Ki-anda-ki eke mere

Point middle finger.

Make new gardens to produce enough food for the festival

4

Ki mala eke mere

Touch the index finger.

Weed the taro gardens and make the vegetable gardens.

5

Ki  su eke mere

Point to the thumb.

Judge and comment upon progress with the gardens.

etc.

 

 

 

14

Ki wina ropa eke mere

Touch the region known as the woman's armlet.

Slaughter all the pigs

 

Beyond calendars, the body-part tally-systems merge with the idiosyncrasies of individual cultures in a variety of ways.  What is actually counted varies widely.  The whole spectrum of objects to be counted seems to be represented.  Examples range from the Huli, who count everything (Cheetham, 1978) to the Loboda, who seem to count nothing but money and use number rhetorically (Thune, 1978).  The Yupno represent a position somewhere in between (Wassmann & Dasen, 1994)   The Yupno count string bags,  grass skirts, pigs , traditional string money, and modern PNG money. They do not count days, people, sweet potatoes, nor betel nuts.

The most detailed account of the cultural use of body-part tally-systems is Biersack's (1982) investigation of the Paiela, who live in the Western Highlands Province.  The Paiela use a 27-cycle body-part tally-system (with the nose as the midpoint) augmented by an additional count at 28 called pondo (two clenched hands knocked together).   Biersack argues that Paiela counting is an "element in a communication process and that its logic is therefore communicational."  (Biersack, 1982 p.816)  She explains that the body tallying of  pigs is actually a two-stage communication process where the body-tallying is the first or private stage and the enumerating of the pigs is the second or public stage.   She says that the logic of Paiela counting is understood not as arithmetic but as a semiotic system.  That is, "the concrete elements -- the vocabulary, tallying procedures, procedures of actual enumeration, and the linkage of these to transactional processes -- are so organized as to generate two levels of signification. At the first level messages of friendship are signified (and counted). But at the second level a statement is made, using a binary metacode, about the certainty or uncertainty of the message and, by implication, the stage of the communication process itself." (Biersack, 1982, p.823)

Paiela counting should be viewed not as primitive counting, but as sophisticated communication.

"Paiela counting behavior thus reveals that the Paiela are conversant with a number of principles associated with pattern and information: that information must be borne on a marker; that it necessarily alleviates uncertainty; that pattern is a relationship between things; and that pattern can be created and destroyed, that pattern is not preserved."  (Biersack, 1982, p.825)

7. Fingers and toes and elbows and brains.

Body-part tally-systems are an important part of the cultural history of mathematics.  As PNG becomes more Westernized , the body-part tally-systems are dying out (Lancy, 1983; Lean, 2002).  Indeed, PNG has a modern educational system and mathematics education follows the western model including mathematics competitions and computer assisted instruction (Sukthankar, 1999). We can hope to record traditional body-part tally-systems and understand them as best we can before they are gone.  But, we should not view body-part tallies as an aberration of mathematical development.  These systems are  natural extensions of counting on one's fingers and toes.  Finger and toe counting is widespread in PNG (Lean, 2002; Mimica, 1992).   There are also traditions of finger counting in mercantile trade in Africa,  in Europe and in Asia  (Ifrah, 2000; Menninger, 1969; Zaslavsky, 1973).  The Venerable Bede even wrote a manual on the process in the 8th century.  Occasional finger arithmetic techniques appear with the promise of making us all better at mathematics (Chisanbop, 1980).  The neuropsychologist Brian Butterworth (1999, p.226) hypothesizes that "as a child grows and develops, the subitizing circuits in the inferior parietal link up with the finger circuits in the intraparietal sulcus. The fingers therefore gain an extended representation by this link: they come to represent the numerosities."  Body-part counting is, therefore, part of our nature.

 

Sources

Biersack, A. (1982). The logic of misplaced concreteness: Paiela body counting and the nature of the primitive mind. American Anthropologist 84, 811-829.

Burton, D. M. (1999).  The history of mathematics: An introduction (4th edition). Boston: McGraw-Hill.

Butterworth, B. (1999).   What counts: How every brain is hardwired for math.  New York: The Free Press.

Carrier, A. (1981). Counting and calculation on Ponam Island. The Journal of the Polynesian Society, 9, 465-479..

Cheetham, B. (1978). Counting and number in Huli. Papua New Guinea Journal of Education(Special Issue), 14, 19-30.

Chisanbop. (1980).  An introduction into the amazing world of Chisanbop. New York: Chisanbop Enterprises, Inc.

CIA.  (2002).  CIA Factbook.  http://www.cia.gov/cia/publications/factbook/geos/pp.html.

Conant, L.L. (1896). The Number Concept.  New York: Macmillan and Co.

D'Ambrosio, U. (1989).  On ethnomathematics.  Philosophia Mathematica,II, v. 4, 3-14.

D'Ambrosio, U. (1992). Ethnomathematics: A research program on the history and philosophy of mathematics with pedagogical implications. Notices of the American Mathematical Society, 39, 1183-1185.

Eco, U. (1979). A theory of semiotics. Bloomington: Indiana University Press.

Ethnologue. (2002).  http://www.ethnologue.com/show_country.asp?name=Papua+New+Guinea   SIL.

Franklin, K. and Franklin J. (1962). The Kewa counting system.  The Journal of the Polynesian Society, 71, 188-191.

Hurford, J.R. (1987). Language and number: The emergence of a cognitive system. Oxford: Basil Blackwell Ltd.

Ifrah, G. (2000). The Universal History of Numbers. New York:  John Wiley & Sons, Inc.

Katz, V. J. (1998). A History of mathematics: An introduction (2nd edition). Reading, MA: Addison-Wesley.

Lakoff, G. and Nunez, R.E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being.  New York: Basic Books.

Lancy, D.F. (1978). Indigenous mathematics systems. Papua New Guinea Journal of Education(Special Issue), 14, 6-15.

Lancy, D.F. (1983).  Cross-cultural studies in cognition and mathematics. New York: Academic Press.

Lean, G. (2002).  Counting Systems of Papua New Guinea, Vols 1-16.  Goroka, PNG: Glendon Lean Centre for Ethnomathematics.

Levy-Bruhl, L. (1926). How natives think. London: George Allen and Unwin Ltd.

Menninger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge: MIT Press.

Mimica, J. (1992). Intimations of infinity: The cultural meaning of the Iqwaye counting and number system.  Oxford: Berg Publishers, Inc.

 NCTM (2000).  Principles and  standards for school mathematics.  Reston, VA: National Council of Teachers of Mathematics.

Panoff, M. (1970).  Father arithmetic: Numeration and counting in New Britain. Ethnology 9: 358-365.

Pumuye, H. (1978).  The Kewa calendar. Papua New Guinea Journal of Education(Special Issue), 14, 47-55.

Saxe, G. (1981).  Body parts as numerals: A developmental analysis of numeration among the Oksapmin in Papua New Guinea.  Child Development,  52, 306-316.

Smith, G. (1978). Counting and culture on Kiwai Island. Papua New Guinea Journal of Education(Special Issue), 14, 57-72.

Strathern, A. (1977).  Mathematics in the moka.  Papua New Guinea Journal of Education, 13, 16-20.

Sukthankar, N. (1999). Cultural factors in mathematics education contributing to the shortage of women professionals in Papua New Guinea.  Educational Review, 51, 173-179.

Thune, C.E. (1978). Numbers and counting in Loboda: An example of a non-numerically oriented culture. Papua New Guinea Journal of Education(Special Issue), 14, 69-80.

Tjon Sie Fat, F. E. (1990). Representing kinship: Simple models of elementary structures. Leiden: Leiden University Faculty of Social Sciences.

Tyler, S. Ed. (1969). Cognitive anthropology.  New York: Holt, Rinehart and Winston, Inc.

Wassmann, J. and Dasen, P.R. (1994).  Yupno number system and counting. Journal of Cross-Cultural Psychology, 25, 78-94.

Wurm, S.A. (1982).  Papuan Languages of Oceania.  Tübingen: Gunter Nar Verlag.

Zaslavsky, C. (1973). Africa counts: Number and pattern in African culture. Brooklyn: Lawrence Hill Books

 

 Home Articles Guidelines