Rauff2003

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).

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