In many high schools in the Western world, it is not uncommon to find students of East Asian descent excelling in the area of mathematics. Some scholars have argued that this phenomenon is linked with the fact that mathematics is very much a part of traditional Sinitic cultures. The invention of the abacus is a case in point. In the realm of communication, an average person in China needs to memorize at least 2,000 characters to be able to read and write efficiently. In the Chinese kinship system, one needs to recognize a relative based on the latter’s seniority and position within a chronological sequence of kinsmen.
Great Asian civilizations have invented some numeration system that has become universal. Say for example, zero has often been attributed to the Hindus. During the Mauryan to Gupta dynasties in India, medieval mathematicians developed sophisticated numerations systems from one to nine digits and zero, the value of pi, trigonometry, geometry and calculus.
On the other hand, traditional Chinese numeration system is a base-ten system employing nine numerals and additional symbols for the place-value components of powers of ten. Before the eighth century A.D. the place where a zero would be required was always left absent. A circular symbol for zero is first found in a document dating from 1247, but it may have been in use a hundred years earlier.
Reflecting on the aforementioned evidences, one may ask, “Do Filipinos also have a tradition of mathematics?” Secondly, “What other early Indian and Chinese influences helped in the development of ethnomathematics Philippine society?”
Ethnomathematics is a subdiscipline of Cultural Anthropology that is concerned with the study of mathematics which takes into consideration the culture in which mathematics arises. Mathematics is often associated with the study of “universals”, however, it is important to recognize that often something we think of as merely universal to those who share our cultural and historical perspectives.
The term “ethnomathematics” was coined by Ubiratan D’Ambrosio, a Brazilian mathematician, to describe the mathematical practices of identifiable cultural groups. These mathematical practices include symbolic systems, spatial designs, practical construction techniques, calculation methods, measurement in time and space, specific ways of reasoning and inferring, and other cognitive and material activities which can be translated to formal mathematical representation.
If we think mathematics as a development of structures and systems of ideas involving number, pattern, logic, and spatial configuration and then examine how mathematics arises and is used in various cultures, we gain a much deeper understanding of mathematics. Thus, there are many human activities which require some form of mathematics The Filipino concept of mathematics is part and parcel of the indigenous knowledge system and practices of the Filipinos.
Rohana Ulluwishewa (1993) defines indigenous knowledge as: Indigenous Knowledge (IK) is a local knowledge that is unique to a given culture or society. It is the basis for agriculture, health care, food preparation, education, environmental conservation and a host of other activities. Much of such knowledge is passed down from generation to generation, usually verbally. IK is the knowledge that people have gained through inheritance from their ancestors. It is a people-derived science, and it presents people’s creativity, innovation and skills.
Thus, the Filipino indigenous concept of mathematics is a type of folk mathematics. It is not learned through formal institutions, such as schools, and may or may not come into conflict with Western-derived mathematical concepts. The mathematics that local people know of is not something that is abstract nor divorced from everyday life.
Some historians and anthropologists speculate that the Philippines was for a time part of the Sri Vijayan Empire from 4th to 10th centuries, which has been described, as Hinduistic in culture, prior to the entry of Islam in Sulu archipelago in 13th century and Catholism in the Visayas island in 16th century.
In the process of Indianization in Southeast Asia, the achipelagic location of the Philippines, has also been Indianized. An apparent instance of the early Indian influence in Philippine political structure was the manifestation of titles and names of local rulers. The rulers of many of the islands were called Rajas, or Rajahs. The central region, Visayas, is said to be named after the last Southeast Hindu Prince Sri Vijaya who converted to Islam after which the local Filipinos were in the process of converting to Islam.
The Brahmanistic elements in ancient Filipino religion and the names of gods and mythological heroes were of Indian origin. The term Bathala (supreme god of the ancient Tagalog) originated from the Sanskrit Bhattara Guru, meaning “the highest of the gods”.
On the other hand, some Filipino customs are of Indian origin. Among them are the following, like the placing a sampaguita flower garland around the neck of a visitor upon his arrival and departure as a symbol of hospitality and friendship. Another is, before marriage, a groom gives a dowry to the bride’s parents and renders domestic services to his future in-laws and when the guests throw rice on the bride and groom after the wedding. Lastly, when a childless couple goes on a pilgrimage to a holy shrine, it is believed that the god of shrine will grant their prayer for fertility.
It was during the Ming Dynasty (1368-1644) that Philippine-Chinese friendship reached its peak. In one of the accounts of Sino-Sulu relations narrated in the Ming Annals, it is recorded that the Sultan of Sulu, Paduka Patara, visited China in 1417 where he was royally received by the Chinese emperor. On his way home after a 27-day visit, the Sultan was stricken ill and died in the city of Dezhou in Shandong province. The emperor honored the Muslim king with the title of Kong Ting (brother) and ordered the building of a handsome mausoleum to mark the tomb of his Filipino friend – the only tomb of a foreign monarch in honor of the 15th century Sulu Sultan (Salvador: 2000).
To take into consideration China’s past glories, in fact, according to Schilling (2001: 75-76) he admits that Asia has enjoyed technological leadership in the past, especially in China, which was the world’s most advanced country a thousand years ago. The Chinese invented paper around 105 CE. Government Service exams were established in 154 BCE. As for her technological innovations, the compass came about 1100 CE, gunpowder around 1000 CE, block printing about the same time, and silk by 1300 CE. Chinese porcelain of unrivaled quality was discovered by Europeans in 1709. He also admires the Song dynasty (960-1279), and in that period, great technological, cultural, economic levels were achieved. At the peak of its power, during the Ming Dynasty (1368-1644), Chinese rulers decided that China had achieved superior culture among other civilizations and so foreign culture were neither needed nor welcome.
Baybayin or Alibata (known in Unicode as the Tagalog script) is a pre-Hispanic Philippine writing system that originated from the Javanese script Kavi. The writing system is a member of the Brahmic family (and an offshoot of the Sanskrit alphabet) and is believed to be in use as early as the 14th century. It continued to be in use during the Spanish colonization of the Philippines up until the late 19th Century. The term baybayin literally means spelling. Closely related scripts are Hanunóo, Buhid, and Tagbanwa.
The ancient Filipino alphabet originated from India. About twenty-five percent of the words in the Tagalog language are Sanskrit terms. For example, with regards to traditional poetry, the Tagalog tanaga, the Hanunoo Mangyan ambahan, the Kalinga ulalim, and the Manobo ulahingan are all heptasyllabic in form, i.e. with seven syllables per line. The Hanunoo Mangyan urukay, on the other hand, has eight syllables per line. Of course, the strict adherence to a given metric line is a mathematical feat by itself.
Interestingly, there are many cultural concepts that have been associated with certain numbers. For example, the word isa (one) has brought about concepts such as pagkakaisa (unity), nag-iisa (alone), and naisahan (took advantage), among others. The number dalawa (two) has been associated with pagdadalawang-isip (state of confusion), “number two” (mistress), and pamamangka sa dalawang ilog (bigamy). The lone word kwatro (four) has related concepts such as upong de-kwatro (cross-legs) and mabilis pa sa alas-kwatro (very fast).
There is also concept of “5-6”, an Indian type of business dealing carried in and still practiced by Indian immigrants in the Philippines, which refers to a type of usury. The word pitu-pito, which can refer to a type of herbal tea and low quality films, is derived from pito (seven). The term syete (seven), which is a loan word for seven, means gossip. Naonse (cheated) was derived from the Spanish onse for eleven.
The rich Indian literature also has a share to Filipino literature and folklore. The Maranao epic Darangan is Indian in plot and characterization. The Agusan legend of a man named Manubo Ango, who was turned into stone, resembles the story of Ahalya in the Hindu epic Ramayana. The tale of the Ifugao legendary hero, Balituk, who obtained water from the rock with his arrow, is similar to Arjuna’s adventure in Mahabharata, another Hindu epic.
One of the characteristics of Philippine languages is its use of decimal-based terms for numerals. This is the characteristic which is shared with other Austroneasian languages. This is not true for Indo-European languages. In English, for example, lexicons such as “eleven”, “twelve”, “dozen”, and “score” are proofs that the said language is not decimal-based. Moreover, one may wonder why a day is divided by twenty-four (24) units, called hours, while an hour is divided by sixty (60) units, called minutes. In French, instead of being assigned specific numeral terms, the number seventy (70) is called soixante-dix. Which literally means “sixty plus ten” while eighty (80) is quatre-vingts, which means “four twenties”.
In Tagalog, counting is by tens. The term pu stands for “ten”, that is why sampu literally means, “one ten” while dalawampu means “two tens”. More than ten, the term labi is used to mean an excess from the number ten. The number “eleven” is labing-isa, followed by labindalawa, labintatlo, etc. An excess from twenty onwards is simply indicated by the term at (and). Thus, dalawampu’t isa stands for “twenty and one”.
The other decimal terms are daan (100), libo (1,000), laksa (10,000), yuta (100,000), angaw (1,000,000), kati (10,000,000), and bahala (100,000,000). Unfortunately, many of these terms are no longer used because of the adoption of Western concepts. Thus, we already refer to 10,000 as sampung libo instead of isang laksa. The term laksa, however continues to thrive in folk literature, such as in the phrase “laksa-laksang ibong lumilipad”.
Despite the decline in the use of indigenous decimal terms higher than one thousand, the use of the decimal system has persisted among the masses. When one bets during a cockfight, he simply extends his finger upward to denote a bet by the tens of pesos, sideways to mean bets by the hundreds, and downwards to refer to bets by thousands.
The number ten is traditionally perceived as a perfect number. This is probably the reason why it is considered polite to end each sentence with a po, which is a variant of the form pu (ten). On the other hand, the number siyam (9) is usually associated with something negative because it is less than perfect. There is pasiyam after the death of a person because it is believed that the soul of the dead travels for nine days before going to the other world. The term siyam-siyam to mean nothing can be achieved from a given situation. In Hebrew, the number seven (7) is a number of perfection while the number six (6) is associated with evil. In Chinese, the number four (4) is unlucky while eight (8) is the lucky number.
The term ilan is used to quantify several entities, such as time, persons, plants and animals, places and objects. It may mean “how many” or “how much” in questions such as Ilang oras ka nang naghihintay? Ilan kayong magkakapatid? Ilan ang alaga mong aso? Ilang bansa na ang napuntahan mo? and Ilang kotse ang pag-aari mo?
The word kailan is usually translated to “when” in English. The term, however, is actually derived from the phrase “ika-ilan” because one is interested in the place of a particular event within a chronological sequence of event, i.e. did it happen before or after another particular activity.
This preoccupation with chronological order is also reflected in the term “pang-ilan” which is quite difficult to translate in a few words in English. Thus, one may ask, “Pang-ilang presidente na ba si Erap?” which, again, poses a challenge to translators because the English language is not so much concerned with chronological sequences.
The Philippine kinship system reinforces the importance of chronological sequences. The position of the Ego vis-à-vis his or her siblings is very important. In Visayan languages, all siblings older than the Ego are referred to as magulang while those younger than the Ego are manghod.
On the other hand, the Tagalogs, Kapangpangans, and Pangasinenses have borrowed Chinese terms for different siblings based on their seniority within the family; thus the terms kuya (eldest brother), diko (second eldest brother), sangko (third eldest brother), ate (eldest sister), ditse (second eldest sister), and sanse (third eldest sister). The presence of several Tagalog terms used to measure things and objects is also another proof of indigenous mathematical concepts.
The concepts related to sukat (measurement) include the domains of haba (length), layo (distance), dami (volume), taas (height), lalim (depth), lawak (area), timbang (weight), halaga (value), and bilis (speed). To measure length, there are terms such as dali, dangkal, damak, bara and dipa, among others. A dali is equivalent to the measure from the tip to the middle joint of the thumb. A dangkal is the measure from the end of the middle finger to the end of the thumb when the said fingers are extended away from each other. A damak is unit of measure from the tip of the forefinger to the tip of the little finger. A bara refers to the length from the extended arm to the middle of the chest. A dipa is measured using one’s outstretched hands.
Distance may be measured through the talampakan (foot), hakbang (step), or the number of mountains one must pass, i.e. kung ilang bundok. Still, one may also resort to the length of time one may finish smoking a cigarette while hiking, i.e., isang sigarilyo lang and layo.
Solid volume may be measured using the salop, gating, or kaban. This is usually associated with measurements for rice and other grains. One may also use the bugkos when measuring through the grip of one hand, such as in the case of bundles of rice and vegetables. The kaing and also tumpok is used to measure the volume of fruits. Liquid measurements (but also applicable for some solid measurements) is through the takal. Today, the baso (glass) and tasa (cup) are also used for liquid measurements.
In terms of timbang (weight), prehispanic standards of measuring gold include the kati, which is approximately 617.6 grams, and suwama, which is about 308.8 grams (Postma 1991), and 16 suwama is equivalent to one kati.
There are also traditional terms related to currency, such as kusing (one centavo), bagol (five centavos), kahati (25 centavos), salapi (fifty centavos). Evidence shows that prior to the coming of the Spaniards, a form of currency was use at least in Manila through the use of piloncitos. The polincito is a gold coin with the Tagalog syllable MA inscripted on it, which probably stood for “Maynila”.
The concept of panahon (time) is also recognized in traditional Filipino society. This is measured through terms such as saglit (‘second’), sandali (‘minute’), araw (day), buwan (month), and taon (year).
The Ifugaos have a concept of a calendar based on the agricultural cycle. Traditionally, days and seasons are indicated through knots made on strings.
The Laguna copper plate, which was found in Lumban, Laguna, shows that early Filipinos used the Śaka calendar year which started in 78 A.D. in southwest India. Within the text of the said artifact are the numerals 822 to refer to the Śaka year 822 which is synonymous to 900 A.D. The first day of the week, soma, is also mentioned in the document as well as the month Waisakha, which falls, more or less, in the period of March to April.
The use of kawi numerals indicate that the early Filipinos not only used a written syllabic script but also had ideographs for decimal-based numerals. Filipinos also have concepts of addition (dagdag) and subtraction (bawas or kaltas). Among market venders, there is an indigenous method of basic addition through the use of one’s fingers and knuckles. On the other hand, the extended fingers facing one another are used for simple multiplication.
In many non-literate Philippine communities, mental computation is often resorted to. It is amazing that among the Ifugaos, there are mombaki (shaman) who can recite their family’s genealogy consisting of up to twenty-five (25) generations of kinsmen merely through the use of their mental databank.
The concept of percentages (bahagdan) is not also alien to the common tao. At present, the use of the modern peso, where one peso is made up of one hundred (100) centavos, enable layman to easily understand the concept of portions of a hundred percentages.
Another case in point is the popularity of number games in the country such as jeuteng, beto-beto, sweepstakes, bingo, lotto, basketball “ending” scores, quiz bee or math Olympiad and a lot more. TV shows are prominent mode of popular culture that embrace number games with huge cash prizes. Among popular game shows that involve numbers are: Deal or No Deal, Pera o Bayong, etc.
The evidences presented above are merely some of the available clues on the nature of Philippine ethnomathematics and the early Chinese and Indian influences in our culture. Evidently, there are still many gaps in the anthropological study of local mathematical concepts and systems. For example, How are mathematical skills learned, imbibed, and passed over from generation to generation? Are the date presented true for all Philippine communities or are these limited to certain ethnolinguistic groups, such as the Tagalog, Kapangpangan, Pangasinan, Visayan and Ifugao? Obviously, there are more questions than answers.