Ethnomathematics in Javanese Death Commemoration

In Java, there is a tradition called death commemoration. In Javanese society, death commemoration consists of geblag (commemoration afther funeral), commemoration at 3 day of the death, 40 day of the death, 100 day of the death, 1 year, 2 years, and 1000 day of the death. This research is a descriptive research. It used ethnography method. The data was collected by interview, observation, and literature study. The result said that to decide the day and pasaran of death commemoration, it is need to use modulo concept. We used modulo 7 to decide the day, because there are seven days in a week and to decide the pasaran we used modulo 5 because there are five pasarans. It means that death commemoration tradition in Javanese society applied mathematical concept. Then, death commemoration tradition in Javanese is a part of ethnomathematics.


METHOD
The method used in this research is an ethnographic method. This study uses an ethnographic method because this research is related to a certain culture, namely the culture of the Javanese people. Ethnographic research methods are considered capable of digging in-depth information with broad sources. Data was collected by interview and literature study. Observation and interviews were conducted with the people of Sumurjomblangbogo village in Pekalongan Regency. Interviews were conducted in depth. Based on Alangui, there are four generic questions in ethnomathematial research. They are where to look, how to look, what it is, and what it means. The framework is presented in Table 1.

RESULTS AND DISCUSSION
One form of Javanese culture is the tradition of death commemoration. Tradition is the process of inheriting or passing on norms, customs, rules, assets [6]. The passing can be changed, lifted, rejected and combined with a variety of human actions. The tradition of commemoration is carried out every time someone dies and is carried out by the bereaved family.
According to Sholikhin[7], the tradition of commemorating death in Java comes from the socio-religious tradition of the Muslim Campa people (inhabiting the South Vietnam area until they were expelled around 1446 and 1471 AD). Bratawdjaja stated that broadly speaking, the death commemoration procession consists of eight activities, namely (1) geblag or commemoration after burial, (2) nelung dina or 3 rd day death commemoration, (3) mitung dina or 7 th day death commemoration, ( 4) matang puluh or 40 th day death commemoration, (5) nyatus dina or 100 th day death commemoration, (6) mendak sepisan or commemoration after one year of death, (7) mendhak pindho or commemoration after two years of death, and (8 ) nyewu or 1000 th day death commemoration [8].
The calculation of the day of commemoration starts from the moment the deceased dies, not from the time he is buried. Therefore, the seven-day salvation is carried out on the seventh day of death or six days after the day of death, the 40-day salvation is carried out 39 days after the day of death, the 100-day salvation is carried out 99 days after the death day, the 1000-day salvation is carried out 999 days after the death day.
Based on the results of an interview with one of the people in the village of Sumurjomblangbogo, the death celebration is usually filled with reading prayers and verses from the Koran which is commonly referred to tahlilan. The funeral ceremony began with the bereaved family inviting neighbors and relatives to attend the funeral. Neighbors and relatives who were invited to take part in the commemoration event were male. Meanwhile, female neighbors or relatives can help in the kitchen. The event ended with the distribution of blessings (food) for the invitees who took part in the salvation event.

Discussion
In the Javanese calendar, pasaran consists of legi, pahing, pon, wage, and kliwon. Meanwhile, in a week there are seven days, namely Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. In mathematics, there is a concept namely modulo. If and are integers, then ≡ ( ) if and only if there is an integer such that = + [9]. In division algorithm, we called as remainder with 0 ≤ < . There are five pasaran, so modulo 5 is used to determine the pasaran. Therefore, to determine the day of commemoration, modulo 7, is used. The order of the day and pasaran must be considered. The order is always the same and rotates according to the order. The form of the calendar that contains the day as well as the pasaran can be seen in Figure 1. The 7 th day of the death The 7 th day death commemoration is held on the seventh day of death or on the 6th day after death. Therefore, the determination of the 7 th day of death commemoration is 6=6(mod 7)=-1(mod 7). This result means that the 7 th day death commemoration falls on 6 days after the day of death or on one day before the day of death. Meanwhile, the determination of the 7 th day in pasaran is 6=1×5+1(mod 5)=1(mod 5). The 7-day salvation falls on 1 pasaran after the pasaran at the time of death. For example, if someone dies on pahing Thursday, the 7 th day death commemoration will be held 1 day before Thursday, namely Wednesday and on 1 pasaran after pahing, namely pon. So, if someone dies on pahing Thursday then the 7 th day death commemoration will be carried out on pon Wednesday.
The 40 th day of the death The 40 th day death commemoration is held on 39 days after death. Days and pasaran for 40 th day death commemoration consecutively are 39=7×5+4(mod 7)=4(mod 7) and 39=5×7+4(mod 5)=4(mod 5=-1(mod 5). This means that the 40 th day death commemoration will fall on 4 days after the day of death and 4 pasaran after the death pasaran or 1 pasaran before the death pasaran. For example, if someone dies on pon monday then the 40 th day death commemoration will be held on Friday, which is 4 days after Monday and on the pahing, which is 1 pasaran before the death pasaran.
Nyatus (The 100th day of the death) The 100 th day death commemoration is held on 99 days after death. The determination of the day and pasaran of nyatus respectively are 99=7×14+1(mod 7)=1(mod 7) and 99=5×19+4(mod 5)=4(mod 5)=-1 (mod 5). These mean that the commemoration will fall on one day after the day of death and on one pasaran before the pasaran death. For example, if someone dies on kliwon Thursday then nyatus will be held on Friday, which is one day after Thursday and on the wage, which is one pasaran before the kliwon.
Mendak pisan (one year after death) In the Javanese calendar, one year consists of 354 days or 355 days. Therefore, the determination of the day in mendak pisan is 353=7×50+3(mod 7)=3(mod 7) or 354=7×50+4(mod 7)=4(mod 7). Meanwhile, the pasaran determination for mendak pisan is 353=5×70+3(mod 5)=3(mod 5)=-2(mod 5) or 354=4×71+4(mod 5)=4(mod 5)=-1(mod 5). These results mean that mendak pisan will fall on 3 days after the day of death or 4 days after the day of death and in three pasaran after the death pasaran or in one pasaran before the death pasaran. For example, if someone dies on pahing thursday then mendak pisan will be held three days after Thursday, that is sunday and on three pasaran after pahing, that is kliwon. If in that year there are 355 days, then mendak pisan will be held on four days after Thursday, that is monday and on one pasaran before pahing, that is legi. So if someone dies on pahing Thursday mendak pisan will be carried out on kliwon Sunday or legi monday.

Mendak pindho (two years after death)
Since one year in javanese calender consists of 354 or 355 days, then mendak pindho commemoration is carried out on 707 days of death or 708 days of death if it passes a leap year. The determination of the day for mendak pindho commemoration is 707=7×101(mod 7)=0(mod 7) or 708=7×101+1(mod 7)=1(mod 7). Meanwhile, the pasaran determination for mendak pindho commemoration is 707=5×141+2(mod 5)=2(mod 5) or 708=5×141+3(mod 5)=3(mod 5)=-2(mod 5). These results mean that the salvation mendak pindho will fall on the same day as the day of the death or one day after the day of death and on two pasarans after the pasaran death or on three pasarans after the pasaran death. For example, if someone dies on kliwon Thursday then mendak pindho will be held on Thursday because it is the same day as the death day and on two pasarans after kliwon that is pahing, then mendak pindho will be held on pahing thursday. If it passes a year that have 355 days, then mendak pindho will be held on Friday because it is one day after the death day and on three pasarans after kliwon that is pon, then mendak pindho will be held on pon friday.

Figure 2. Death Commemoration
Nyewu (The 1000th day of the death) Determination of the day and pasaran for the nyewu salvation respectively are 999=7×142+5(mod 7)=5(mod 7)=-2(mod 7) and 999=5×199+4(mod 5)=4(mod 5)=-1 (mod 5). Based on these results, nyewu will be held on 5 days after the day of death or on two days before the day of death and on four pasarans after the death pasaran or one pasaran before the death pasaran. For example, if someone dies on legi Wednesday, nyewu will be held on Monday, which is two days before Wednesday and at the kliwon, which is one pasaran before legi.
The following is an example of determining the day and pasaran for each death commemoration if someone dies on kliwon Friday. In summary, the formula for the day of death commemoration can be seen in Table 3. Meanwhile, the formula for pasaran of death commemoration can be seen in Table 4.

CONCLUSION
The tradition of death commemoration in Java consists of (1) geblag or commemoration after burial, (2) nelung dina or commemoration of three days of death, (3) mitung dina or commemoration of seven days of death, (4) matangpuluh dina or commemoration of 40 days of death, (5) nyatus dina or commemoration for 100 days of death, (6) mendak pisan or commemoration for one year of death, (7) mendhak pindho or commemoration for two years of death, and (8) nyewu or salvation for 1000 days of death. The Javanese calendar uses the day and also pasaran. Pasaran is a Javanese day which consists of only 5 days, namely kliwon, legi, pahing, pon, and wage. The order of the day and pasaran is always the same and it rotates according to the order. So, to determine the day and pasaran for death salvation, modulo 7 and modulo 5 are used, respectively. Modulo 7 is used to determine the day of death. Meanwhile, modulo 5 is used to determine the death pasaran.
The 7 th day of death (mitung dina) falls on 1 day before the day of death and 1 pasaran after the pasaran of the death. The 40 th days of death falls on 4 days after the day of death and 1 pasaran before the pasaran of death. The 100 th day of the death falls on 1 day after the day of death and 1 pasaran before the death pasaran. Mendak pisan (one year death commemoration) falls on three days after the death day and three pasaran after the death pasaran or on four days after the death day and one pasaran before the death pasaran. Mendak pindho (two years death commemoration) falls on the same day as the death day and two pasarans after the death pasaran or on one day after the death day and three pasaran after the death pasaran. While nyewu falls on 2 days before the death day and 1 pasaran before the death pasaran.

Suggestion
The existence of Javanese culture in the form of a death salvation tradition which is a form of ethnomathematics can be used by teachers in schools as an example of the application of mathematics in life and used to organize fun mathematics learning.