Remembering the hindu festivities mathematically by the balinese using integer operations and least common multiple

Ethnomathematicsis considered as a new study in mathematic education. As a study, numerous regions in this world starts to explore through ethnomathematics, including Indonesia. As the intersection between mathematics and mathematical modelling and culture, ethnomathematics exists in various society’s cultural elements, including in the Balinese Hindus’ festivities. To find the mathematical concept used in determining the festivity days, the researcher(s) conducted ethnographic research in Bali Mula society in Kintamani District, Bali. Participation observation, in-depth interview, and literature and documentation were used in collecting the data. As the result, the researcher(s) revealed that the mathematical concept used is integer operations, least common multiple, mixed fraction, and number sequences. Since it contains mathematical concept used in junior high, thus ethnomathematics of “4-hindu’s festivities” may be used as context in mathematics learning. By using ethnomathematics as the context, the researcher(s) expect that it will help teachers in motivation their students to learn mathematics.


Introduction
Bali is the name of an island in Indonesia. The majority of people in this island are Hindus Balinese tribes. There, religion and customs unite themselves and transform into the unique characteristics of Balinese people. Also, supported by its extraordinary nature, Bali becomes both national and international tourism destination. Its abundant cultures at least affects Balinese people's mindset, including the ability in mathematics. However, someone's ability in mathematics is affected by his cultural background, and hung on what he sees and feels or experiences [1].
Mathematics, as one of disciplines, is definitely correlated with culture. There are people's daily activities with mathematical applications within. For example, people in Kudus has used the concept of circle, triangle, trapezium, rectangle, block, square pyramid and triangular prism in their daily life [2].
The correlation between mathematics and culture is called as Ethnomathematics. It combines mathematics practiced, adopted and combined within cultural practice by numerous communities ordinarily [3]. Mathematical concept can also be learnt in each stage that allows the learner to attach it in the cultures of Balinese society or others.
To date, there are a lot of researches on Ethnomathematics in mathematics and mathematics education. As the research area, Ethnomatematics may refer to a research to describe how mathematics is created, taught, emphasized and specialized in cultural system [4]. Thus, such researches may be in the form of exploration or its use.
As the attempt to use ethnomathematics, the thing we can learn from mathematics teacher is how to use ethnomathematics in mathematical learning. It is possible since children who go to school basically do not establish themselves in "tabula rasa". They have abundant collection of ideas and experiences according to cultural aground and daily activities of the integral part of cognitive function, and both of them are visible and popping up [5].
Ehtnomathematics may also be advantageous in the development of mathematics curriculum. This is in line with a thought mentioning that it very makes sense to integrate ethnomathematics an mathematics curriculum, because such integration may result in a comprehended and respected noble value within a culture [4]. Besides, the use of facts in students' daily life makes mathematical learning be more meaningful and exciting [6].
The above mentioned experts' perspective implies the significance of the investigation of the noble value contained in society's culture to improve the education quality, therefore, exploration by ethnomathematics is the initial step to figure out the concepts of mathematics taught at school in the cultural activities of Bali Mula Society. As conducted by the previous researchers, the research(s) observed each cultural activity to find out the types of applied mathematical concepts.
The Ethnomathematics of Balinese Sculptors applied similarity, shifting, and rotation; meanwhile, the Ethnomathematics of Balinese Calendar Compiler adopted repetition and confluence [7] On the other hand, ethnomathematics applied in Balinese webbing may be found in the use of tessellation in webbing pattern. The existing pattern uses the concept of rectangle, square, and regular tessellation [8].
According to the above notion, the research(s) conducted an ethnographic research to explore the ethnomathematics deeper, especially in the society of Bali Mula. The religious system in determining festivity days is the main focus in this research. Furthermore, this research aimed to figure out the types of mathematical concepts used by Balinese society in determining their festivities. After the researcher(s) found the mathematical concepts in the cultural elements, it was mapped with the mathematics materials used in the corresponding education levels. It is because Ethnomathematics will be relevant to mathematics learning used at schools [9].

Method
This research is conducted in qualitative way. A qualitative research may refer to as approach to explore and comprehend the meaning of social issues within the society, either individually or in group [10]. Specifically, this research is an ethnographic research with Bali Mula society in an area as a study. From March to June 2017, the researchers collected the data by using participant observation technique, in depth interviews, and documentation. In detail, the researchers described the data collection as follow:

Result and Discussion
There are many festivities in Hindu. There are daily festivities, which are based on wewaran, pananggal and panglong, pawukon, and sasih. Daily festivities are in the form of Yadnya Sesa and Tri Sandya. Yadnya Sesa refers to festivities of food offerings (in small amount) before eating or after cooking.
Generally, wewaran-based festivities are divided into two categories: based on 3-days week (triwara) and 5-days week (pancawara); and 5-days week (pancawara) and 7-days week (saptawara). Wewaran refers to the rhythm of days based on the rhythm of universe. In wewaran, there are 10 types of week that has been preserved by Balinese society. The naming is according to the number of days in each week, they are 1-day week (ekawara), 2-days week (dwiwara), 3-days week (triwara), and so on until 10-days week (dasawara). Meanwhile, for society living in several areas in Java island and West Java, they commonly knows two of those week types: 5-days week and 7-days week.
Each one cycle of saptawara has its own name. There are 30 names for saptawara popularly called pawukuan or pawukon. So, one wuku occurs in 7 days started from Sunday to Saturday. Galungan and Kuningan are festiviries conducted based on pawukon. Meanwhile, festivities of Siwalatri dan Nyepi are commemorated based on sasih (month) in pranatamangsa, in which the former occurs in sasih kepitu (7 th month) while the latter in sasih Kesanga (9 th month).
Balinese Hindus depends such festivities on Desa-Kala-Patra. Such terms mean Place-Time-Situation. The term verily affects the festivities, mainly for Balinese Hindus living outside Bali Island. Other than the influence of Desa (place), Kala (time) also plays critical role. While time, in its largest scope, may refer to the exact time to conduct such festivities.
Considering numerous festivities, a mean for the purpose of remembering or calculating them are surely needed. Self calculation, moon's position observation, wall calendar check, internet access, or android-based application are the means to determine the festivities. In this section, the researchers present how Bali Mula society in Kintamani figure out the 4-Hindus festivities (Buda Kliwon, Tumpek, Buda Cemeng, and Anggara Kasih).  The general question that new Hindus parents mostly find is "how old is your child?." To properly answer such question, the parents will start to count it. The degree of difficulty in calculating such age can be categorized as follow: between the age of 1-210 days (0-6 months) and more than 6 months. For example, people who ask about the child's age in following days: In this case, parents tend to tell their kid's age in this month unit: "7 days to five months" 2) > 6 months  On Redite Umanis Ukir Ukir exactly occurs after Landep. It is quite easy to answer question related to child's age when such question comes on the day of Redite Umanis Ukir or the other day in Ukir wuku. Parents will celebrate the child's birth day (otonan) by creating upakara according to their respective capability. Therefore, question related to age comes on the day of Redite Umanis ukir for a children who was born on the day Anggara Umanis Landep may be easily answered. The distance between Anggara Umanis Landep to Anggara Umanis Landep is 6 months (1 oton) or its multiple. Anggara Umanis Landep occurs 5 days before Redite Umanis Ukir. Thus, the child is 6 months 5 days or 12 months 5 days or 18 months 5 days on the day of correspondingRedite Umanis Ukir.
b. Literate society 1) Self-calculation. People who are literate and have books as their reference may count the festivities days by themselves. There are standard rules specified in a printed book on how to count the 4-Hindus Festivities. Specifically, the method to determine Buda Keliwon, Tumpek, Buda Cemeng, and Anggara Kasih is by using the following formula [12]: The number of Wuku should be divided by 5, if The remainder is 1, thus it means the festivities day of Buda Keliwon The remainder is 2, thus it means the festivities day of Tumpek The remainder is 3, thus it means the festivities day of BudaCemeng The remainder is 4, thus it means the festivities day of Anggara Kasih The remainder is 5, thus it means of Pengembang For example, Wuku Sinta containing 1 as the wuku number, then 1 : 5 = 0 with 1 as the remainder. So, in Wuku Sinta the festivities to celebrate is Buda Kliwon. The calculation above may be simplified by ordering all of wukus in sequence by using fingers. It starts with folding the thumb for Wuku Sinta, index finger for Wuku Landep so on until the five wukus calculated.  (4)   presented in the square as the confluence of row and column. For example, in Wednesday row (Buda), the second column showing the date of 10 May 2017 refers to the day of Kliwon (Pancawara). Therefore, the festivities day of Buda kliwon occurs in such day.

3) Modern application
The information may be accessed on a website. Other than database-based on website, this information can also be accessed through application. There are various choices of androidbased application containing the information of Hindus festivities available on playstore.

Conclusions
Balinese Hindus society have many methods to figure out their festivities days. The differences between background and literacy also determine their distinctive methods. For illiterate society, they use their environmental observation as their guideline in determining the festivities days. Meanwhile, the literate society have more methods in determining such events, such as selfcalculation, wall calendar check, moon's position observation, or internet access and android-based application. The calculation of the 4-Hindus Festivities (Buda Kliwon, Tumpek, Buda Cemeng, dan Anggara Kasih) may be done in mathematical concepts, they are: integers operations, least common multiple, mixed fractions and sequence. Since it fully contains mathematical concepts, the determination of the 4-Hindus Festivities may be used as the context in mathematical learning at junior high school. The researchers hope that such utilization may motivate the students in learning mathematics as mandated in the regulation of minister of education.