Vol. 10, No.1, pp. 91-104, 2020 DOI: 10.23960/jpp.v10.i1.202011

Jurnal Pendidikan Progresif e-ISSN: 2550-1313 | p-ISSN: 2087-9849 http://jurnal.fkip.unila.ac.id/index.php/jpp/

Ethnomathematics in Temple and Its Relevance in Mathematics : A Literature Study

Wiwit Kurniawan1, Tri Hidayati3 1Department of , Universitas Pamulang, 2Department of Informatics Engineering, Universitas Pamulang, Indonesia

*Corresponding email: [email protected]

Received: 03 February 2020 Accepted: 04 March 2020 Published: 02 April 2020 Abstract: Ethnomathematics in Borobudur Temple and Its Relevance in . Objectives: The purpose of this literature study is to reveal the forms of mathematical culture in temples in Indonesia, especially in the Borobudur Temple. Methods: The object of this study is the scientific literature on the theme of ethnomathematics and Borobudur Temple. The stages of this research are determining the research problem, finding and evaluating relevant literature, conducting data analysis, and compiling the results.The data analysis technique used was hermeneutics analysis. Findings: In this study, it has been revealed that Borobudur Temple is a building that has been designed with great mathematical techniques. There is a numbering system, so-called watakwilangan and primbon in Javanese culture that regard the number does not only refer to certain quantity, but also a character affecting human life. In addition, geometric shapes appear on the structure, both basic geometric shapes, and fractal shapes. Conclusion: There are unique mathematical concepts in the form of Javanese number systems, logic, and space configurations that can create a beautiful and persisting building.

Keywords: ethnomathematics, borobudur, culture, mathematics, education.

Abstrak: Etnomatematika pada Candi Borobudur dan Relevansinya pada Pendidikan Matematika. Tujuan: Penelitian studi literatur ini bertujuan untuk mengungkap bentuk-bentuk matematika budaya pada candi di Indonesia, khususnya Candi Borobudur. Metode: Objek kajian ini adalah literature yang bertema etnomatematika dan candi Borobudur. Tahapan penelitian ini adalah penentukan masalah penelitian, mencari dan menilai literature yang relevan, melakukan analisis data dan menyusun hasil. Temuan: Pada penelitian ini terungkap bahwa Candi Borobudur adalah bangunan yang dirancang dengan teknik matematika yang tinggi. Sistem bilangan pada budaya jawa tidak hanya mewakili jumlah tertentu, tetapi juga suatu karakter yang mempengaruhi kehidupan manusia. Selain itu, bentuk geometris muncul pada bagunan itu, baik bentuk geometris dasar maupun bentuk fraktal. Kesimpulan: Terdapat konsep matematika yang unik baik berupa sistem bilangan jawa, logika dan konfigurasi ruang yang bisa membuat bangunan menjadi indah dan bertahan.

Kata kunci: etnomatematika, borobudur, budaya, pendidikan matematika

To cite this article: Kurniawan, W. & Hidayati, T. (2020). Ethnomathematics in Borobudur Temple and Its Relevance in Mathematics Education: A Literature Study. Jurnal Pendidikan Progresif, 10(1), 91-104. doi: 10.23960/jpp.v10.i1.202011.

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 INTRODUCTION Indonesian culture is vibrant and diverse; A universal application and value-free view this cultural richness should be used as a learning of mathematics makes mathematics be medium. In the ethnomathematics view, there are recognized as something independent from the various forms of local mathematics found in every cultural root, local context, and daily life. This culture, and Indonesian culture is no exception. understanding causes mathematics education be Various mathematical activities and concepts such presented in abstract terms, and seemingly it does as measuring, calculating, and classifying may not touch the realm of application, so students emerge in multiple aspects of Indonesian culture. find mathematical concepts is difficult to For example, numerous temple building understand. Gilsdorf explained that a certain constructions in Central and Yogyakarta have concept of mathematics is was generated from a unique patterns and sophisticated geometric certain culture (Gilsdorf, 2012). This kind of shapes. Regarding the building, there are learning makes mathematics be difficult for underlying patterns in the form of geometric students because the concepts do not originate shapes such as circles, triangles, squares, and or emerge from their own culture. Western rectangles. Furthermore, the Javanese temple mathematical concepts are incompatible with what pattern design signifies a form based on fractal is in the local culture. geometry(Situngkir, 2018). These abundant On the other hand, local culture has its own Indonesian mathematical heritage should be a mathematical concepts that have been used in appreciated knowledge that enriches mathematics local society. However, due to the dominance of learning at school. Western education, local mathematical concepts Marcia Ascher explained that mathematical were discarded and not recognized. As a result, ideas involve numbers, logic, or spatial the local mathematical concepts configuration and, in particular, the combination (ethnomathematics) will be marginalized and or organization of these into systems or structures. disappear because they are not considered a valid (Ascher, 2017, p. 2). Ascher is a pioneer and a form of mathematics. Ethnomathematics helps to key thinker in ethnomathematics discourse. She shape meta-awareness of the role of mathematics has provided an elementary concept of in society and culture so that the mathematics as an anvil to trace local mathematics ethnomathematics program can reveal how in non-western cultures. So, these primary traditional mathematics plays a role in its own concepts would guide ethnomathematics culture (Rosa & Gavarrete, 2017, p. 3). researchers to find indigenous mathematical Mathematical education requires a radical reform concepts in a form that may be considered that enables cultural and critical aspects different from modern and formal mathematics. embedded in the philosophical ground of Rosa and Gavarrete explain that the concept mathematics teaching (Nunez, 2015). of ethnomathematics research can be understood Mathematics learning cannot be separated from as to how mathematics is made of many the culture and historical context that has been historically rich, diverse, and distinct traditions shaped it. Mathematics (how mathematical concepts are formed by rich, needs to deliver mathematical views that are inter- varied and diverse cultural traditions (Rosa & connected with the local culture and customs of Gavarrete, 2017, p. 4). In this case, it is essential students so that they can understand mathematics to emphasize that cultural diversity is vital in seeing through understanding their own local culture. how a mathematical concept is formed and Kurniawan & Hidayati, Ethnomathematics in Borobudur Temple... 93

understood by humans being. This idea provides of the interior of local societies. Mathematics, in multicultural spectacles in understanding this case, ethnomathematics, may appear in every mathematics. environment and everyday life. Rosa & Orey explained that D’Ambrosio saw that various forms of ethnomathematics attempts to establish relations activities considered as non-academic and do not between mathematical ideas and daily activities pertain to expert actually may contain elements embedded in local practices (emic) and academic of cultural mathematics. Many professions, such conceptual frameworks (etic). Ethnomathematics as farmers, traders, and laborers, have their are attempts to form relationships between concepts in dealing with problems, and they can mathematical ideas and procedures that exist in carry out mathematical conceptualizations of the local practices and frameworks of academic reality they understand. However, with a narrow conceptual (Rosa & Orey, 2016, p. 5). In this view of mathematics and an understanding of understanding, ethnomathematics is better academic mathematics as the “true “mathematics, understood as an attempt to explore how people the mathematical forms that emerge in daily life understand mathematics in everyday life. Rosa are excluded from criteria recognized as and Orey have provided that the mathematics. As explained, there are ethnomathematics is a way to comprehend other mathematical notions of peoples that written mathematical forms. Also, Rosa & Orey have put history has hidden, frozen, or stolen. Including the ethnomathematics as a bridge for mathematics these ideas makes it clear that what is labeled learners who have local mathematical concepts “Western” mathematics is more accurately called to understand formal mathematics, and vice versa “world mathematics” (Powell & Frankenstein, (Rosa & Orey, 2016). 1997, p. 8). Ethnomathematics forms have Various studies examine ethnomathematics occasionallydifferent philosophical foundations and its role in mathematics education. Most of with formal mathematics. In the term of learning, them, using ethnomathematics as a perspective It can be used as enrichment material to introduce to uncover how local mathematics exists in a different kind of mathematics to students as Amit Indonesian culture. These studies show that & Quoder stated that ethnomathematics seeks Indonesian culture has many aspects of to establish relationships between mathematical mathematics that need to be revealed. and cultural content from learners and sometimes Imswatama & Lukman (2018) has conducted a the appropriate that is in accordance study entitled “The effectiveness of mathematics with local needs and local culture. However, it teaching material based on ethnomathematics.” may differ from planned teaching programs (Amit In this study, it has been proven that & Abu Qouder, 2017). ethnomathematics has a positive impact on The ethnomathematics concept from students’ ability to learn mathematics subjects. D’Ambrosio takes the emphasis on openness and Abi (2017) discussed how and what it looks equality. Much further, according to D’Ambrosio, like to integrate ethnomathematics into the ethnomathematics is a way of celebrating diversity mathematics curriculum and learning models. This in the realm of mathematics. In addition to the study shows that ethnomathematics can bridge issue of equality, ethnomathematics, in mathematics and daily life, which is easier to learn D’Ambrosio’s view, is ubiquitous mathematics. by students. Suwito & Trapsilasiwi (2016) has Mathematics does not only appear in the cultures conducted a research aimed to develop 94 Jurnal Pendidikan Progresif, Vol. 10, No. 1, pp. 91-104, April 2020

innovative learning tools that can foster and relevance to mathematics education is original and preserve culture through learning mathematics. has never been done before. Moreover, Muhtadi (2017) revealed various On the various problems and theoretical forms of Sundanese ethnomathematics, which are studies above, this research discusses grouped into three activities, namely: estimating, ethnomathematics. Ethnomathematics is basic measuring, and making patterns. In the three forms mathematical concepts such as patterning, of ethnomathematics, there was concepts in the calculating, and predicting that exist in local form of cubic (a unit to measure volume), brick cultures. The object of the study is the Borobudur (a unit to measure land area), and a lead-way Temple, its discourse, history, and culture written pathway (swing model). Rani (2018) has in academic literature. This temple is the largest conducted a study, “Ethnomathematics at Ratu in Indonesia and is considered one of the Boko Temple as a Realistic Mathematics wonders of the world. This research explores Learning.” In his research, Rani argued that in ethnomathematics form in Borobudur Temple and the architecture of the Ratu Boko Temple building, its relevance to mathematics learning. Mathematics several buildings have a mathematical education in ethnomathematics perspective is relationship, namely flat geometry material such mathematics education that empowers the as triangles, squares, rectangles, trapezes, knowledge and local wisdom of Indonesian parallelograms, and circles. Rani concluded that culture to be synthesized into mathematics ethnomathematics acts is a bridge between teaching. Indonesian culture explored is restricted abstract mathematical concepts and reality or to material and non-material aspects. Material concrete objects in everyday life. cultural elements in the form of Borobudur temple Research from Abi and Siwito & architecture, while non-material culture is local Trapsilasiwi has provided evidence that knowledge of Javanese temple architecture. This ethnomathematics exists in Indonesian culture and study has served innovative knowledge that can simultaneously be used in school mathematics enriches mathematics education by providing learning. The study from Muhtadi and Rani, in reconciliation of local culture and modern more detail, reviewed the concepts of ethnic mathematical concept. Most important thing is mathematics that exist in Sundanese and Javanese that the students can learn mathematics through cultures. Muhtadi and Rani described how the their own culture. essential elements of mathematics emerge in different ideas in Sundanese and Javanese culture.  METHODS Although Rani presents ethnomathematics in the This research uses a qualitative approach, Javanese culture of the Boko temple, it did not and the method devised was a literature study or yet mention other more significant and iconic literature review. A qualitative approach is a temples such as Borobudur. Although Abi and research concept that emphasizes the description Siwito & Trapsilasiwi have discussed how of quality over quantity, as well as revealing the ethnomathematics in Indonesia in the realm of meaning of a phenomenon under study (Taylor, learning, both have not yet explored Bogdan, & DeVault, 2015). In this study, a ethnomathematics in Javanese culture for qualitative approach is used to explain how education interest. Based on the analysis of the mathematical forms and concepts exist in the existing literature, the study about construction of Borobudur Temple. A ethnomathematics in Borobudur temple and its literaturestudy or literature review is a method Kurniawan & Hidayati, Ethnomathematics in Borobudur Temple... 95

that scrutinizes and analyzing specific themes and method requires analysis of historical contexts and discourses. The data and pieces of information figures where the artifacts/temples were made; studied are available on books, article journals, then, the current context is also presented in and documents relating to the research theme. interpretation. By this, the phenomenon in the form According to Templier & Paré (2015),the stages of a temple would be understood in terms of of literature study methods are: formulating the history and present. What was shown in the research questions and objectives, searching the interpretation was the historical context of extant literature, screening for inclusion, assessing Javanese mathematical culture, as well as the the quality of primary studies, extracting data, present context in the form of formal mathematics. and analyzing data. In this way, the phenomenon of ethnomathematics in the Borobudur Temple could be well Research subject comprehended because it firmly roots in The subject of this research was the authenticity in the past and, at the same time, gives discourse of Borobudur Temple, including meaning to the present. Javanese culture and the architectural design of the Borobudur Temple. In the temple architecture  RESULTS AND DISCUSSION design, it involves the shape, size, design, motifs, The perspective used to uncover and philosophies contained therein have a mathematical knowledge of these cultural artifacts mathematical concept. is the ethnomathematics concept of Ascher and Data collection D’Ambrosio. In the understanding of Data collection methods used in the form ethnomathematics from Ascher, forms of of literature review and documentation. The mathematics outside formal mathematics are literature review was used to reveal the theories recognized as ethnomathematics. In local cultures, about ethnomathematics, the historical idea on mathematical ideas do not always take the form Borobudur Temple, and Javanese culture that of modern mathematics. Ascher mentioned that relates to the mathematical concept and the ethnomathematics in non-modern cultures could temple. The documentation method was carried be seen in the concepts of numbers, logic, space out to examine various record sources related to configuration, and systematization, or pattern the research subject, including reconstruction of formation. Of course, the idea will be different Borobudur record and particular pictures from what is in modern mathematics. Therefore, portrayed the temple and its detail. it is necessary to have sufficient cultural horizons and strong historical knowledge in interpreting the Data analysis forms of ethnomathematics in the Borobudur The data analysis method used was Temple. Hermeneutic analysis. This analysis delivers the Besides Ascher, the ethnomathematics possibilities of interpretations in thinking. In so concept used as a lens of thought was doing, methods of hermeneutics analysis can D’Ambrosio’s ethnomathematics theory. Unlike uncover and present the structure of the meaning Ascher, who saw ethnomathematics narrowly, of lived experience (Ho & Leung, 2017). With D’Ambrosio comprehended it in a global and vast this analysis, the mathematical concepts that exist view. D’Ambroio stated that the formation, in Javanese culture in the form of Borobudur systematization, and use of local mathematical Temple would reveal. In understanding Javanese knowledge are an integral part. Therefore, mathematical phenomena, the hermeneutic D’Ambrosio gives three ideas as a continuous and 96 Jurnal Pendidikan Progresif, Vol. 10, No. 1, pp. 91-104, April 2020

bound process of ethnomathematics, namely In ancient Javanese culture, the concept of literacy, matheracy, and aristocracy. numbers adopted the Indian number system, In the first part, it will be discussed how which is a ten-based number. Brahmins and the ethnomathematics concept of Borobudur Pandita widely used this number in various Temple in the Ascher perspective, in the form of activities, both scientific, worship, and daily. The numbers, logic, space, and patterns. The concept of numbers in Javanese culture differs numerical system in ancient Javanese culture is from modern number systems even though both the numbers adopted from Indian culture, namely of them are based on ten. From the perspective a ten-based number system. The logic that of Javanese culture, the symbol of numbers does appears in the Borobudur Temple artifact is about not only indicate quantities but also symbolizes the attachment and cause-effect between humans, nature and character. In the Javanese tradition, symbols, and world realities. The space this concept is called numerical character or configuration in Borobudur Temple is in the form watakwilangan. It means that a certain number of fractal geometry, which imitates the natural does not only refer to an amount but also a form. The systematization process, in the particular character affecting human life and their Borobudur Temple, is the use of destiny. This concept commonly appears in ethnomathematics to assembly a temple that is Primbon, a Javanese numerology, which used architecturally successful and contains spiritual as guidance to everyday activity, including meaning. The following discussion is the constructing a building. Each number symbolizes elaboration of each concept. a particular character, such as 2 symbolizes men and 3 symbolizes women (Prabowo, 2010). The Concept of Numbers Besides, numbers in Javanese cosmology are Borobudur Temple is a remarkable building closely related to the reality they believe. and requires careful planning in its construction. Therefore, their numerical system had formed the Because this building is large and contains concept of primbon. It is a calculation of extraordinary details, there is certainly a concept quantification (the conversion of phenomena into of numbers that support its creation. The number numbers) of certain properties in daily life, which of stupas in the Borobudur Temple is 73 units, results as a guide to action. The measurement of with more detail is one main stupa, 32 stupas on Borobudur architecture was relied on this the first circular terrace, 24 stupas on the second numerical concept. Its width and length reflect circular terrace, and 16 stupas on the third circular certain characters that supernaturally bless the terrace. The number and level/terrace were neatly society and human life. arranged and it systematically indicates that they In addition, numbers in Javanese already have an excellent numerical system and cosmology do not only represent a certain amount calculation technique. In an apparent view, the but also objects or realities. All three are a different shape of Borobudur Temple, when it is seen from but mystically interconnected reality. In Javanese the side or top, it always looks symmetrical, culture, a quantity is not only written in the form although it is not entirely accurate. This of numbers but also words. This concept is called symmetrical shape indicates that there are candrasengkala or suryasengkala. For sophisticated measurements and calculations in example, enunciating 1600 as a saka year can its architecture. be written through the sentence “sirna Kurniawan & Hidayati, Ethnomathematics in Borobudur Temple... 97

ilangkertaning bumi”. The four realities are into a single building. Making Borobudur Temple representative of the numbers. Bumi represents was a different technique; it was done by number 1, kertaning represents number 6, ilang resembling the rocks and following the original represents number 0, and sirna represents size of the stones. It is like putting together a number 0. This year is the moment of the puzzle or a mosaic painting. Because it vanishment of the Majapahit Kingdom. This usedmosaic techniques, the standard and precise description shows that there are three measurement concepts such as in the modern interconnected entities in the number system in world are not needed. Javanese culture, namely numbers, character, and Logic reality. Logic is simply interpreted as reasoning for In the measurement system, the Javanese cause and effect on reality. In Javanese culture, tradition uses tala, hasta, and depa especially in the construction of Borobudur measurements. The size that often appears and Temple, the logic concept about cause and effect widely used is tala. One tala is equivalent to the appears in the development of the architecture. length of the tip of the head to the chin. There is Because they require a building that lasts forever, also an opinion that the length of the tip of the the architecture of the Borobudur Temple decides middle finger to the base of the palm. The size of the most logical structure for this purpose. The the tala was used to determine the length of basic structure that can be so durable is the various Javanese architectures. Unlike modern structure of the pyramid, which is wide below measurements that are standardized for their and pointed at the top. This construction makes measurements, the Javanese culture measurement the temple building resistant to various natural is relative to everyone. This is because the length disturbances such as storms, earthquakes, and of the head or the length of the hand is different floods. for each person. This inaccuracy gave the impact One logical strategy at Borobudur Temple to Borobudur shape and size. The shape and size to defend from rainwater by making waterways of the Borobudur Temple look perfectly (Munandar, 2018; Setyawan & Gunawan, symmetrical, but it is not. In further examination, 2018). Therefore, in Borobudur Temple, there is the size of the Borobudur Temple is not exactly a drainage system. The function of waterways symmetrical, but similar and close to (Situngkir, is to flow the rainwater from the top court to the 2018). The size of each stone, the stupa’s bottom of the temple. Various strategies diameter, and size, as well as the projection of undertaken to achieve the objectives of the temple this temple, are not accurately the same. construction rely profoundly on logic. Therefore, Hokky Situngkir said that making the concept of cause and effect on the Borobudur Borobudur Temple is similar to making Temple mega project was beneficial. In addition patterns. The concept of similarity in the pattern to philosophical and religious reasons, the shape does not use a standard rule, but with precision of the temple, location, and additional structures relying on cognition or mind (Situngkir, 2018). are logical strategies in the architecture of the Furthermore, the pattern of making the Borobudur Temple. Borobudur Temple is like a mosaic arrangement which assembly existing original stonessize. So, Space Configuration it is unlike modern building construction, that The concept of space or geometry in the makes the same brick size and then arrange them Borobudur Temple is very complicated; this is 98 Jurnal Pendidikan Progresif, Vol. 10, No. 1, pp. 91-104, April 2020

because Borobudur does not only contain the The self-similarity geometrical patterns concept of Euclidian geometry, but there is also prevalently exist in Borobudur architecture. The the concept of fractal geometry (Situngkir, 2018). form of stupas and ornaments were generated in If we look at the building of Borobudur Temple, a fractal pattern. Pattern shapes that are similar we will see square, rectangular, circular, and to each other can be seen in the following image. beam shapes. These various forms fall into the category of Euclidian geometry. Square and circle-like shapes can be seen in the Borobudur structure in the following image.

(a)

Figure 1. Combination of angular shapes and curves Nevertheless, if traced further, there are very complicated ornaments, and various shapes are similar to each other to form regular patterns. The concepts of similarity and pattern formation are the basis for forming fractal geometry patterns. The fractal pattern is a geometrical shape that is similar to itself but appears on different scales. (b) The similarity could be exact, but in a particular Figure 3. (a) real; (b) illustration pictures of case, it is quasi-exact. The following picture is fractal patterns in Borobudur (Situngkir, 2018) one example of various kinds of a fractal patterns. Another exciting thing is that, although they have similar and patterned shapes, they do not have an accurate similar size (Situngkir, 2018). This form is in accordance with the fractals structure in natural shape, like the existing patterns in the conch shell, flowers, and leaves in plants. The various forms in nature are symmetrical and resemble each other, but it is not the same. The formation of Borobudur Temple is also possible Figure 2. Fractal Pattern to follow the foundation of such fractal geometry. Kurniawan & Hidayati, Ethnomathematics in Borobudur Temple... 99

The measurement of tala and hasta is not extension styles are called birds, notches, shafts standardized in length. Then the mosaics stone and holes, and grooves, and tongues stone structure has shaped the figure of Borobudur extension (Puspitasari, Setyawan, & Rini, 2016). Temple into a beautiful fractal dimension, not Mastery over the shape and space allows them precise but orderly patterned. to make building structures composed of stone without cement and adhesive, which relies on the Systematization stone extension technique. The photo and name At Borobudur Temple, there are at least of the extension stone technique, according to four forms of extension between stones. The Puspitasari, Setyawan, & Rini, are as follows.

(a) (b)

(c) (d) Figure 4. (a) shaft and hole extension; (b) bird tail extension; (c) grooves and tongue extension; (d) notch extension (Puspitasari, Setyawan, & Rini,2016)

The various forms of the connection allow Fractal Geometry in Borobudur for the preparation of a large number of aids but To perceive Javanese cultural remain robust and sturdy. Form the pictures ethnomathematics knowledge in Borobudur above, it can be concluded that Javanese culture temples, we cannot merely lay on modern had recognized the geometrical form such as mathematical knowledge. Every culture and era square, rectangle, cone, and spherical shape. has its own mathematical method that is Furthermore, they had utilized those geometrical appropriate for its people. Therefore, the form formations to create intelligently capable devices of ethnomathematics in temples in Indonesia enabling them to resemble stones firmlywithout cannot be seen through the concept of modern epoxy cement. mathematics and Euclidean geometry. There must 100 Jurnal Pendidikan Progresif, Vol. 10, No. 1, pp. 91-104, April 2020

be another understanding of geometry that can numbers, the logic of causation, understanding explain the mathematical phenomena in the of form, and space. Borobudur temple. In Indonesian culture, Matheracy Process Situngkir explained that the geometry used was The process of matheracy is the formation not geometry provided with rulers and protractors of existing understandings, symbols, and ideas into (Situngkir, 2018). Geometry concepts in various a unity in the form of theories that have narratives. artifacts, both batik, and temple, are not based Borobudur Temple is proof of mind ingenuity and on the basic form of Euclid’s geometry. spiritual elegance. Both are united into a mega Indonesian culture understands geometric shapes, structure that lasted for thousands of years. such as natural forms. Therefore, understanding Various sites in the Borobudur Temple have their of geometry in Indonesian culture cannot be own meaning. reduced to Euclidian geometry. What is in nature is far more complicated than that. However, it is Technocracy Process much more irregular. The boundary between sea The process of aristocracy is the utilization and land is not a line; mountains are not cones, of existing knowledge for practical purposes. clouds are not curved lines (Situngkir, 2018, p. From searching history books, it is found that xxi). there were many obstacles in the construction of There is a 4: 6: 9 length ratio for the overall Borobudur Temple. However, due to the mastery size of the legs, body, and head of the temple, of concepts and good mathematical knowledge, both vertically and horizontally (can be seen at these difficulties could be overcome. A variety of figure. 3.b). The height and width of the temple existing knowledge allowed the construction of parts called “feet,” “body,” and “head” roughly an extraordinary Borobudur Temple. meet numerical comparisons of 4 to 6 and 9 (Situngkir, 2018, p. 52). This comparison does Borobudur Ethnomathematics in not only show the size and choice of architectural Mathematics Learning techniques but also an expression of art and From the above study, it can be seen that beauty. In nature, many similar similarities appear. the concept of ethnomathematics in Borobudur Therefore, the Borobudur Temple building is a Temple is in the form of numbers, logic, space building that is in harmony with nature’s design, configuration, systematization, and fractal or in other words, its construction mimics how geometry. In terms of the process, there are also nature has been made. This is an unusual method processes of literacy, matheracy, and of mathematics known by modern society technocracy. Various ethnomathematics concepts (Situngkir, 2018, p. 58). are very relevant for learning mathematics. In the case of Borobudur Temple, the use Literacy Process of ethnomathematics concepts is not only limited In ancient Javanese culture and society, to the introduction of various basic geometric there had been a robust literacy process. Literacy shapes and the number of stupas, statues, or is not only meant as the ability to read and write height and width of buildings. Ethnomathematics but the ability to capture natural phenomena and is much further than that. With the interpret them as concepts of ideas. When looking ethnomathematics process, the students are at the architecture of the Borobudur Temple, both expected to be able to learn how a certain number history and cultural point of view of Java, various system, how Indonesian culture understood the concepts play a role. Many concepts embody as abstraction process that produces geometric

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shapes, and how that kind of knowledge was A knowledge building is not generated from used to build monumental buildings and had a culturally void space, but it constructed by a solved various real problems. series of adaptations of various elements ranging In geometry, there is a process of from antediluvian cultures to hitherto existing abstraction, namely the formation of ideal ideas cultures. Mathematical education must be such as points, lines, and geometric shapes from harmonized and suitable for the particular student real reality. This process emerged in Greek coming from various cultural backgrounds culture. In Indonesian culture, especially (Knijnik & Wanderer, 2015). So, it is the Javanese, there is also a process of abstraction importance of paying attention to cultural and like that which gives rise to forms that are social relevance, and then it focuses on students’ classified as fractal geometry. It is a form having interests as one of the aspects of relevance characteristics of symmetry or self-resembling and (Kazima, 2013). It is vital to enable the teacher repeatedly forming patterns. This form appears to development of opulent mathematical ideas in the building of the Borobudur Temple. In within a context that was familiar to the students addition, the logic and connection system on the and the community (Civil, 2010). stone temple can be used as a learning process Ethnomathematics is always providing vast of technocracy. Namely, students are taught how perspective, soit can be said that being inclusive the mastery of geometry can be used to build and accepting the presence of “The Other” is the temples. spirit of ethnomathematics. Ethnomathematics learning is both reinforcements inward and outward exploration.  CONCLUSIONS Strengthening inward is taking ethnomathematics Ethnomathematics is an inclusive knowledge to sharpen mathematical concepts mathematical concept. Understanding that already exist and firmly established. The ethnomathematics provides awareness that there concept of geometry in Borobudur temple is able are other forms of formal and modern mathematics to enrich the knowledge of the school that we learn in school. Also, ethnomathematics mathematics material. On the other hand, the use describes the awareness that mathematical of ethnomathematics also aims to explore cultural concepts are something that is interconnected reality. This means that students are taught to with the cultural context in which mathematics is explore the knowledge existing in real life. It also formed. Thus, mathematics is not an abstract leads students to expand their mathematical concept that is separated from real life. understanding by acclaiming local mathematical Mathematics, in the view of ethnomathematics, knowledge and also to develop critically and is thought that originates from the struggle of life. meta-cognition (Herawaty, 2018). Widada Therefore, to develop existing mathematical (2019) has proved that the mathematical knowledge, it is necessary to explore how problem-solving abilities of students after being mathematical concepts exist in culture and social given ethnomathematics with outdoor learning life. models were higher than before being given the In the perspective of D’Ambrosian, learning models. So, ethnomathematics is suitable Borobudur Temple is a building that contains for the student to increase their problem-solving symbols and concepts of spirituality. This shows ability in a real-life situation. that Javanese culture at that time had an influential 102 Jurnal Pendidikan Progresif, Vol. 9, No. 2, pp. 10-23, November 2019

literacy, namely the ability to form ideas in symbols and its diverse approaches for and concepts of world understanding. The shape, mathematics education (pp. 23–50). symbol, and size, as well as its architecture, are a https://doi.org/10.1007/978-3-319- world understanding of ancient Javanese culture 59220-6_2 and Buddhism. The process of matheracy or the Ascher, M. (2017). Ethnomathematics: A formation of mathematical concepts (in the form multicultural view of mathematical of symbols or ideas) can be perceived in the ideas. Routledge. architectural concept of Borobudur Temple. The Civil, M. (2002). Culture and mathematics: A symmetrical shape and structure of the fractal community approach. Journal of pattern, as well as the various symbols and their Intercultural Studies, 23(2), 133-148. interpretation at each site in the Borobudur Gilsdorf, T. E. (2012). Introduction to cultural temple, has proven that they can narrate their mathematics: With case studies in the mathematical concepts. In the process of Otomies and Incas. John Wiley & Sons. technocracy, Indonesian culture has shown its Herawaty, D., Widada, W., Novita, T., Waroka, reliability with the establishment of the Borobudur L., & Lubis, A. N. M. T. (2018, Temple. Technocracy is the process of using September). Students’ metacognition on mathematical knowledge for practical purposes. mathematical problem solving through With the monumental Borobudur is a Temple ethnomathematics in Rejang Lebong, building which is symmetrical, sturdy, and resistant Indonesia. In Journal of Physics: to adverse natural effects, as well as calculations Conference Series (Vol. 1088, No. 1, p. of the location and availability of resources. So, 012089). IOP Publishing. it can be concluded that the Borobudur Temple Ho, K. H., Chiang, V. C., & Leung, D. (2017). is the result of technocracy in ethnomathematics Hermeneutic phenomenological analysis: knowledge. The concept of Borobudur Temple the ‘possibility’beyond ‘actuality’in is supported by the number system, spiritual thematic analysis. Journal of advanced philosophy, location determination, and nursing, 73(7), 1757-1766. architectural design to be able to realize the Imswatama, A., & Lukman, H. S. (2018). The Borobudur Temple building, which has survived effectiveness of mathematics teaching for centuries.That vast knowledge is a vibrant material based on ethnomathematics. learning resource which generates joyful and International Journal of Trends in engaging culture-based mathematics lesson. Mathematics Education Research, 1(1), 35-38.  REFERENCES Kazima, M. (2013). Students’ preferences for Abi, A. M. (2017). Integrasi etnomatematika contexts and their relevance to school dalam kurikulum matematika sekolah. mathematics in Malawi. African Journal Jurnal Pendidikan Matematika of Research in Mathematics, Science Indonesia, 1(1), 1–6. and , 17(3), 244- Amit, M., & Abu Qouder, F. (2017). Weaving 254. culture and mathematics in the classroom: Knijnik, G., & Wanderer, F. (2015). Mathematics the case of Bedouin ethnomathematics. In education in Brazilian rural areas: an M. Rosa, L. Shirley, M. E. Gavarrete, & analysis of the Escola Ativa public policy W. V. Alangui (Eds.), Ethnomathematics and the Landless Movement . Ancho, Inero Valbuena, Designing an Approachin Unlearning Negative ... 103

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