Pop: A Study of the Ethnomathematics of Globalization Using the Sacred Mayan Mat Pattern : Pop: A Study of the Ethnomathematics of Globalization Using the Sacred Mayan Mat Pattern Daniel Clark Orey Milton Rosa


Introduction: Introduction Before the present era of globalization, the world’s continents were separated by vast expanses of ocean and sea. Ancient peoples knew of the existence of others only through myth, legend, and the stories of conquerors and travellers. Most of humanity lived in isolated and self-sufficient cultural groups and lived and died in the same place. Recently, the world’s peoples have been linked together through extensive systems of communication, migration, trade and production.


Globalization: Globalization Globalization is an ongoing historical process that has, at its roots, the very first movement of peoples from their original homelands. Explorers, conquerors, migrants, adventurers, and merchants have always taken their own ideas, products, customs, and mathematical practices with them in their travels.


Globalization: Globalization The analysis of many of the great events of human history such as conquests by Caesar, Alexander, Cortez; the adventures of Marco Polo, the Portuguese Naval School of Dom Enrique, and the navigation of Columbus, all occurred primarily for economic reasons.


Globalization: Globalization Imperialistic adventures determined the colonial social-cultural characteristics through the imposition of non-native customs on local and diverse indigenous peoples. This form of colonialism was practiced primarily by European nations and is often referred to as the Europeanization of the world.


Globalization: Globalization In order to maintain and govern their colonies, Europeans required enormous amounts of capital and power, and settled most questions of cultural difference by force. This increased a certain amount of awareness of non-Western cultures by the colonizers, and raised many new questions for scholars about the nature of society, culture, language, and knowledge


Globalization: Globalization From the first years of the colonization, Spanish missionaries were aware of the need to learn the languages of the Indians in order to communicate with them directly and to “instruct” them in the Christian doctrine. The first Bishop of Guatemala recommended that friars and secular clergy study native dialects and compose their preaching and sermons in the mother tongues of the natives.


Globalization: Globalization Emerging theories of social evolution allowed European scholars to organize this new knowledge in a way that justified political and economic domination of others. Colonized people were considered less-evolved, thus giving the powerful sense of justification to the colonizers as they came to believe themselves more evolved. Nevertheless, an effective administration required some degree of understanding of other cultures.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge We do not really know when an interest in the mathematical practices of other cultures was first expressed. The earliest observations of distinct mathematical practices probably occurred in tandem with the first travels to different regions of the world. Travellers who came in contact with local cultures observed different customs that no doubt included different mathematically-related practices such as counting and measuring.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The Greek historian, Herodotus (484-425 BC) wrote one of the earliest accounts during his travels across his known world. In 440 BC, he wrote a book called The Histories, in which he shared his observations of the different cultures, practices, customs, habits, and mathematical practices of the peoples he met.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The globalization of mathematical, scientific, and technological knowledge brought accelerated technological progress to various parts of the world. The invention of zero and the notion of place value have been attributed to the Hindus around the 9th century, and was transmitted to the Arab peoples through religious expansion and commercial activities, war, and conquest.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge When in the 15th century the Arabs invaded Europe, they brought with them the mathematical knowledge that they acquired from India (thus the term Hindu-Arabic numeration system). Medieval Europe was influenced by the exchange food, customs, culture, science and technology. In turn, when they conquered and colonized the peoples who lived in the Americas, the Europeans introduced this system into the there.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The number system used by the Greeks and Romans was cumbersome and impractical for many uses. The adoption of the decimal number system used by the Hindus and brought to Europe by the Arabs made perfect sense. This improved ability to calculate allowed for growth in the western sciences.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The Hindus also took advantage of this same cultural interchange by learning important concepts of Greek mathematics by way of the Arabs. Despite this “Eastern” globalization, the earliest systematic use of a symbol for zero in a place value system was used by the Mayans centuries before the Hindus began to use a symbol for zero.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge Ifrah found similar results: “What is quite remarkable is that Mayan priests and astronomers used a numeral system with base 20 which possessed a true zero and gave a specific value to numerical signs according to their position in the written expression”. “So we must pay homage to the generations of brilliant Mayan astronomer-priests who, without any Western influence at all, developed concepts as sophisticated as zero and positionality, and despite having only the most rudimentary equipment, made astronomical calculations of quite astounding precision”.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge At the end of 15th and the beginning of the 16th centuries, explorers provided descriptions of different aspects of the “exotic” cultures they encountered in Asia, Africa, and the Americas. Early chroniclers of the Americas reported observations and registered data collected in relation to the cultures they encountered in their explorations.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge Using a process that can be considered ethnomathematical in nature, Juan Diaz Freyle published, in 1556, the first book of arithmetic of the new world entitled Sumario compendioso de las quentas de plata y oro que en los reinos del Pirú son necessarias a los mercadores y todo genero de tratantes: Con algunas reglas tocantes al arithmética. Translation: A Compendium Summary of the Accounts of Silver and Gold that in the Kingdoms of Peru are Necessary to Merchants and All Kinds of Dealers: With Some Rules Concerning Arithmetic.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge Freyle described the arithmetic practiced by the indigenous people of the Americas, he first described the process of the indigenous people’s assimilation of the conquering people’s mathematical knowledge. This can be perceived as a transformation of the native mathematical system through a global and cultural dynamic perspective.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge When Europeans invaded and conquered the northern part of the Americas during the early 16th century, they “began to apply commercial arithmetic to the purchase of citizens in North America from local chiefs and kings, and the later sale of those still alive, to entrepreneurs and landowners across to the Americas”.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge They too made little effort to conserve the culture of either slaves or of the indigenous tribes. Nevertheless, the latter have managed to maintain a repertoire of mathematical theories, not only in arithmetic, geometry and astronomy… but especially in connection with skills such as archery and in games of chance involving the throwing down of rods and sticks decorated in various ways.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The ascension of the Portuguese, Spanish, French, Dutch, English, and Belgian Empires in 18th and 19th centuries contributed to increasing contact with the cultures they colonized. This context allowed for an increased development of global commerce, a greater spread of the growing capitalist economy, and the industrialization of Europe. These aspects led to the present day social, cultural, and economical transformations of all societies and cultural groups on the planet.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge The newly industrialized countries continued their search for new lands as sources of supply, cheap manpower and the raw materials to be manufactured at low costs. At the same time, millions of Europeans from the lower classes were encouraged to immigrate to the newly established colonies in promise of better lives. These further exchanges allowed for a continued accumulation of data and information of distinct cultural groups that were “found” and subjugated in the colonies.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge In the 19th century, the first forms of what would become modern anthropology began to be systematized. According to some experts, as different cultures were studied during the ongoing processes of assimilation and colonization, the customs and mathematical practices of diverse cultural groups also became objects of study by many early European anthropological societies.


The Globalization of Mathematical Knowledge: The Globalization of Mathematical Knowledge In the 20th century, a growing and increasingly sensitive understanding of mathematical practices and ideas from diverse cultural groups became increasingly available through the growth of the fields of ethnology, culture; history, anthropology, linguistics, and the development of ethnomathematics. Insights from many theoretical studies signal the possibility of the sensitive internationalization of mathematical practices and ideas expressed in different cultural contexts.


The Perspective Offered by Ethnomathematics: The Perspective Offered by Ethnomathematics Ethnomathematics recognizes that all cultures and all people develop unique methods and sophisticated explications to understand and to transform their own reality. It also recognizes that the accumulated methods of these cultures are engaged in a constant, dynamic, and natural process of evolution and growth in every society. In this context, culture is a complex whole that includes knowledge, beliefs, art, laws, morals, customs and any other practices and habits assured by a member of a society.


Culture is…: Culture is… Culture is… everything you believe and everything you do that enables you to identify with people who are like you and that distinguishes you from people who differ from you. Culture is about groupness. A culture is a group of people identified by their shared history, values, and patterns of behaviour… culture is a problem-solving resource we need to draw-on, not a problem to be solved”. Lindsay, Robins, and Terrel, 2003, p. 41 Ethnomathematics looks at the mathematics of this problem-solving resource.


The Perspective Offered by Ethnomathematics: The Perspective Offered by Ethnomathematics Another presupposition of ethnomathematics is that it validates all forms of mathematical explaining and understanding formulated and accumulated by different people and cultures. This knowledge is regarded as part of an ongoing evolutionary process of change that is part of the same cultural dynamism present as each group comes into contact with each other in this even more global reality. As alternative forms of practical mathematics emerge, researchers in ethnomathematics seek to understand, explain, comprehend, and analyze practical problems in the daily lives of non-western peoples.


The Perspective Offered by Ethnomathematics: The Perspective Offered by Ethnomathematics A basic tenet of an ethnomathematics paradigm is that all cultural groups have developed unique ways to look for and accumulate knowledge. All cultures have, by necessity, evolved unique ways to quantify, count, classify, measure, explain and model the phenomena of their own daily occurrences (Borba, 1990).


The Perspective Offered by Ethnomathematics: The Perspective Offered by Ethnomathematics Some cultural groups have evolved particular ways to find solutions to everyday problems. A study of the different ways in which people solve problems and the practical algorithms on which they base these mathematical perspectives becomes relevant for any real comprehension of the concepts and the practices in the mathematics that they have developed over time.


The Perspective Offered by Ethnomathematics: The Perspective Offered by Ethnomathematics For example, when we speak of patterns and sequences, we know that humanity utilized different numeric and geometric patterns to make music, dance, or create basketry, ceramics, rugs, and fabric. Many times, these patterns possessed religious and spiritual aspects that sought to connect their own human perspective with the “divine” around them.


Ethnomathematics and Anthropology: Ethnomathematics and Anthropology One of the most important concepts of ethnomathematics is the association of the mathematics found in distinct cultural forms. Ethnomathematics as a program is much wider than traditional concepts of multicultural mathematics and ethnicity. In this case, D’Ambrosio (1990) refers to “ethno” as that related to distinct cultural groups identified by cultural traditions, codes, symbols, myths, and specific ways of reasoning and inferring.


Ethnomathematics and Anthropology: Ethnomathematics and Anthropology The focus of ethnomathematics consists essentially of a serious and critical analysis of the generation and production of knowledge (creativity), intellectual processes in the production of this knowledge, the social mechanisms in the institutionalization of knowledge (academic ways), and the diffusion of knowledge (educational ways). In this holistic context, the study of the systems that form reality and look to reflect, understand and comprehend extant relations among all of the components of the system require constant analysis of their reality. D’Ambrosio has defined ethnomathematics as the intersection of cultural anthropology, mathematics, and mathematical modelling which is used to translate diverse mathematical practices.


Ethnomathematics as an Intersection of Three Disciplines: Ethnomathematics as an Intersection of Three Disciplines


Ethnomathematics in the Process of Globalization: Ethnomathematics in the Process of Globalization All individuals possess both anthropological and mathematical concepts; these concepts are rooted in the universal human endowments of: curiosity, ability, transcendence, life and death. They characterize our very humanness. Awareness and appreciation of cultural diversity that can be seen in clothing, methods of discourse, religious views, morals, and our own unique world views can allow us to understand each aspect of the daily life of human beings.


Ethnomathematics in the Process of Globalization: Ethnomathematics in the Process of Globalization The culture of each group represents the set of data related to acquired and collected understandings of the world. It represents a set of values and the unique way of seeing the world as it is transmitted from one generation to another. The principal focus of anthropology that is relevant to our work includes such aspects of culture as language, economy, politics, religion, art, gender, sexual orientation, and our daily mathematical practices. Since, cultural anthropology gives us the tools to increase our understanding of the internal logic of a given society; an anthropological study of the mathematics of distinct cultural groups allows us to further our understanding of the internal logic and beliefs of different peoples.


Ethnomathematics in the Process of Globalization: Ethnomathematics in the Process of Globalization Knowledge is generated and intellectually organized by individuals in response to their own social, cultural, and natural environment. This knowledge is socially organized and used to recognize and explain activity in the daily lives of people. According to D’Ambrosio, observers, chroniclers, theoreticians, sages, and professionals expropriated this knowledge, and then classified, labelled, diffused, and transmitted it across generations.


Mathematical Practices as Diverse Cultural Forms of Knowledge : Mathematical Practices as Diverse Cultural Forms of Knowledge


Mathematical Practices as Diverse Cultural Forms of Knowledge: Mathematical Practices as Diverse Cultural Forms of Knowledge Both ethnomathematics and a new globalized mathematics must take care not to trivialize other cultures based on the misrepresentations of their scientific and mathematical ideas or structures. It is also important to uphold a balanced analysis that maintains a group’s cultural integrity while accurately portraying its scientific, mathematical and technological contributions.


Slide39: We have outlined here, one example of how this future scholarship might proceed…


The Great Architect: The Great Architect Many cultures share the belief that a “Great Architect” of the universe possessed certain mathematical characteristics. This “Great Architect” is, according to many Mediterranean traditions, God, Yahweh, and Allah, and, according to the Mayan tradition, named Tzakol. The knowledge of this “Great Architect” was learned and captured by many Mediterranean and ancient non-Western civilizations. Since there is more than one religious practice in both the ancient and modern world, more than one system of values, more than one name for the “Great Architect”, there is, as well, more than one way of explaining, knowing, and understanding these diverse realities.


The Mathematics of Indigenous Peoples: The Mathematics of Indigenous Peoples A study of the mathematics of indigenous peoples who were “discovered” and colonized by Europeans allows us to introduce mathematical ideas of cultural groups who have been excluded from traditional mathematical discourse. It is in this context that an ethnomathematical perspective can be used to challenge what is often known as an ethnocentric view of diverse cultural systems. Complex social organizations are typically thought of as having advanced technology and thus, a more “complicated” mathematical system; yet, indigenous cultures such as the Mayans, developed equally complicated mathematics which had an equally conscious effect on the world around them.


The Mayan Civilization: The Mayan Civilization The Mayan civilization has lived for more than 3000 years in the region now called Central America. The Mayan people are best-known by their distinct architecture, the patterns they found in their observations about the universe, the development of mathematical relationships, and a symbolic and sacred system that they developed to represent these patterns. About 7 million Mayan people are dispersed in urban and rural communities in Southern México, Belize, Guatemala, Honduras and El Salvador. With centuries of persecution, cultural insulation, and disrespect of Mayan traditions, beliefs and religion, most Mayan people now live in crushing poverty.


The Mayan Process of Globalization: The Mayan Process of Globalization For indigenous Mayan people, the violent encounter with globalization began in 1524 with the arrival of the Spanish conqueror Pedro de Alvarado. With the invasion of Central America by Europeans, the world of the Mayans, like all the other Indigenous societies in the hemisphere, came to an abrupt and extremely brutal end. Although medieval Europe was in many ways less developed than the Mayans, the conquerors arrived with an enormous military advantage: gunpowder, steel swords, and horses.


The Mayan Process of Globalization: The Mayan Process of Globalization At the same time, indigenous societies were weakened by diseases against which they had no immunity. It was the superior European technology and firearms that proved a vital factor in the conquest of the Americas by Europeans. In a quest for wealth, European invaders defeated the Mayans, and destroyed their libraries that were possibly the greatest repositories of indigenous science in the Western Hemisphere.


The Mayan Process of Globalization: The Mayan Process of Globalization Some surviving texts were carried to safety by Mayan priests. Among them was the hieroglyphic source for the Popol Vuh, which is considered by some to be the “Mayan Bible”, and the Dresden Codex, which reveals the sophistication of Mayan knowledge of astronomy and mathematics.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond The Mayans made use of a series of sacred geometric-numeric patterns that they transmitted from generation to generation. The utilization of these patterns probably originated with a species of rattlesnake Crótalus durissis, found in the region (Nichols, 1975; Diaz, 1995; Grattan-Guiness, 1997). Rattlesnake skins possess a unique diamond pattern; this particular species is called the “diamond backed rattle snake” in English.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond The contemplation of this form and geometric pattern seems to have inspired Mayan art, geometry, and architecture. The images of these rattlesnakes are constantly found in many aspects of Mayan culture. They symbolize the birth and life changes of the ancient Mayans as they enliven and crawl their way across time.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond The significant and purely abstract, patterns found in geometric rattlesnake forms are found in the fabrics and in the façades of numerous ancient buildings, monuments and architectural structures though out the ancient Maya territories. These structures aided inhabitants of the region to compute, track, trace and mark the movements of the Sun, moon, and the stars.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond Crótalus durissis


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond A rhombus representing the geometric form on the skin of the rattlesnake


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond It is possible to observe that the degrees of slope of Mayan pyramids are extremely steep and are difficult to climb comfortably. The easiest and most comfortable way to climb Mayan pyramid stairs is to climb the steps in diagonal or in a zigzag.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond The trajectories formed by the movement of the priests ascending and descending of the pyramids have the same form and geometric patterns found in the rattlesnake skin. In this case, Mayan priests ascended and descended pyramids in a criss-cross ritual that reproduced the diamond pattern of the rattlesnake.


The Geometric Pattern of the Mayan Diamond: The Geometric Pattern of the Mayan Diamond From what we understand of the Mayan cultural perspective, numbers, symbols, and words could direct the priests to deities of corresponding numerical values. This ascribed a multidimensional aspect to the art, literature and mathematics of the time. The Mayan culture used numbers based on the snakeskin pattern for a type of numerology; using the numbers from 1 to 9 could have had a sacred value and a specific significance.


The Sacred Significance of the Numbers : The Sacred Significance of the Numbers


The Sacred Mayan Mats: The Sacred Mayan Mats The word Popol present in the title in of the sacred book Popol Vuh contains the prefix Pop that is the Maya Quiché word for mat. According to Recinos (1978), Ahpop is the Mayan word that means mat. The Gods that were represented in the monuments of numerous Mayan pyramids sat on top of Pop patterns built over sacred mat patterns. The monuments themselves were constructed over mats that had magic or mystical power and used number values – providing a spiritual foundation to accompany the physical buildings.


The Sacred Mayan Mats: The Sacred Mayan Mats Diaz de Castillo affirmed that the priests and the Mayan nobility also sat on top of sacred mats for ceremonial and festivities. He also described that in the time of the conquest of the Mayans by the Spanish, important meetings were made between Spanish leaders and the Mayan nobility and priests. In these meetings, the Spanish leaders sat on sacred mats that were offered by the Mayan nobility. However, they covered the mats with cloth that contained values that neutralized any mystical power and blessing that emanated from the numbers presented in the mats. The geometric patterns repeated in the sacred mats demonstrated the beauty and power of these patterns.


Different Geometric Patterns of the Sacred Mayan Mat: Different Geometric Patterns of the Sacred Mayan Mat These patterns were sculpted in stones and used in jewellery and cloth. They are still used in the clothing of 21st century Maya descendents in Guatemala, Southern Mexico, Belize and Honduras. Through much of their weaving, the present magic of the designs in the vestments are connected with ceremonies that are promoted by their modern ancestors.


The Universal Diamond: The Universal Diamond In the universal diamond the four fields represent the frontiers between space and time in the Mayan universe. The small diamonds that are in each field represent the cardinal points of this universe; the east is placed where the sun rises, the west is placed below and represents the end of the day, the north is placed on the left and the south on the right. The Mayan spatial orientation of the four corners of their universe is not based on the cardinal points of the western compass.


The Universal Diamond: The Universal Diamond Frequently, the diamonds are placed so eastern and western fields are colored blue to represent the Caribbean on the east and Pacific Ocean on the west. The center of each large diamond is placed so that a small diamond represents the sun. Sometimes, a fine line is placed on the design that connects the east and west and represents the trajectory of the sun across the sky.


The Universal Diamond: The Universal Diamond The present-day Mayans weave and sew many of the same designs and motifs that have been popular since the classic period of Mayan culture between 3rd and 10th centuries. Many of the pictures found on ceramics, lintels, stela and murals also contain the same patterns and geometric forms that are utilized in the Mayan weavings.


Huiple: Traditional Mayan Dress: Huiple: Traditional Mayan Dress


The Universal Diamond: The Universal Diamond Wall of a Mayan Temple in the Yucatan, México


The Universal Diamond: The Universal Diamond The diamond shape was considered extremely important, indeed sacred because it represented the light reflected with brilliance in a polished or refined diamond. This diamond shape brought a sense of order and light, and reminded them that all need to live in harmony. The attraction of the diamond form was in concord with the sacred numbers of the Gods; it was divine power that implied the numbers of 1 to 9 .


Decoding Mayan Messages : Decoding Mayan Messages According to Nichols, the patterns X’s or XX’s found on many Mayan mats (Pop) contained information. The numbers placed on these mats progressed sequentially and zigzagged diagonally. The first number is positioned on the right vertice of the first square that composed the mat. For example, on a mat of 3 lines by 2 columns, the numbers are placed as in the diagram (right). According to Girard, “when the King spreads his legs and lifts his arms over his head, he assumes a posture that can be called a cross and which is nothing more nor less than the representation” of the glyph of kin or glyph of the sun.


Decoding Mayan Messages: Decoding Mayan Messages The final numerical number of this matrix might be calculated in the following manner: We add the corresponding numbers of each line of the matrix. 1 + 6 = 7 5 + 2 = 7 3 + 4 = 7 Consulting the table, the result 7 has the value: God in Divine Power. Adding all the results we get: 7 + 7 + 7 = 21 We then add the digits resulting in the ultimate value of: 2 + 1 = 3 According to the table, the number 3 corresponds to Creature and Life.


Decoding Mayan Messages: Decoding Mayan Messages A possible interpretation of the message of this result can then be: God utilizes His Divine Power to give life to all creatures in the world. Objects found in some of the most important archaeological sites of Guatemala such as Tikal or Quirigua reveal that Mayan priests made certain decisions based on sacred mats because they contained significant sacred numbers that were based on ultimate values for each pattern. For example, to find a solution for a given situation, a priest needed to make a decision towards codifying a mat that contained the ultimate value 6 which signifies “Life and Death.” In this context, the Mayan priests were charged with maintaining the spiritual, religious, scientific, and mathematical knowledge of Mayan civilization.


The Mayan Number System of the Divine Creation: The Mayan Number System of the Divine Creation The Quiché codex begins by referring to the creation of the universe. Divinity –pre-existent to its works – creates the cosmos, which extends through two superimposed, quadrangular planes – heaven and earth – their angles delimited and their dimensions established. Thereby is established the geometric pattern from which will derive the rules for cosmology, astronomy, the sequential order in which events occur, and the marking out and use the land, which for the Maya are all reckoned from that space-time scheme. (p. 28).


The Mayan Number System of the Divine Creation: The Mayan Number System of the Divine Creation The Mayans developed a sacred and magical number system through the construction of mats, elaborated in diverse patterns. According to Mayan theosophy, the creation of the world was closely associated with mathematical concepts. Diaz (1995) stated that the creation of the four corners of the Mayan universe was governed by the geometric pattern of the rhombus which represents the geometric pattern on the skin of the rattlesnake Crótalus durissis.


Decoding Mayan Messages: Decoding Mayan Messages In this perspective, the Mayan people have an interesting geometric mathematically-based creation story of their universe. In this story, the god Tzakol’s used his supernatural intervention in the creation process by applying the sacred-symbolic power of the numbers. The first record of the creation of the Mayan universe seems to be related to sacred numerical values as described in the book Popol Vuh (Recinos, 1978). It can be interpreted by the following mathematical pattern: [1] According to Diaz (1995), “the root of Tz’akol is Tsa or Tza, that is Tzamná or Itzamná, which comes from Tzab, rattlesnake, which is onomatopoeic with the sound of the rattle” (p.8).


Number 0: Number 0 This is the first account, the first narrative. There was neither man, nor animal, birds, nor forests; there was only the sky. … Nothing existed. (Recinos, 1978, p. 81). It was like a seed phase because all was in suspense, all calm, in silence, all motionless, and the expanse of the sky was empty. Thus, the Mayans used a seed symbol for zero.


Number 1: Number 1 Tzakol, known as Huracán, is the first hypostasis of God. He planned the creation of the universe, the birth of life, and the creation of man (Recinos, 1978).


Number 2: Number 2 The Creator brought the Great Mother (Alom) and the Great Father (Qahalom). Alom is the Great Mother and represents the essence of everything that is conceived. Qahalom is the Great Father who gives breath and life.


Number 3: Number 3 Then came the three: Caculhá Huracán (the lightning), Chipi-Caculhá (the small flash) and Raxa-Caculhá (the green flash) that represent life and all creatures.


Number 4: Number 4 Diaz (1995) states that the Venus Goddess, called Kukulkan is represented by number 4 because it corresponds to the four sides of the rhombus. His view is that the number 4 is “in the designs on the skin of the Crótalus” (p. 8).


Number 5: Number 5 The gods delegated their power to the priests. The priests were considered as the hands of the god because they gave to the Mayan people the gods’ answers to their prayers. In Mayan ceremonies, the priests held ceremonial rods decorated with rhombuses in the center and a snake head on top and they were “the mathematical insignias of the wise priests that ordered the construction of the Mayan temples” (Diaz, 1995, p. 8).


Number 6: Number 6 In Mayan cosmology, bones are like seeds because everything that dies goes in the Earth and then new life emerges from the Earth in a sacred cycle of existence.


Number 7: Number 7 The Mayans believed that the divine power of the gods reorganizes the order of the cosmos and reunites the human world with the supernatural and mystical worlds.


Number 8: Number 8 Everything on and of the Earth relates to material reality (the body) and spiritual reality (the soul).


Number 9: Number 9 Alom made nine drinks with the milling of yellow and white corn. With these drinks she created the muscular body and the robustness of men.


The Symbolism of Mayan Numerology: The Symbolism of Mayan Numerology Mayans perceived that natural events occurred in accordance with numerical patterns, as in the annual sequence of the lunar cycles. Numbers were related to the manifestations of nature and for this reason it was possible to determine that the universe obeys laws that allowed them to measure and anticipate certain forms of natural events.


The Symbolism of Mayan Numerology: The Symbolism of Mayan Numerology Despite the advanced mathematical knowledge of the Mayan people, they incorporated concepts of theogony[1] with concepts of numbers by utilizing symbolic elements to express their ideas about the creation of the universe. In this context, the Mayan theology posits nine cosmic manifestations that are perceived in nature and through which the Mayan people infer the abstract manifestations of God. [1] The genealogical account of the origin of the gods.


Final Considerations: Final Considerations The theogonic philosophy of the Mayans exceeds the limits of mathematical knowledge because it relates to the numbers of the abstract manifestations of God, with the objective of explaining, understanding, and comprehending the organizational principles of the creation of the universe. According to Girard (1966), the Mayans also developed ways for which numbers were symbolically transformed into others. For example, the binomial mother-father is transformed into the number three when a child is added to the family.


Final Considerations: Final Considerations There exists a belief that ideas and the mathematics produced by non-Western cultures are irrelevant for both economic and technological development in this modern globalized world. Many mathematical practices produced by non-western cultural groups have served peoples for hundreds of years and are still dynamic and alive. Ethnomathematics is a way of understanding the unique differences among the mathematical practices of diverse cultural groups. From a global perspective, ethnomathematics can be considered an academic counterpoint to globalization, and offers a critical perspective of the internationalization of mathematical knowledge through attempts to connect mathematics and social justice.


Final Considerations: Final Considerations There exists a belief that ideas and the mathematics produced by non-Western cultures are irrelevant for both economic and technological development in this modern globalized world. Many mathematical practices produced by non-western cultural groups have served peoples for hundreds of years and are still dynamic and alive. Ethnomathematics is a way of understanding the unique differences among the mathematical practices of diverse cultural groups. From a global perspective, ethnomathematics can be considered an academic counterpoint to globalization, and offers a critical perspective of the internationalization of mathematical knowledge through attempts to connect mathematics and social justice.


Final Considerations: Final Considerations D’Ambrosio (2000) stated that mathematics is integrated with a modern globalized civilization that has conquered and dominated the entire world. The only possibility of building a fair and just planetary civilization depends on restoring the dignity of the “losers” and together, both the winners and losers move towards social justice and peace. It is also possible to perceive ethnomathematics as the academic articulation between cultural globalization of mathematical knowledge and diverse non-western cultural groups.


Final Considerations: Final Considerations Through a study of the mathematical practices of non-western people, as found in the Mayan sacred mat and geometric diamond patterns, it is possible to demonstrate one use of an ethnomathematical, anthropological, and global perspective in which we might recreate, study and preserve a portion of the wisdom and knowledge of these unique and resilient peoples. artefacts From the perspective of all cultural groups, globalization reveals a vast patchwork of cultures that have been destroyed, colonized, integrated and differentiated during the long history of human interaction, travel and trade.


Final Considerations: Final Considerations Defining globalization only as the spreading of Western mathematical knowledge over other cultures especially is partially inaccurate; conquerors are almost always influenced by mathematical practices of the peoples that they have conquered and assimilated. Conventional beliefs hold that the globalization of mathematical knowledge has been driven by Western expansion.


Final Considerations: Final Considerations Many non-Western contributions to the development of mathematical knowledge are becoming more and more apparent, partially due to work in ethnomathematics. If individuals in different cultural groups are going to understand the overall importance of their own mathematical knowledge, they may also need to expand the scope of this knowledge through collaboration with diverse cultural groups by sharing the different mathematical practices that are part of a developing new context of globalization.


Final Considerations: Final Considerations When discussing, sharing, and internationalizing mathematical practices and the ideas used by other cultures, it is necessary to recast them into an individual’s Western mode. Modelling allows us to translate these practices into western mathematics. It is possible to distinguish between the mathematical practices and ideas which are implicit and those which are explicit, between western mathematical concepts and non-western mathematical concepts which are used to describe, explain, understand, and comprehend the knowledge generated, accumulated, transmitted, diffused, internationalized, and globalized by people in other cultures.


Bibliography: Bibliography Borba, M. (1990). Ethnomathematics and education. For the Learning of Mathematics, 10(1), 39-43. Borba, M. (1997). Ethnomathematics and Education. In Power, A.B. & Frankenstein, M. (Eds.), Ethnomathematics: Challenging eurocentrism in mathematics education (pp. 261-272).New York, NY: State University of New York. Cajori, F. (1993). A history of mathematical notations: two volumes bound as one. New York, NY: Dover Publications. Coe, M. D. (1966). The Maya. New York, NY: Praeger Publishers. Coe, M. D. (1992). Breaking the Maya code. New York, NY: Thames and Hudson. Coe, M. D., Kerr, J. (1988). The art of the Maya scribe. New York: Harru N. Abrams. Georgetown University Child Development Program, Child and Adolescent Service System Program. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44-48. D’Ambrosio, U. (1990) Etnomatemática [Ethnomathematics]. São Paulo, SP: Editora Ática. D’Ambrosio, U. (1993). Etnomatemática: um programa [Ethnomathematics: A program]. A Educação Matemática em Revista, 1(1). Blumenau, SC: SBEM, 5-11. D’Ambrosio, U. (1998). Ethnomathematics: The art or technique of explaining and knowing (P. B. Scott, Trans.). Las Cruces, NM: ISGEm (Original work published in 1990). D’Ambrosio, U. (1999). Educação para uma sociedade em transição [Education for a society in transition]. Campinas, SP: Papirus Editora. D’Ambrosio, U. (2000). Ethnomathematics: A step toward peace. Chronicle of Higher Education, 12(2), 16-18. D’Ambrosio, U. (2001). Etnomatemática: entre as tradições e a modernidade [Ethnomathematics: A link between traditions and modernity]. Belo Horizonte, MG: Autêntica. D’Ambrosio, U. (2002, August). Ethnomathematics [Etnomatemática]. Closing lecture delivered to the II Congress II Congress on Ethnomathematics, Ouro Preto, Minas Gerais, Brazil. Deuss, K. ( 1981(1981). Indian costumes from Guatemala. Twickenham, Great Britain: CTD Printers Ltd. Dias de Castillo, B. (1983). Historia verdadera de la conquista de La Nueva España [True History of the Conquest of New Spain]. Ciudad de México: Porrúa. Diaz, R. P. (1995). The mathematics of nature: The canamayté quadrivertex. ISGEm Newsletter, 11(1), 5-12. Gerdes, P. (1985). How to recognize hidden geometrical thinking? A contribution to the development of anthropological mathematics. For the Learning of Mathematics, 6(2), 10-12, 17.


Bibliography: Bibliography Girard, R. (1966). Los Mayas [The Mayans]. Ciudad de Mexico: Libromex Editores. Girard, R. (1979). Esotericism of the Popol Vuh: The sacred history of the Quiché-Maya. Pasadena, CA: Theosophical University Press. Grattan-Guiness, I. (1997). The rainbow of mathematics: A history of the mathematical Sciences. London: W. W. Norton & Co. Jr. Merick, L. C. (1969). Origin of zero. In National Council of Teacher of Mathematics (Eds.), Historical topics for the mathematics classroom (pp. 49). Washington, DC: NTCM. Ifrah, G. (1998). The universal history of Numbers: From prehistory to the invention of the computer. New York, NY: John Wiley & Sons, Inc. Lindsey, R. B.; Robins, K. N. & Terrel, R. D. (2003). Cultural proficiency: A manual for schools leaders. Thousand Oaks, CA: Corwein Press, Inc. Morales L. (1993). Mayan Geometry. ISGEm Newsletter, 9(1), 1-4. Mtetwa, D. K. (1992). “Mathematics” & ethnomathematics: Zimbabwean student’s view. ISGEm Newsletter, 7(1), 1-4. Nichols, D. (1975). The Lords of the mat of Tikal. Antigua, Guatemala: Mazda Press. Orey, D. (1982, February). Mayan math. The Oregon Mathematics Teacher, 1(1), 6 – 9. Oweiss, I.M. (1988). Arab civilization. New York, NY: State University of New York Press. Powell, A. B. & Frankenstein, M. (1997). Ethnomathematics praxis in the curriculum. In Powell, A. B. & Frankenstein, M (Eds.), Challenging eurocentrism in mathematics education (pp. 249-259). New York, NY: SUNY. Recinos, A. (1978). Popol Vuh: The sacred book of the ancient Quiché Maya. (D. Goetz & S. G. Morley, Trans.) Oklahoma: Norman University of Okalahoma Press. (Original work published 1960). Rosa, M. (2000). From reality to mathematical modelling: A proposal for using ethnomathematical knowledge. Unpublished master’s thesis, California State University, Sacramento. Orey D. C. & Rosa, M. (2003). Vinho e queijo: Etnomatemática e Modelagem! [Cheese and wine: Ethnomathematics and modeling]. BOLEMA, 16(20), 1-16. Rowe, A. P. (1981). A century of change in Guatemalan textiles. New York, NY: The Center for Inter-American Relations. Sen, A. (2002, January). How to judge globalism [33 paragraphs]. The American Prospect [On-line serial], 13(1). Available: http://www.propsect.org/print/V13/1/sen-a.html Toffler, A. (1980). The third wave. New York, NY: William Morrow and Company, INC. Wilkinson, D. (2002). Silence on the mountain: Stories of terror, betrayal, and forgetting in Guatemala. New York, NY: Houghton Mifflin Company.


Obrigado! Thanks!: Obrigado! Thanks!