Volume 9 Number 1 December 1993
Universidad de San Carlos de Guatemala
This article is a study of the geometry that is found in the various aspects of the daily activities of the Maya, such as design of cities, and the shape of buildings, ceramics and weavings. A geometry legacy is also found in the Mayan languages. Finally, an axiomatic geometry such as those of Occidental origin will be presented, but using Mayan elements, of a similar nature to those found in present-day indigenous weavings. Thus geometries of this type can be taught in elementary schools.
As happens with the study of other sciences developed by the Maya, in geometry we find that Mayan knowledge was integrated and developed for the collective good. In studying the layout of Mayan cities an impressive relationship to Astronomy is found. "The Mayan spatial orientation of the four corners of their universe is not based upon our cardinal directions..., but probably to either our intercardinal points..., or toward two directions in the east and two in the west, that is to say, sunrise at winter and summer solstices, and sunset at the same two solstices." (Vogt, cited in Leon-Portillo, 1988, p. 130). Also, there are many examples that show the alignment of the temples with celestial bodies. Some such examples are given by Vinette (1986), Morley (1968), and Morley and Brainerd (1983). The alignment of the Stelas 10 and 12 in Copán, Honduras, indicate the time of year in which fields are burned in preparation for the planting of crops. This also indicates that such monuments had a secondary function in addition to the primary function presented in their inscriptions.
In the oral tradition the priests declared that much of their knowledge came from the corn. It is from the ears of corn (la mazorca in Guatemala) that the shape of the temples is derived. And their grand staircases come from the rows of kernels. Corn also offers other kinds of knowledge. Many of the calculations on the Mayan calendar come from the period of cultivation and its various stages: planting (siembra), banking up of soil around the corn stalks (calza), weeding (limpia), etc.
The great majority of Mayan temples are truncated tetrahedrons, rectangular prisms, or, in some cases, cylinders, such as those found in the archeological site at Ceibal.
The relationship that these architectural works have with celestial bodies indicates that they were carefully planned before their construction (Morley, 1983, p. 294), as does an observation of the evolution of the elements, such as the Mayan false arch, used in their architectural designs (Morley, 1983, p. 267).
There is also evidence that paintings were carefully planned. A good example is the symmetry in the murals of Coba (Vinette, 1986, p. 389). These planos (plans), as they are called today, were kept and in some cases served as property titles. Such is related in the book, Sobre los Indios de Guatemala (On the Guatemalan Indians), "and they showed you, for your interpretation, two paintings in which the natives of said town (Atitlán) have painted their houses and were antiquities of those who were caciques ... Paintings that were over 800 years old, by which I was able to find out information about the Quichés" (Carrasco, 1982, p. 72-73).
Many civilizations have left much information about their cultural development in their ceramics. Most archaeological excavations reveal the remains of ceramics, either as whole or reconstructible objects. These, generally, provide significant information to studies of geometry. Besides their shape, a collection of curves and other geometric figures are present adorning the exterior, and sometimes the interior, of the vessels. In Maya ceramics "three basic forms are found: jugs, bowls, glasses, plates and vessels with a restricted mouth" (Rubio, 1992, p. 6), and each of those categories is different from the other precisely because of its geometric shape.
For decorative curves the Maya used human figures, animal shapes, flowers, inscriptions and dates. Among the curves there was a predilection for intertwined curves. Spiral curves also appeared frequently. The concept of curves and lines seems to have existed naturally. The phrase "they were placed in a straight line" is found in verse 651 of the Popol Vuh, and in the examples that are presented below in the languages K'ekchí and Chortí expressions are found for line, align, row, in rows, side, edge of and many other terms.
Much of indigenous knowledge is transmitted in oral form. This method of study and conservation of indigenous culture is very well exemplified in the book, El Ladino Me Jodió (The Ladino "Harmed" Me) (a Ladino may be anyone who is culturally non-Indian). (Saravia, 1986). Given that even today the oral tradition is used to maintain the cultural heritage, it is undeniable that researchers also need to consider the use of that methodology. Thompson (1965, p. 123) pointed out that "...there is more, my contacts with our Mayan works of San Antonio and the long conversations with Faustino in the course of our travels, helped me realize that the modern descendants of the ancient Mayans still conserve many of the old customs."
Because the Maya are so conservative and equilibrated you can be well-assured that fundamentally they act today as they did a thousand years ago, and from there you can deduce much about the past by studying the present" (Thompson, 1965, p. 124).
In order to support this thesis, a study of geometric terms present in various Mayan languages was undertaken. The following geometric terms were taken from Nuevo Diccionario de las Lenguas K'ekchí y Española (1955):
|to play dice||bulic, buluc|
|in a row||chitzol, tzoltzo|
|shape of a ball||t'ort'o|
|shape of a bundle||bolbo|
|one side||jun pac'al|
|various sides||q'uila pac'al|
|one row||jun kerel|
|half a measure||jun bas|
|middle of two||yibej|
|in the middle||sa'xyi, yitok|
|two and a half||cuan rox|
|three and a half||cuan xca|
|to measure by handspan||c'utu banc|
The book, Método Moderno para Aprender el Idioma Chortí: Una Gramática Pedagógica presents some terms in the Chortí language that indicate the existence of a Geometry. It is apparently more metric and topological than that found in the K'ekchí language, which seems to be motivated more by shapes.
|on top of||tor|
|along side of||tuti'|
It can be concluded from the above examples, that given the great quantity of geometric terms that exist in these Mayan languages (here taken at random), it can be observed that these elements were used and continue to be used by the Maya people.
The Popol Vuh, verse 237, the describes the tasks for children as being: "playing the flute, singing, writing, painting, sculpting, ...". Nowadays weaving and embroidery have been added to those tasks. It is in weaving that many of the designs, that were once present only in ceramics, can be found.
In Mayan weavings for both personal and domestic use a wide variety of mosaic designs can be found. The mosaics have many different interpretations. The work of Anderson (1978) provides a good guide to this area.
Let's take a look at a mosaic (Figure 1):
Notice the repeating triangles in rows or chains, either horizontally or diagonally.
Consider another example (Figure 2):
Broken lines seem to be repeated, but by analyzing those lines you will notice that they form the sides of rhombuses.
A final example (Figure 3):
The elements < and > are repeated in a horizontal row. These mosaics suggest the general idea of geometry in indigenous weaving. They are still present today and are a part of everyday clothing.
From Paulos Gerdes' little book, Desenhos da Africa, suggested the possibility of a mathematization of the designs that appear in weavings. A generating element was sought to which various operators could be applied: translation, rotation, ***homostasis. Composition of that basic element is used to develop different shapes and the composed shapes are used to develop chains that are then used to form the mosaics.
The Element: The undefined element that serves as the foundation for this geometry was found to be a common denominator among the various shapes that appear in Guatemalan weavings. It is similar to the symbol for "less than"
<To this element various operators are applied, such as:
Dilation: Dilation acts on size, thickness, and positive or negative state:
thin positive small
thick negative large
Rotation: This acts on one or both sides, changing the angle or the orientation:
Shapes: A shape is defined as a set of two or more elements with a certain orientation. The elements used in the shapes can be simple or can be the result of the application of an operator, for example:
Two elements joined at their vertices:
Two elements joined at their vertices, but negative:
Chains: A chain is defined as the union of one or more shapes, for example (Figures 4, 5, 6, 7, and 8):
Mosaics: Mosaics are defined as the union of 1 or more chains. Let's considered a complete example:
We start with an initial element:
We define a shape:
We construct a chain:
With that chain we can form various mosaics (Figures 9 and 10):
Here are two more examples of mosaics (Figures 11 and 12):
As was indicated at the beginning, the objective of this article is to introduce the reader to the study of the geometry of mosaics that are found in Guatemalan weavings in order to raise the self-esteem and to praise this cultural treasure.
Anderson, Marilyn, Guatemalan Textiles Today, Watson-Guptil Publications, New York, 1978.
Carrasco, Pedro, Sobre los Indios de Guatemala, Seminario de Integración Social Guatemalteca, Publication 42, Editorial José de Pineda Barra, Guatemala, 1982.
de León, Carlos and López, F., Popol Vuh: Libro Nacional de Guatemala, CENALTEX, Ministry of Education, Guatemala, 1985.
Esparza, David, Cómputo Azteca, Editorial Diana, Mexico, 1976.
Gerdes, Paulos, Desenhos da Africa, Editora Scipione, Brazil, 1990.
Landa, Fray Diego de, Relación de las Cosas de Yucatán, Editorial Pedro Robredo, Mexico, 1938.
León-Portilla, Miguel, Time and Reality in the Thought of the Maya (2nd Ed.), University of Oklahoma, 1988.
Lubeck, John and Cowie, Diane, Método Moderno para Aprender el Idioma Chortí: Una Gramática Pedagógico, Instituto Lingüístico de Verano, Guatemala, 1989.
Morales, Italo, U Cayibal Atziak: Imágenes en los Tejidos Guatemaltecos, Ediciones Cuatro Ahua, Guatemala, 1982.
Morley, Sylvanus, La Civilización Maya, Fondo de Cultura Económico, México, 1968.
Morley, Sylvanus and Grainerd, G. W., The Ancient Maya (4th ed.), Stanford University Press, California, 1983.
Rubio, Rolando, Introducción a la Arqueología Maya, Cuaderno de Trabajo, Museo Popol Vuh, Universidad Francisco Marroquín, Guatemala, 1992.
Saravia, Albertina, El Ladino Me Jodió, CENALTEX, Ministry of Education, Guatemala, 1986.
Sedat, Guillermo, Nuevo Diccionario de las Lenguas K'ekchi' y Española, Tipografía Nacional, Guatemala, 1955.
Thompson, J. Eric, Arqueólogo Maya, Editorial Diana, México, 1965.
Vinette, F., "In Search of Mesoamerican Geometry", in Michael Closs (ed.) Native American Mathematics, University of Texas Press, 1988.
Shape Hunting: A Different Experience
María Victoria Ponza
In ICME-7, Québec, 1992, Geometry was given special consideration. Perhaps Collette Laborde of France summed it up best: "Geometry is not dead; it's alive and it behaves well." Some say that Geometry is a useful tool with which to see the world's beauty. It is the part of Mathematics that is most related to reality. That is why the teaching of Geometry cannot be limited to formalisms and symbols, but has to take into account touching, seeing, drawing and handling. And it has to relate the student to the world of patterns, shapes and movements, and from there to abstractions. In other words, it is necessary to start with the students' surroundings.
The best resources to learn Geometry are in life, culture, art and play. First, students must learn watching, touching and feeling. One possible strategy is to start from what the student knows and, through experience and real problem solving, attain an understanding of the concept. This has been the strategy used with Shape Hunting. It offers the kind of experiences that lead to Ethnomathematics in the best sense: Mathematics for everyone.
Secondary schools in Argentina offer five years of studies for students from approximately 12 to 18 years old. Large class sizes and an absence of teaching materials characterize most public schools. Therefore teachers need to have great imagination and make a special effort if they are to achieve their goals.
This Experience was realized with 48 in one second year class in Mariano Moreno School in Río Ceballos, Córdoba, Argentina. The concept being studied was polygons. The first stage was to review some of the basic knowledge that they brought with them from elementary school: polygonal shapes, parallelograms, planes, perimeter and area, concave and convex polygons.
The second stage was to invite them to go out hunting. Their first reaction was to ask what kind of weapons they should take: rocks, rifles, etc. I answered that the only weapons they would were intelligence, and paper and pencils. I told them that the aim was "To look for geometrical shapes in the world which surrounds us." Outdoor work was to be done on an athletic field that is surrounded by hills with an amphitheater as the operations center. I further explained that the two principal tasks were to: a) draw all the polygonal shapes they find and to take down everything about the place where they found them, and b) estimate the area and perimeter of a triangular piece of land.
They organized into cooperative groups and appointed a coordinator. They gathered their data using informal measurements such as branches, shoes, steps. They organized their data, estimated the area, and experimented with Heron's formula. They discussed discrepancies in the various estimates of the area. They classified the shapes they had hunted down into convex polygons, concave polygons, and other.
The amount of geometric shapes in nature amazed them. They said, "Not even a month would be enough for taking note of all the shapes in this piece of the world. How would in be in the whole world?"
They went back outside for a second time to reestimate the area and to check the classifications they had made.
Except for one girl, all to the students worked in a disciplined way and had fun.
The group coordinators functioned very well. Two boys who had studied very little in class gave excellent suggestions for proving theorems and solving problems.
One group was walking around a flower bed. I told them: "Be careful with the plants." They answered, "we have to feel the shape." I shut my mouth and felt very happy with this work.
It is necessary to lead pupils to research, allowing them to enjoy it. Theorems are important; we have to teach them. But we must not limit ourselves to simply lecturing. To learn the qualities of a rectangle by watching and touching a bench at the athletic field or the bark of a tree, is very important too. If it is possible to take the learning to natural or manmade environments the learning can be richer and a respect for nature can be promoted. To find unknown elements is the best that can happen
UNESCO Documents Available
The following titles are free of charge upon request from ED/ECS/SE/STE, UNESCO, place de Fontenoy, 75700 Paris, FAX 33- 14065-9405:
The Influence of Computers and Informatics on Mathematics and its Teaching (UNESCO/ICMI Study) edited by B. Cornu and A. Ralston. STEDS 44, 1992, English only.
Educación Matemática en las Américas VIII, edited by R. Scott (IACME 8 Conference, Miami 1991). STEDS 43, 1992, Spanish only.
Educación Matemática en las Américas VII, edited by E. Luna and S. Gonzalez (IACME 7 Conference, Santo Domingo, Dominican Republic, 1987). STEDS 37, 1990, Spanish only.
Mathematics, Education, and Society, edited by Keitel, Damerow, Bishop and Gerdes (ICME 6 Fifth Day). STEDS 35, 1989, English, Spanish and Russian versions.
Evaluation and Assessment in Mathematics Education, edited by D. Robitaille (ICME 6 Theme Group). STEDS 32, 1989, English only.
Innovations in Science and Mathematics Education in the Soviet Union, STEDS 24, 1987, English only.
Mathematics for All, edited by Damerow, Dunkley, Nebres and Werry (ICME 5 Theme Group). STEDS 20, 1986, English and Spanish.
Other documents are under preparation. One to appear later this year, also free of charge:
Factors Influencing the Learning of Mathematics, edited by A. Bishop (prepared by ICMI Study Group PME).
Updated Annotated Bibliography of Multicultural Issues in Math Ed
Patricia Wilson of the Mathematics Education Department at The University of Georgia announces that a January 1994 update of the Annotated Bibliography of Multicultural Issues in Mathematics Education is available. It contains new annotations, new articles, and several corrections. The Bibliography is not intended to include all related works, but seeks to represent a variety of areas of work and ideas.
To get a copy of the Annotated Bibliography write:
Dr. Patricia Wilson
The University of Georgia
105 Aderhold Hall
Athens, GA 30602-7124 USA
ISGEm Communications Network Now Operational
The ISGEm Communications Network is now operational and available to anyone with an Internet connection. The network provides a forum for members of ISGEm to pose questions, offer opinions, further discussions, relay research results, and announce meetings and conferences on Ethnomathematics.
The ISGEm Communications Network operates like most electronic bulletin boards. Subscribers may post their message to the ISGEm e-mail address. The message is then relayed to all subscribers to the network.
To subscribe to the ISGEm Communications Network send an e-mail message to:
No subject is needed. Your message should contain the word SUBSCRIBE and your name. Once you are on the network ISGEm messages will be automatically sent to you. You can send messages to the network using the ISGEm address: firstname.lastname@example.org
The ISGEm Communications Network is managed by ISGEm member James Rauff (Department of Mathematics, Millikin University, Decatur, IL 62522). Millikin University has donated the network facilities for ISGEm.
China Regional Conference on Math Ed
The ICMI - China Regional Conference on Mathematics Education will be held in Shanghai, China, August 16-20, 1994, with the theme Teacher Preparation in Mathematics. Persons interested in information can write:
Jerry P. Becker
Curriculum and Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Correction for May 1993 ISGEm Newsletter
Gilmer, et. al. are not the authors of Multiculturalism in Mathematics, Science, and Technology: Readings and Activities as reported on page 8 of the May 1993 issue of the ISGEm Newsletter. Addison-Wesley is the author.
The work of Gloria Gilmer, Mary M. Soniat-Thompson, and Claudia Zaslavsky is entitled Building Bridges to Mathematics Cultural Connections. For further information see Have You Seen below.
Journal of Women and Minorities in Science and Engineering
The Journal of Women and Minorities in Science and Engineering is seeking submissions for publi cation. The first issue will be published in the Winter of 1994. The purpose of the Journal is to publish original, peer-reviewed papers that report innovative ideas and programs, scientific studies, and formulation of concepts related to the education, recruitment, and retention of underrepresented groups in science and engineering. Issues related to women and minorities in science and engineering will be consolidated to address the entire professional and educational environment. Subjects for papers can include:
To receive guidelines for manuscript preparation contact:
Kathy Wagner, Editorial Assistant
Journal of Women & Minorities in Science & Engineering
Women's Research Institute
Virginia Polytechnic Institute and State University
Sandy Hall Room 10
Blacksburg, VA 24061-0338 USA
Phone:703/231-6269 Fax:703/231-7669 E-mail:email@example.com
Subscriptions can be obtained by sending a letter of interest and check for $40.00 payable to:
Begell House, Inc.
79 Madison Avenue
New York, NY 10016-7892
II CIBEM in Brazil
The Second Iberoamerican Congress on Mathematics Education will be held in Blumenau, Brazil, from July 18 to 22, 1994. For further information write:
Maria Salett Biembengut
Universidade Regional de Blumenau - FURB
Rua Antonio de Veiga, 140
CEP 89012900, Blumenau, SC BRAZIL
Fax: (0473) 22-8818
ISGEm Activities at the 1994 NCTM Annual Meeting
Below are the days and times of some of the main ISGEm activites at the annual 1994 NCTM Annual Meeting in Indianapolis. Please check the Message Center at the Meeting for room assignments.
Thursday 14 April 1994
7:00-8:30 AM ISGEm Executive Board Breakfast
2:00-3:00 PM SIG Meetings
Curriculum & Classroom Applications
Research in Culturally Diverse Environments
3:00-4:00 PM SIG Meetings
Out of School Applications
4:30-6:30 PM ISGEm Business/Program Meeting
Topic: Mathematics in the Cultural Context
Claudette Bradley, Yupic Math
Clo Mingo, Anasazi Math
Friday 15 April 1994
7:00-8:30 AM Breakfast Meeting to organize
Joint Committee on Mathe in the Cultural Context
7:00-9:00 PM ISGEm Executive Board Meeting
Research Pressesion on Ethnomathematics at the NCTM Annual Meeting in Indianapolis
For Tuesday, April 12, 1994 at the Research PreSession of the NCTM Annual Meeting in Indianapolis, Joanna Masingila has organized a three-hour session entitled "What Does Mathematics Practice in Everyday Situations Have to Do with Teaching and Learning Mathematics in the Classroom?" The individuals involved are: Joanna O. Masingila (organizer and presenter), Susana Davidenko (presenter), Ewa Prus-Wisniowska (presenter), Frank K. Lester, Jr. (discussant), Ubiratan D'Ambrosio (discussant), Gloria Gilmer (presider)
This thematic session includes a presentation on the mathematics practice in several everyday contexts, a comparison of how persons in these contexts solved problems and how secondary students solved problems from these contexts, and a model for connecting everyday and school mathematics. The model and related issues will be discussed by the two discussants.
ISGEm/HPM Joint Meeting
ISGEm and International Study Group on the Relations Between History and Pedagogy in Mathematics (HPM) will hold a joint meeting immediately following the NCTM Annual Meeting in Indianapolis, April 16, from 1:00-3:30.
Ethnomathematics: Linkages to Pre-Columbian Cultures
Luis Ortiz-Franco will present a session entitled Ethnomathematics: Linkages to Pre-Columbian Cultures at the NCTM Annual Meeting in Indianapolis on Friday, April 15, from 3:00 to 4:00 PM.
Culture in Sessions at the NCTM Meeting
In addition to Ortiz-Franco's session mentioned above there are many sessions announced for the NCTM Meeting in Indianapolis that stress culture in the teaching of mathematics. Below are some examples:
Wednesday, April 13
10:30-12:00, Jennie M. Bennett, Multicultural Connections
3:00-4:00, Patricia S. Wilson, Thinking about Culture and Mathematics: Activities for Your Classroom
Thursday, April 14
9:00-10:00, Alverna M. Champion, Link Mathematics with Culture: Explore Ways to Infuse Culture into Your Classes
10:30-12:00, Mary Ellen Hynes & Hilary Buckridge, Linking Multicultural Activities to Alternative assessment
12:30-2:00, Beverly J. Ferrucci, Multicultural Activities to Enhance Your Mathematics Classes
3:00-4:00, Maria A. Reid, Strategies Used to Teach Mathematics in the Multicultural/Multilingual Classroom
Friday, April 15
8:30-10:00, James J. Barta & Alice Hosticka, Reconnecting Elementary Math &Culture: Multicultural Perspectives
10:30-11:30, Enrique Ortiz, Linking Mathematics to Music and the Latin American Culture
10:30-11:30, P. Uri Treisman, Curriculum, Culture, & Community: Essential Linkages for Action
10:30-12:00, Kay Gilliland, Symmetry across Cultures: EQUALS Math & Art Activities from Many Countries 12:30-2:00, Cynthia M. Garnett & Lucille Croom, The Cultural Connection: Mathematics & the Young Child
3:00-4:00, Myra Lee Demeter, Mathematics & the Cultural Arts of Hawaii
Saturday, April 16
10:00-11:00, Toby T. Castillo, Apache Mathematics, Past & Present
10:00-11:00, Mary E. Gilfeather, Math Power through a Multicultural Prism of Manipulatives & Games
Have You Seen
"Have You Seen" is a regular feature of the ISGEm Newsletter in which works related to Ethnomathematics can be reviewed. We encourage all those interested to contribute to this column.
Gilmer, Gloria; Soniat-Thompson, Mary; and Zaslavsky, Claudia. Building Bridges to Mathematics Cultural Connections, Addison-Wesley, 1992.
This work is a set of grade level activities designed to strengthen mathematics skills in the context of culture. It presents activities that capitalize on the social nature of students from a variety of backgrounds working together on projects that explore our world. The entire set consists of one kit for each K-8 grade level. Each kit contains one activity per chapter of Addison-Wesley Mathematics. Each activity is presented on eight duplicate cards so that a teacher can have the entire class work on the same activity in eight cooperative learning groups. A Teacher's Guide for each grade level is included in the kit so that the teachers using other programs will also be able to integrate these activities into their curriculum.
Nelson, David; Joseph, George Gheverghese; and Williams, Julian. Multicultural Mathematics: Teaching Mathematics from a Global Perspective, Oxford University Press, Walton Street, Oxford OX2 6DP, ENGLAND.
This work explores ways of helping school children understand the universality of mathematics. From the premise that "Mathematics is a truly international language and field of study that knows no barrier between race, culture or creed" it attempts to provide answers to the question: "How can we exploit its rich cultural heritage to improve the teaching of mathematics and educate our children for life in a multicultural society?" The chapters are:
Rationales for a Multicultural Approach to Mathematics
Teaching Mathematics from a Multicultural Standpoint
Ten Key Areas of the Curriculum
Simultaneous Equations: Numerical Approach from China
Geometry and Art
Statistics and Inequality: A Global Perspective
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Math Tech, Inc.
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Alverna Champion, 2nd VP
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Luis Ortiz-Franco, 3rd Vice President
Dept of Math, Chapman University
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Maria Reid, Secretary
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College of Education, UNM Dept of Mathematics
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Dept of Mathematics, Towson State U
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Jerome Turner, NCTM Representative
St. Xavier University
Antigonish, NS B2G 1C0 CANADA