Everardo Lara González and Jose Francisco Lara Torres

UNAM magazine

introduction

Through mathematics it is possible to understand the vision of the world, the laws of nature since the origin of the universe. In addition, they allow us to enjoy the harmonic message of the numbers. The ability to handle them facilitates understanding and communication in different fields of everyday life, and applying them with imagination, amazing horizons are reached.

The nahuatl language is dual. It is composed of opposite meanings that are associated and whose hidden reality is the metaphor.

In the course of history several peoples are renowned for their concern in the study and teaching of mathematics, a discipline that arises when human beings began quantifying objects and natural phenomena. They created and developed different symbols to represent ideas, forms, or messages. It was how developed writing and, consequently, the appearance of numerals that represent a way of thinking.

Mesoamerican peoples created their own numerical systems on the basis of a philosophical foundation that coincides with the vision of other great cultures of the world, that believed that the principles of mathematics were the principle of order of all beings.

The application of the nahuatl mathematical model allows expanding the vision of mathematical reasoning so that future generations have access to the structure of the ancient nahuatl thought system, a relevant socio-cultural organization guide. So they know its meaning and enrich the solution and logical interpretation of a set of symbols and situations, such as:

The ancient nahuatl representation systems.

The theory of the origin of the universe.

The symbolism of pyramids in the ceremonial centers.

Rituals and events: apparent parties and traditions in which we currently participate and belong to the ceremonial corn count.

The social practices of nahuatl mathematical system.

The nahuatl theory of zero.

The lunar count of Mexico Tenochtitlan.

Dual nahuatl language

To understand and comprehend the nahuatl cosmovision is required to enter the world of their language, since language is the key to get into the immensity of the philosophy. It's a language that was perfected to achieve a spiritual refinement in which the mathematical order thinking is a collective union where individualism is not admitted. It has survived despite colonialism imposed, death and humiliation of our original peoples. The language is spoken daily by millions of people around the world through words that subsist on a daily basis, for example: night and day; cold and hot; man and woman.

The nahuatl language is dual. It is composed of opposite meanings that are associated and whose hidden reality is the metaphor. The union of the whole universe, is dimensioned in a quadrangular, therefore, nahuatl numbering chooses the first four numbers to symbolize the beginning of the order.

Interpretation of mathematics figurative forms of nahuatl culture.

The nahuatl numbering system corresponds to observation, contemplation, visualization and communication of a figurative symbolic code of human activity. Four square dimension ideograms depict an ascending ordering of expression and content of higher thinking, which is currently visible in the socio-cultural activities of the peoples of the continent and in other regions of the world.

Some chroniclers** [1]** mention the existence of instruments with beads of gold and precious stones that were used to keep record of statistical memory and perform advanced mathematical calculations. David Esparza Hidalgo, IPN researcher of the, produced this illustration of the Nepohualtzintzin based on observations of the mentioned chroniclers references.

Carlos Casas and Crespo drew up in 1998 the following design of this instrument using accounts in the form of corn to ease the use and set in the user the image of the main element of the nahuatl cosmovision, which is the corn.

The concept of the Nepohualtzintzin is the philosophical principle of the nahuatl mathematical science structure thought, which is interpreted as: Ne, the person; Pohualli, count; Tzitzin, transcend. The substantial idea included therein is: the person who has knowledge of the simplicity account of harmony to transcend to the origin of creation.

It is a dual mathematical construct (theoretical construct that is developed for solving certain scientific problems) which contains: the quantitative calculation of a formal model, the account of figurative model and the nahuatl language structure of an ascending dimensional measure. This corresponds to a precise, clear, and beautiful harmony of advanced mathematical thinking, of scientific creativity that seeks the being evolution, by subtly connecting our spirit with the essence in balance of the universe, where the moon becomes a cosmic center of the magnetic account that gives mankind the satisfaction of existing.

The mathematical model corresponds to the memory of a social construction process of a mathematical knowledge, based on observation, contemplation, visualization, communication, and implementation of socio-cultural activities which were systematized in nahuatl culture schools. Associated, logically, with the nahuatl language numeric system and the figurative model. The quadrangular communication is observed in social practices: verbally in narratives, myths, chants; and non-verbal in rituals, dances, pyramids architecture, traditional games, music, medicine, etc.

The construction of process of counts in the memory registry of the Nepohualtzintzin. Its relationship with the Moon at the base is the number 7; four pieces from the left side and three on the right side (4+3) that correspond to a lunar phase. The perimeter of the right-hand side corresponds to the cycle of the Moon: 7x4 = 28.

The Nepohualtzintzin design contains a total of 91 elements that correspond to the numerical sum of its 13 ascending levels. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 91

These figures correspond to accounts related to the lunar cycle.

Counts related to the human being and the ceremonial count of corn when we use the Nepohualtzintzin in perimeter form is obtained from the number 40, representing:

- the complete count (20) of men and a full account (20) of women (20 + 20).

- Women purification period after childbirth (quarantine).

- The two 20 scores of the solar calendar related to spiritual connection fast and corn seed, prior to its planting in the area of Tenochtitlan.

By multiplying 20 with 13-dimensional scale, we get 260. 260 = count of the human being destinations (horoscope) and the ceremonial count of corn.

The Nepohualtzintzin accounts related to astronomy

On the left side, the number of counts is 52 (13 x 4), which corresponds to the astronomical cycle called new fire. Every 52 years the Pleiades** [2]** are observed passing over the zenith of the Tenayuca pyramids and the Cerro de la Estrella, located to the North and East of Mexico City, the Pleiades being an association of stars that form the remarkable constellation of the boreal hemisphere known as Taurus.

On the right side we have 39 counts (13 x 3). The numeric addition from 1 to 39, the result is 780: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 780 representing the Mars orbit around the Sun with a perspective from Earth.

Mathematics in the current curriculum in Mexico

In the current curriculum from 2011, one of the fields for basic education refers to mathematical thinking, the development of mathematical thinking begins in preschool and its purpose is that children use the principles of counting; recognizing the importance and usefulness of numbers in everyday life, and start solving problems and the implementation of strategies that involve adding, collect, remove, equalize and compare collections. These actions create algorithm notions for adding or subtracting (agreement 592, VI.2.1).

To further the mathematical thinking development in grammar and high schools, teaching is oriented to learning to solve and ask questions that will can be useful mathematical tools. In addition, the need that students themselves justify the validity of the procedures and results is emphasized, through the use of this language (SEP 592 Secretarial agreement, VI.2.2).

Accordingly, it seeks mainly that children reason and not that just memorize rules or results. Throughout basic education it is sought that students are responsible for building new knowledge from their previous knowledge, which implies:

- Develop and validate conjectures.

- Pose new questions.

- Communicate, analyze and interpret resolution procedures.

- Search for arguments to validate procedures and results.

- Find ways to solve problems.

- Efficiently manage techniques. (agreement 592, VI.2.2)

Performance in mathematics for basic education students in Mexico.

Unfortunately, the academic performance of the majority of Mexican students in the field of mathematics has been poor. The results of national and international tests show results that leave much to be desired. For example, Mexico ranks among the last places in the OECD countries in math performance, according to the results of the PISA examination (program for international assessment of students). PISA 2009 results. Performance in mathematics Mexico from an international perspective:

In Mexico, only 5% of the students is grouped in high levels, 44% in intermediate levels, and 51% in the lower levels.

National perspective:

The entities that achieved superior performance on the national average are the Federal District, Nuevo Leon, Chihuahua, Aguascalientes, Colima, and Jalisco. However, none reached the OECD average of 496 points. San Luis Potosí, Tabasco, Guerrero, Oaxaca and Chiapas are States below the national average.

Results of the mathematics test *Prueba Enlace*

The national evaluation of achievement in scholar centers (Prueba de Enlace), represents the most extensive evaluative action in the national education system, both by the number of different tests that includes, as for its coverage. In the school year 2011-2012, were applied in grammar schools tests on Spanish, mathematics and natural sciences. The evaluation includes all students from 3^{rd} to 6^{th} grade, both in public and private schools. In relation to mathematics the following table is depicted a view comparative from 2006 to 2012 of the results obtained by the students evaluated at this educational level.

(Falta tabla)

Use of the Nepohualtzintzin as a teaching resource for the development of mathematics competencies in system educational national

David Esparza Hidalgo, civil engineer graduated from the *Instituto Politécnico Nacional*, made an important research from 1960 among persons belonging to the indigenous peoples located in the center of the country, in order to develop information about the existence of calculation instruments of the ancient inhabitants of Mexico. From his written work are highlighted: Aztec computation (Editorial Diana); Nepohualtzintzin, current prehispanic computer (Editorial Diana).

Since 1996 the Nepohualtzintzin began to be used in grammar schools at the invitation of the Directors interested in the project.

The Secretariat of public education, through the General coordination of Intercultural and bilingual education, designed and published in 2009 a guide for teaching the Nepohualtzintzin for the development of mathematical skills, which has been used by teachers in Mexico City and several States of the Mexican Republic, mainly at the level of basic education.

More specific follow-up has been given to several grammar schools that have used the Nepohualtzintzin in Mexico City of Mexico, including the following:

- 09DPR2032U school Álvaro Obregón, Milpa Alta, D. F. by Profra. Ma. Elena Telésforo Cruz, in collaboration with school teachers and with the support of the Nepohualtzintzin community, in particular by Aydée Nayeli Hernández Ortiz. Students of the technological and the CETIS, located in the Milpa Alta delegation, who developed their social service through the strategy of socializing education through the Nepohualtzintzin have also contributed.

- 09DPR0102L School Efraín Huerta, Iztapalapa, D. F. by Profra. Dolores Marmolejo Rodriguez, with the support of the School Council of Social participation, chaired by Ms. Claudia Soto and in collaboration with the campus teachers and mothers.

- 09DPR2014E School Ignacio Rodriguez Galvan, evening shift, by Prof. Adrián Flores Sandoval, in collaboration with school teachers.

Comments on the *Prueba de Enlaces* test in schools that use the Nepohualtzintzin.

Subject to being corroborated with further results, students tend to get fewer scores in the categories of elementary and inadequate; the groups get progressively better results in mathematics, equivalent or higher than the average scores on State and national levels.

Specific examples of successful preliminary test results obtained by groups of the schools participating in the project Comparative information by level of achievement in mathematics shows the average score achieved by students of a Group with reference to the average score of the school, institution and country, in the same grades, subject and type of school.

Conclusions

some observations and lessons obtained on the use of the Nepohualtzintzin in basic education are the following:

- Its use improves the self-esteem of the child, when recognizing himself as heir of the nahuatl mathematical science and by allowing them to have a greater understanding of mathematics and better results in this area.

- The Nepohualtzintzin has two premises: a formal mathematical model of quantitative numerical representations, and its complement, where the structure of the philosophical thought through figurative representations.

Thanks to this, it is more understandable for the student to comprehend the message of the total order, i.e., the holistic aspect of the nahuatl worldview.

Students who use it manifest that they have less math suffering and most of them expressed satisfaction when using it.

- Improves school climate and collaborative work. Even some teachers who have implemented it, express that the bullying manifestations have declined in their classrooms, by students who behaved violently.

- Better results are achieved as teachers dare using it systematically with their students. Some teachers say that it is difficult for them to follow the fast pace of the children with the use of the Nepohualtzintzin.

- Students find easier different paths and methods to solve mathematical problems that must be resolved. This requires teachers to assume a less directive role and more facilitative of the autonomous learning of their students.

- Excellent results are achieved when all members of the school community are convinced of its usefulness and engage resolutely in their use (students, parents teachers and education authorities).

It is currently in analysis process of the results of a pilot study in four schools in the Distrito Federal in the school year 2011-2012, which will yield results on the impact achieved by using the Nepohualtzintzin (Nepo) along with another series of activities aimed to knowledge and recognition of the cultural and linguistic diversity of Mexico, as well as the contextualization of calculation instrument. Three grammar schools operated the instrument in three school groups (2°, 5° and 6°) and one operated as control group working without the Nepo. Contextualization intervention consisted in the realization of courses on the history of the calculation instrument and the philosophy of Mesoamerican culture; also demonstration classes were given to teachers, parents and students to encourage their implementation and workshops for the development of Nepos and other representative objects of ancestral traditions and customs. The results of this pilot program will guide and support the proposed comprehensive intercultural education that includes the implementation of the Nepohualtzintzin, when comparing results with the progress of students in mathematics and the Organization and the interactions in the school community.

Adrian, L. G. (2009). *Manual didactico del Nepohualtzitzin para el desarrollo de las competencias matematicas.* Mexico: CGEIB-SEP.

Caso, A. (2000). *El Pueblo del Sol.* Mexico: Fondo de Cultura Economica.

Chapman, A. (2006). *Los Hijos del Copal y la Candela. 2 V.* Mexico: UNAM.

Clavijero, F. J. (1974). *Reglas de la lengua mexicana con un vocabulario.* Mexico: UNAM.

Girard, R. (1972). *Esoterismo en el Popol Vuh.* Mexico: Editores Unidos Mexicanos.

Leon Portilla, M. (1979). *La Filosofia Nahuatl.* Mexico: UNAM.

Lopez Austin, A. (2008). *Cuerpo humano e ideologia. Las concepciones de los antiguos nahuas, 2 v.* Mexico: UNAM.

Lucio, C. (2008). *Lecciones y ejercicios del idioma ateka, Mexico: edicion particular.* Particular.

Romerovargas Yturbide, I. (1964). *Moctezuma el Magnifico.* Mexico: Romerovargas y Blasco, Editores.

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