RECENT THESES & DISSERTATIONS ON ETHNOMATHEMATICS

Ubiratan D'Ambrosio

A number of dissertations on Ethnomathematics are reported in different parts of the world. A number have been submitted in Brazil and in Spain. These are effective research contributions which add significantly to the area. This is a report on a number of them.

On May 1995, Maria Luiza Oliveras Contreras presented a doctoral dissertation at the University of Granada, Spain, with title Etnomatemáticas en Trabajos de ArtesanÍa Andaluza: Su Integración en un Modelo para la Formación de Profesores y en la Innovación del CurrÍculo Matemático Escolar [Ethnomathematics in the Artisanal Work in Andalusia. Its Integration in a Model for Preservice Teacher Training and in Innovation of School Mathematics Curricula]. This important work is the result of more than 10 years of research on the Mathematics identified in artistic artifacts typical of Granada. Three kinds of these were chosen for the research: empedrados (stone pavement), taraceas (marquetry) and alfombras (carpets). A very original ethnography is proposed by the author to identify the Mathematical contents of these beautiful handworks. An ethnomathematical theoretical framework allowed the recognition of important styles of doing Mathematics which would be unrecognizable with the prevailing views of academic mathematics. An important aspect of the theses is researching the way the techniques of work are transmitted among artisans, the masters and the apprentices. This was very appropriately called "ethnodidactics" by the author. And the methods there observed were important in proposing a structure of teacher training through projects. We recognize there a model of training teachers to act as researchers. This important contribution to Ethnomathematics will probably become a book in the series published by the Department of Didactics of Mathematics at the University of Granada.

In March 1995 Gelsa Knijnik submitted to the Faculty of Education of the Federal University of Rio Grande do Sul, in Porto Alegre, Brazil, a thesis under the title Cultura, Matemática, EducaÁ“o na Luta pela Terra [Culture, Mathematics, Education in the Struggle for Land] . This very important work is the result of several years of research among teachers of the so-called "Movimento dos Sem-Terra". This is a political action with the objective of occupying the lands which, according to Brazilian constitution, are subjected to expropriation for land reform. The effective possession of these large tracts of land after the occupation implies several legal démarches which may take years, normally about five years. Meanwhile, those occupying the lands are confined to these areas and have to develop their own social structures: schools, medical assistance and production. They cannot leave the territory and the support they receive is nonpermanent, obeying humanitarian demands. In this period of confinement they have to rely on their own resources. These rural populations have a minimal education and have to run their own surveying and land demarcation practices, and the production system, as well as their schooling. There is so much mathematics in all these activities. The ethnographic research of Gelsa Knijnik focused on identifying the Ethnomathematics of these processes and giving the supporting instruments to integrate these practices in a school mathematics curriculum relevant for their immediate needs and allowing the transition to the official school system after overcoming the legal obstacles. How to conduct the teacher training for this parallel educational systems, relying, of course, on the human resources provided by uneducated confined population, is a major challenge. The thesis of Knijnik presents a socio-political and pedagogical study of these issues, always stressing the Mathematical content in every step of the process. The theoretical framework includes a thorough discussion of conceptual aspects of Ethnomathematics.

In April 1995 Adriana Isler P. Leite presented a dissertation to the Programa de Pos-GraduaÁ“o de EducaÁ“o Matemática of the Universidade Estadual Paulista/UNESP at Rio Claro, under the title A Brinadeira é Coisa Seria: Estudos em Torno da Brincadeira, da Aprendizagem e da Matemática [Playing is serious: Studies about playing, learning and Mathematics] . The dissertation was the result of an extended ethnographic research over three years involving children aged between 5 and 8 years old. The focus was understanding the way children play spontaneously and recognizing the Mathematics contents of these activities. The theoretical framework was Ethnomathe-matics and the ethnography adopted, with the analysis of about 60 hours of video taping, lead to an important contribution to understanding the formation of mathematical concepts in early childhood. It is of much importance to the conceptual discussion of the nature of Ethnomathematics in view of the theories of cognition and learning, particularly of Vygotski.

Marianna Kawall Leal Ferreira submitted a dissertation to the University of S“o Paulo on Da Origem dos Homens a Conquista da Escrita: Um Estudo sobre Povos Indigenas e EducaÁ“o Escolar no Brasil [From the Origin of Men to the Conquest of Writing: A Study of Indian Peoples and School Education in Brazil] dealing with the construction of knowledge in an Amazonian tribe. Very careful research was conducted among a number of different tribes of the Parque IndÍgena do Xing™. A variety of cultures provided the author with the opportunity to understand the historical and philosophical ground upon which these tribes build their knowledge. Several aspects of Indian culture, as seen in the schools of the tribe, are analyzed, focusing on the educational process which give emphasis on the transmission of "official" knowledge and values.

SÙnia Maria Clareto worked in a small fishing community on the seashore (caiÁara) in the State of S“o Paulo. The dissertation was an ethnographic study of the space perception of school children after taking classes of Geography. Specifically, what was the perception of the child as "standing upside down" after being exposed to a terrestrial globe. A most interesting dissertation entitled A crianÁa e seus dois mundos: A representaÁ“o do Mundo em crianÁas de uma comunidade caiÁara [The child and its two worlds: The representation of the World by children of a "caiÁara" community] based on this research was submitted to the Universidade Estadual Paulista/UNESP at Rio Claro.

Samuel Lopez Bello submitted a dissertation on EducaÁ“o Matemática IndÍgena -- Un Estudo Etnomatemático dos Indios Guarani-Kaiová do Mato Grosso do Sul [Indigenous Mathematical Education -- An Ethnomathematical Study of the Guarani-Kaiovaa Indians in the State of Southern Mato Grosso] . The dissertation refers essentially to questions about Education, particularly Mathematical Education, among Indian communities in a somewhat remote State in Western Brazil. The main objectives were to identify and recognize different ways of explaining and knowing in the Guarani culture and to relate these with the strategies of formal schooling. The ethnographic research gave origin to new methodologies and techniques on participant observation. New interpretations of cognitive models among indigenous cultures resulted from the research. An important result was the recognition of the role of the history of the individuals and of the communities in the cognitive processes. Among the variety of topics discussed, particularly important were questions about shapes, measures and counting.

General comments

The thesis of Gelsa Knijnik was published, with slight modifications, as a book with title Exclus“o e Resistencia: EducaÁ“o Matemática e Legitimidade Cultural [Exclusion and Resistance: Mathematics Education and Cultural Legitimacy], Artes Médicas, Porto Alegre, 1995. The thesis of Maria Luiza Oliveras Contreras will also appear as a book. The mathematical part of the dissertation of Mariana K. Leal Ferreira became a booklet: Com quantos paus se faz uma canoa! A Matemática na vida cotidiana e na experiÍncia escolar indÍgena [With how many logs one can make a canoe! Mathematics in the daily life and in Indian school experiences], MEC/Assessoria de EducaÁ“o Escolar IndÍgena, Brasilia, 1994. The others will appear only as papers presenting partially the important results. The fact that they are in Portuguese and Spanish limit, in a sense, the accessibility to these important contributions to Ethnomathematics. Indeed, a considerable amount of research in these fields comes out of research in Latin America, as well as in Lusophone countries in Africa and in Portugal and Spain, but language is still a barrier. Fortunately, much of the important work of Paulus Gerdes has been translated into English and French.

These works reveal the large scope of Ethnomathematics. Indeed, we can hardly classify these as Mathematical works. This is, in a sense, a sort of "epistemological aggression". The distinction between Ethnomathematics and Ethnoscience, Ethnohistory, Ethnomusicology, Ethnomedicine, Ethnopsychiatry, Ethnomethodology, becomes very artificial and difficult to establish. Even in the Mediterranean civilizations and as recently as the XV century, Mathematics and Religion, the Sciences and the Arts, are difficult to separate.

This leads us to look into different ways, styles, techniques of explaining, of understanding, of coping with the surrounding natural and cultural environments as the essence of the History of Ideas. In order to organize these studies, we have to coin a few words to express the above: different ways, styles, techniques [tics], of explaining, understanding, coping with [mathema] the surrounding natural and cultural environments [ethno]. Thus we have the word Ethnomathematics, which in this conception obviously incorporates Ethnoscience.

Ways, styles, techniques of explaining, of understanding, of coping with the surrounding natural and cultural environments have been developed and accumulated throughout the entire history of mankind in different cultures, with different objectives and following different patterns of thought. We can hardly fit knowledge recognized in a variety of cultural environments into the current academic classification of knowledge which comes from the civilizations around the Mediterranean. With the increasing attention to -- and respectful attitude towards -- different cultures, broader epistemologies are needed. There is a general acceptance of multiculturalism in Education. But this can be voided if we do not adopt a multicultural approach to the History of Ideas. The Program Ethnomathematics is an answer to this. The theses and dissertations mentioned above fit into this program.