4- National Presentation formated.doc

NATIONAL PRESENTATION AT THE PME34: MATHEMATICS EDUCATION IN BRAZIL

Maria Aparecida Viggiani Bicudo

UNESP – University of State of São Paulo

The objective of this round table is to present an overview of research and practice in mathematics education in Brazil and, to the degree possible, outline what we believe to be the most important contributions if this practice and research to the national and international mathematics education scenario.

Toward this end, we will make a succinct presentation of the Brazilian educational system, of the way mathematics education is present in this system, and of the general panorama of mathematics education research in Brazil. We will highlight what we see as being the most significant research in the national and international context from the perspective of innovation and emphasizing the education of different marginalized groups. We are pleased to have with us three researchers who work with the Education of Young Adults, education of indigenous Brazilians, and with communities of landless people, whose themes are aligned with the national discussion on mathematics education, with strong influences from anthropology and ethnomathematics, linked to the theme of the congress, “Mathematics in different settings”.

GENERAL DESCRIPTION OF THE BRAZILIAN EDUCATIONAL SYSTEM

According to the Law of Directives and Bases for National Education (Lei de Diretrizes e Bases da Educação Nacional - LDB), n° 9394-96, school education is composed of:

Basic education, which includes pre-school, elementary school, and high school;

Higher education.Article 22 of the LDB states that the goal of Basic Education is “to develop the student, guaranteeing the common education indispensable for exercising citizenship, and providing the means for progress on the job and in later studies”. It can be offered in the traditional manner or in the modalities of youth and adult education, special education, or professional education, the latter of which can also be offered at the level of higher education.

In Brazil, both Basic Education and Higher Education can be provided by public schools, which are free, or private schools.

Basic Education includes:

1- 12010. In Pinto, M.M.F. & Kawasaki, T.F.(Eds.). Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. XXX-YYY. Belo Horizonte, Brazil: PME.

Bicudo, Knijinik, Fonseca, Domite

Infant education (pre-school): 3 to 4 years of study for children 0 to 5 years of age. It is offered in schools in the public education system and day care centers for free, as well as in paid private schools and day care centers.

Elementary Education: 9 years of study for children aged 6 to 14 years (standard age).

It also includes Special Education and Youth and Adult Education - EJA. The former is preferentially integrated into the regular teaching, although it can also be integrated into specialized schools and services. Youth and Adult Education is for people 15 years of age and older.

High School: 3 to 4 years of study for students 15 years of age and older (theoretically). In practice, students as young as 13 are accepted as long as they have completed their elementary education. It includes courses with the objectives of complementing elementary education studies and preparing students for higher education and professional education. Studies are integrated with science and technology with the aim of educating citizens for a productive life. Youth and Adult Education are also included in this category for individuals with a minimum age of 18 years who, for various reasons, were unable to follow the standard educational pattern.

Higher Education encompasses:

Undergraduate studies: the goal is to educate professionals in various fields of knowledge and professions. Duration is from 3 to 6 years. It includes sequential courses and professional education.

Graduate studies: As the name implies, these are courses for students who have completed their undergraduate degrees and what to continue their studies. They include:

Stricto-Sensu Graduate programs: These are Masters and Doctoral degree programs. Those programs that have received a positive evaluation by the agencies responsible can also offer post-doctoral studies. This educational level has been expanded and, since the end of the 1990s, includes Masters of Science programs, which also work with masters students in mathematics education.

Lato-Sensu Graduate programs: includes specialization programs (certificate programs).

Brazil has been engaged in a deliberate effort to reduce the illiteracy rate in the country. According to data provided by the Brazilian Institute of Geography and Statistics (IBGE), the rate of illiteracy decreased from 33.6% in 1970 to 13.6% in 2000. However, these are general data and can certainly undergo changes according to conceptions of illiteracy adopted for the surveys.

Brazilian educational policy has emphasized education for youth and adults as the objective aiming to increase chances for those who have reached 15 years of age without dominating important aspects of their performance as citizens, for example, reading and writing in their native language, mathematics, science, and technology.

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In light of the important work being carried out in Youth and Adult Education, this round table will discuss practice and research in this area from the perspective of mathematics education.

MATHEMATICS EDUCATION WITHIN THE BRAZILIAN EDUCATIONAL SYSTEM

Mathematics education is present in all courses at all levels in the Brazilian educational system. At the pre-school and early elementary education level (first to fifth grades), activities involve aspects related to spatial position and direction, sequences, counting, arithmetic operations, and geometric notions. Activities are developed using games, manipulation of concrete materials, fun activities, films, computer resources, and writing and drawing with paper and pencil.

However, the quality of this work varies considerably from school to school and region to region in Brazil. This variability is related to the teachers’ education, the salaries they receive, the pedagogical project of the school, and the socio-cultural context in which the students who attend certain schools live. With respect to pre-school and elementary school, in larger cities, private schools generally offer a better quality education, including mathematical education.

High school also varies with respect to the quality of the education teachers provide to their students. 0However, the total number of students who continue on to high school is smaller, indicating that the system itself selects the “more capable” of handling the demands of schooling, as this phase of learning is somewhat more demanding in terms of what students, teachers, and families expect at this level of education.

Challenges, strengths, and weaknesses of mathematics education in Brazil

The challenges faced by those engaged in mathematics education in Brazil are numerous and complex.

A first and very important challenge is the size of Brazil – size in the sense of geographic extension and number of people in school, including the professionals who work there, students who study there, and people circulating around it because they are family members or would like to have studied there now or in the past. As an example, I will cite the state of São Paulo, considered to the richest state in the Union, and with a unique history in terms of educational ideas compared to the rest of the country. The State Secretary of Education works with a labour force that includes teachers, pedagogical team members, and others who care for the maintenance of the buildings and infrastructure totalling 434,038 professionals who provide services to 4.369,736 students1 (Secretaria da Educação do Estado de São Paulo, 2010). This situation, together with differences in infrastructure of the schools, the socio-cultural reality in which they are imbedded, the education of the teachers, and the cultural and

1 According to Dr. Maria Ines Fini. Technical Counselor of Secretaria da Educação do Estado de São Paulo. March 2010.

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economic devaluation of teachers, constitutes a dense complexity that reveals a totality which is difficult to work with.

Within this complexity, the question of teacher education in general, and mathematics education, in particular, stands out. Other obstacles faced include: devaluation of teaching, growing urban violence, and lack of long term educational policies.

Among the positive aspects of mathematics education in Brazil is the growing understanding that mathematics is more than just another exact science dedicated to numbers, calculations, and the study of spatial relations of geometric objects. Proposals for mathematics teaching and learning activities are many, and little by little, they have become incorporated into the everyday life of the school. They are activities based on constructivist, critical-dialectic (and on a smaller scale, phenomenological) conceptions of knowledge. The presence of these conceptions and proposals in the scenario of mathematics education research has created fertile ground for innovative pedagogical proposals.

Research in Mathematics Education in Brazil and its main contributions for the national and international community

The mathematics education research community in Brazil has been growing strong and is characterized by diverse areas of investigation. The number of strict-sensu Graduate programs offering Masters and Doctoral degrees, including Professional Masters degrees, is large and they are distributed among the different regions of the country. The majority, however, are concentrated in the southeastern region (Minas Gerais, Rio de Janeiro, and São Paulo).

Graduation program on Mathematical Education, Teaching of Mathematics,

Teaching of Sciences.

Total

Master, Doctoral and Professional Master Programs 40

Master Programs 18

Doctoral Programs 9

Master and Doctoral Programs 8

Professional Master Programs 22

Table 1: Graduation program on Mathematical education, Teaching of Mathematics, Teaching of Sciences. (CAPES, 2010).

This fact is related to the roots of mathematics education in Brazil which, without a doubt, lead to Ubiratan D’Ambrósio, as well as educators like Paulo Freire and Joel Martins. These are important people on the Brazilian educational scenario, and beginning in the 1970s, they intensified a critical discourse related to the construction of knowledge and the conception of science and of education. These conceptions provide the foundation for investigations about the teaching and learning of

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mathematics, for understanding the teacher’s stance as being primordially dialogical, and therefore, reflecting respect for the other; for criticism of Western science; and for acceptance of the other as different. Underlying the stance of researchers in this community is an appreciation for the importance of history and philosophy for understanding science and education.

It is a community that is open to technological advances and their presence on the educational scenario, and that collaborates with international mathematics education research centres. Many Brazilian mathematics education researchers received their advanced degrees in centres located in the Unites States, England, Germany, and France, enabling the establishment of professional ties and exchange between these centres and the centres in Brazil where these researchers now develop their work.

The diversity of themes and roots that has proliferated, manifesting itself in Brazilian thought in mathematics education inquiry, is revealed in the very organization of the International Seminar in Mathematics Education Research (SIPEM) organized by the Brazilian Mathematics Education Society (SBEM), which is held every three years with the objective of discussing research in the field. The names of the working groups, which currently number twelve, exemplify this diversity: WG 1 - Mathematics Education in the Early Grades; WG 2 and WG 3 - Mathematics Education in the later elementary school grades and high school; WG 4 -Mathematics Education in Higher Education; WG 5 - History of Mathematics and Culture; WG 6 - Mathematics Education, New Technologies, and Distance Education; WG 7 - Mathematics Teacher Education; WG 8 - Evaluation of Mathematics Education; WG 9 - Cognitive and Linguistic Processes in Mathematics Education; WG 10 – Mathematical Modelling; WG 11 – Philosophy of Mathematics Education; WG 12 – Teaching of Probability and Statistics.

The subjects addressed that permeate the research presented in these 12 WGs are presented in Table 2, below. These data can be found in A philosophical exercise in Mathematics Education research in Brazil (Um exercício filosófico sobre a pesquisa em educação Matemática no Brasil) (Bicudo & Monteiro, 2009), presented in the IV International Seminar in Mathematics Education Research in Brasília in 2009. In this study, we analysed 216 research texts presented and discussed in the III International Seminar in Mathematics Education Research in 2006 in Águas de Lindóia, in the state of Minas Gerais. The column “number of times” in the Table 2 refers to the number of times the author was cited in the total of 1461 research texts analysed.

Nucleases of themes Authors Nº of times Total

Philosophy Morin, E. 5 20

Husserl, E. 4

Foucault, M. 6

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Bachelard, G. 5

Philosophy of

Mathematics

Davis &Hersh 410

Caraça, B.J. 6

Philosophy of EducationBicudo, M.A.V. 11

13Bicudo and Espósito 2

Philosophy of

Mathematics Education

Skovsmose,O. 16

49

Skovsmose, O. and Borba, M.C. 1

Alro, H. and Skovsmose, O. 1

Garnica, A. V. M. and Bicudo, M.A.V. 8

Ernest, P. 5

Machado, N. J. 7

Maria A.V.Bicudo 11

Mathematics Education

(ethnomathematics,

interdisciplinarity, and

other themes)

D’Ambrosio, U 37

37

Education and

Educational Sciences

Freire,P 17

69

Martins, J. 3

Nóvoa, A 9

Bourdieu, P. 6

Coll, C. et al. 4

Coll, C. and Teberosky, A 1

Giroux, H 4

Vigotsky, L. S 11

Piaget, J 7

Bakhtin, M. M., 7

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Nucleuses of themes Authors Nº of times Total

Qualitative Research

(different modalities)

Biklen, S. K. 3

47

Bogdan, R.C.and Bicklen, S. K 10

Bogdan, R. C. 3

Bicudo and Espósito 2

Martins; Bicudo 1

Lüdke, M.; André, M.E.D.A. 5

André, M.E. D. A. 9

Araújo, J. L.; Borba, M. C. 1

Bardin, L 8

Connelly, M; Clandinin, J. 4

Fiorentini, D.; Lorenzato, S. 1

History of

Mathematics

Boyer, C. 5

13Eves, H 4

Ferreira, E. S. 4

Ethnomathematics

Domite, M. C. 4

15Knijinik, G. 7

Ferreira, E. S. 4

History of

Mathematics

Education

Miguel, A, A; Miorim, A, 3

16Miguel, A 7

Miorim, A 6

Problem solvingPolya, G 8 13

Dante, L. 5

Official documents -

LDB, PCN, EJA.BRASIL 36 36

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Nucleuses of themes Authors Nº of times Total

Mathematical

Modeling

Barbosa, J. 15

46

Bassanezzi, R.C. 13

Biembengut, M.S. and Hein, N. 7

Biembengut, M.S. 8

Almeida, L. M. W. and Borssoi, A. H. 3

Statistics

Godino, J. D. 4

21

Godino, J D., Batanero, C., 1

Batanero, C 4

Lopes, C 8

Jacobini, O. R. and Wodewotzki, M. L. L. 1

Jacobini, O. 3

Nucleuses of Themes Authors Nº of times Total

Technology in

Mathematics

Education

Boavida, A. M. 2

37

Penteado, M.G. 3

Borba and Penteado 3

Villareal, M. E 2

Borba, M. C. 8

Borba; Villareal 3

Skovsmose, O.; Borba, M. C. 1

Borba, M. C and Araújo, J. L. 2

Araujo, J. L. 6

Araújo, J. L.; Salvador, J. A. 1

Gitirana, V. 6

Teacher Education Perez, F.; Castillo, D. 1

124Perez, G. 1

Costa, G.L.M., Viel, S.R 1

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Schön, D 10

Almeida, M E 4

Tardif M. 12

Alarcão, I 4

Hargreaves, A. 4

Hargreaves, A. & Dawe, R. 1

Lorenzato, S. E Vila, M. 1

Fiorentini, D. 27

Fiorentini,D.; Fernandes, F.L.P; Cristóvão, E. 1

Pinto and Fiorentini 1

Fiorentini, D.; Castro, F. C. 1

Fiorentini, D., Miorim, A.and Miguel, A . 3

Fiorentini, D., Miorim, A. 3

Ponte, J. P. da; Serrazina, M. 2

Ponte, J. P. da 22

Boavida, A. M.; Ponte, J P. 2

Ponte et al 1

Shulman, L 11

Curi, Edda 4

Garcia, C. M. 7

Nucleuses of Themes Authors Nº of times Total

Mathematics Teaching and Learning

– including Didactics, Algebra,

Geometry, Calculus, and Mathematics

Teaching and Learning at different

educational levels

Bishop, A. 4 148

Vergnaud, G 15

Baldino, R.R. 5

Douady, R. 4

Artigue, M. 7

Barufi, M. C. B. 3

Barufi, M.C.B. Lauro, M. M. 1

Almouloud, S. A. 5

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Lerner, D 6

Alves-Mazzotti, A. J. 4

Kamii, C. 3

Kamii, C. ; D’Clarc, G. 1

Lins, R. & Gimenes, J. 2

Lins, R. C. 4

Almeida, L. M. W.; Brito, D.

S.3

Ball, D. L. 6

Magina, S.M.P. 5

Balacheff, N 5

Lorenzato, S. 3

Monteiro, A. 7

Nunes, T 9

Carraher, D.W. 3

Carraher, D.; Carraher, T.N;

Schliemann A. L. D 2

Carraher, T.N. 1

Chevalard, Y. 12

Chevalard, Y; Bosch, M;

Gascón, J. 3

Campos, T 5

Brouseau, G. 9

Duval, R 11

Evaluation Esteban, H 4 30

Cury, H. 8

Buriasco, R. L. C. 5

Buriasco, R. L. C.; Cyrino, M. C.; Soares, M.

T. C. 1

Borasi, R 4

Brito, M 6

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Ludke, M. 2

Table 2: Themes and authors cited

In this table, one can observe that the authors cited most were those who work with the theme of teacher education. It also illustrates how this theme constitutes a Nucleus of Ideas woven into an interdisciplinary network that includes education, philosophy, philosophy of mathematics, philosophy of mathematics education, history of mathematics education, history of mathematics, mathematics education, cognition theory, various aspects of teaching, ICTs, different approaches to mathematics, the influence of international schools, and concern for teaching at different educational levels. Permeating the influences of this network, two authors are evident: Dario Fiorentini of Brazil and João Pedro Mendes da Ponte of Portugal. One also observes in the ground in which these ideas are planted, the ideas of Schon, Tardif, and Shulman, who are also referred to in the works of Fiorentini and Ponte.

From a national and international perspective, the view of the constitution of knowledge, and the interdisciplinary, dialogical, and critical position of mathematics education researchers in Brazil, have led to a strengthening of research focusing on “the different” in relation to the standard considered to be common in Western civilization and which has spread throughout school culture. Research in ethnomathematics stands out, much of which involves work with indigenous populations and with the landless movement, as well as Youth and Adult Education. In light of its relevance, three important Brazilian researchers who focus on these themes will present their work at this National Presentation: Dr. Maria do Carmo Domite, Dr. Gelsa Knijnik and Dr. Maria da Conceição Ferreira Reis Fonseca.

Dr. Knijnik works with ethnomathematics and has been developing her own way of conceiving of this area of practice and investigation. Her focus for many years has been the way of life of communities known here in Brazil as “landless”. At the end of this text it is presented an abstract of Dr. Knijnik work.

Dr. Fonseca has developed research about Youth and Adult Education. This distinction is not based only on age. The Youth and Adults at whom the initiatives are aimed bring to the discussion the educational project of the country, and of each school, presenting the demands of a substantial part of the population which has been excluded from the educational system. It works with “exclusion” is both a reflection and a causal factor of a much broader mechanism of the denial of rights and opportunities by a society marked by inequalities of many kinds. She says that it is not possible to talk about Youth and Adult Mathematics Education in Brazil without also taking into consideration the seriousness and diversity of the challenges of Social Inclusion, as well as the defining character of these challenges to the constitution of Mathematical Education research in this country, such as it is today. At the end of this text it is presented an abstracted of Dr. Fonseca work.

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Dr. Domite works with ethnomathematics and has developed research with indigenous communities, prioritizing the work of the teacher. She seeks to understand the way indigenous teachers proceed and what conceptions can be understood through analysis of their experiences. Her research has enabled her to outline a set of ideas and proposals regarding directions to be taken by indigenous teacher educators. These ideas resulting from their own experience as an indigenous teacher educator represent one point of view and one direction for transformations that are required in the development of educators who are external to indigenous culture.

According to Dr. Domite, her research must be understood as a set of ideas and proposals for indigenous teacher education as a line of research and educational action when taking into account the culture that one brings inside oneself. These ideas – resulting from her experience as a (mathematics) educator of indigenous teachers – must represent, as a whole, one point of view and one direction for the transformations required in the education of educators who are external to the indigenous culture, to the “other” group. Even as she reflects on cultural and pedagogical practices that exist in teacher education, she argues that it is important to be cautious regarding such reflections because:

When reflecting about indigenous teacher education, we are only focusing on one aspect of its confluence between what is going on in the community, in the (professional) life of the teacher, in the formative dynamics, and in the curriculum of teacher education, among others. However, while our intention is to encourage teacher educators to recognize and value the intuitive/experimental knowledge of the indigenous teacher, we also emphasize the challenging and dangerous aspects of this approach;

When we consider the boundaries involving a set of questions concerning mathematics education, culture and teacher education, it is not easy to draw a line that helps us to make a distinction between mathematics and teacher education as well as mathematics and questions related to culture.

The impact of mathematics education research on educational issues raised by the educational system in Brazil and which have led to advances in mathematics education.

The impact of mathematics education research in the Brazilian educational system is most clearly felt in the legislation relevant to education, mainly mathematics education. The discourse declared in legal texts incorporates the language of the research and, to the extent possible, these documents suggest activities that reflect proposals present in the studies carried out by these researchers. In the sphere of classroom practice, however, the reality reveals a gap. In pre-school, elementary and high school education, the influence of research is unclear. It is present in a diluted form as a trend, and in more advance schools, its presence may be more visible. In higher education, in undergraduate programs aimed at the education of mathematics teachers, there is a divide between professors who teach mathematics courses, such as algebra, geometry, and analysis, and those who teach didactics of mathematics and

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history of mathematics, for example. The former generally work with a view of mathematics understood as a science, focusing on informational activities related to contents and presentation of procedures to produce this knowledge. Professors in the second group, who are involved with pedagogy and mathematics education, focus on the teaching and learning of mathematics, using an approach that brings knowledge from the field of mathematics education.

The presence of mathematics education is strong in graduate programs that work in this field of knowledge, forming a locus wherein research and debate on mathematics education take place.

References

Bicudo, M.A.V. & Paulo, R.M. (2009) Um exercício filosófico sobre a pesquisa em educação Matemática no Brasil. WWW.sepq.org.br/sócios

BRASIL. Congresso Nacional. Lei nº 9.394 de 20 de dezembro de 1996. Estabelece as diretrizes e bases da educação nacional. 1996. Disponível em: http://portal.mec.gov.br/arquivos/pdf/ldb.pdf Acesso em (dia) (mês) 2010.

SISTEMA EDUCATIVO Nacional de Brasil: 2002 / Ministério da Educação de Brasil (MEC/INEP) y Organización de Estados Iberoamericanos (OEI). Madrid España. Disponível em: http://www.oei.es/quipu/brasil/ . Acesso em : (DATA)

CAPES - Coordenação de Aperfeiçoamento Profissional de Nível Superior.Disponível em: <http://conteudoweb.capes.gov.br/conteudoweb/ProjetoRelacaoCursosServlet?acao=pesquisarIes&codigoArea=90200000&descricaoArea=MULTIDISCIPLINAR+&descricaoAreaConhecimento=ENSINO&descricaoAreaAvaliacao=ENSINO+DE+CI%CANCIAS+E+MATEMATICA >

São Paulo. Secretaria da Educação do Estado de São Paulo. Assessora . 2010.

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http://conteudoweb.capes.gov.br/conteudoweb/ProjetoRelacaoCursosServlet?acao=pesquisarIes&codigoArea=90200000&descricaoArea=MULTIDISCIPLINAR+&descricaoAreaConhecimento=ENSINO&descricaoAreaAvaliacao=ENSINO+DE+CI%CANCIAS+E+MATEMATICA




GELSA KNIJNIK’S RESEARCH ON ETHNOMATHEMATICS

Gelsa Knijnik

University of Vale dos Sinos

The starting point of Dr. Knijnik trajectory as a researcher was inspired by the work of Ubiratan D’Ambrosio and Eduardo Sebastiani Ferreira, from whom she learnt not only about how mathematics education is intrinsically connected to culture, but also about the ethics involved in the act of researching. Knijnik’s first studies were performed in the peripheries of the capital of Rio Grande do Sul State (Knijnik, 1988) and they are focused in pedagogical experiences she developed with pre-service mathematics teacher. In what can be considered the second phase of her trajectory, Knijnik’s work assumed two new characteristics. From the empirical side, it was centered in rural areas of that Brazilian State. It was the beginning of her work with peasants of the Landless Movement that at that few years before emerged precisely in that part of the country. From the theoretical side, she put special attention to a sociological perspective based on Pierre Bourdieu, Claude Grignon and Claude Passeron’s theorizations about the inter-relations between popular and academic knowledges. At that point of her trajectory, she conceptualized her ethnomathematics approach as the investigation of the traditions, practices and mathematical concepts of a subordinated social group and the pedagogical work which was developed in order for the group to be able to interpret and decode its knowledge; to acquire the knowledge produced by academic Mathematics; and to establish comparisons between its knowledge and academic knowledge, thus being able to analyse the power relations involved in the use of both these kinds of knowledge (Knijnik, 1997). In fact, the pedagogical work she was performing with the Landless peasant of the South of Brazil favoured investigating and interpreting their own methods to carry out practices as the “cubação da terra” – to find the area of a surface that corresponds to a portion of a land (1999, 2002).

The notion of culture which gave support to that ethnomathematics approach understood it as a human production, which is not fixed, determined, closed in its meanings for once and for all. On the contrary, culture is considered a conflictive, unstable and tense terrain, undermined by a permanent dispute to impose meanings through power relations. Knijnik (2003: 4) argued that to operate with a concept of culture marked by power relations implies considering the cultural practices that are the subject of the ethnomathematics’ research not as a body of “traditional” knowledges which do not re-update their meanings over time, an inert set of knowledges that is transmitted from generation to generation, as though it were cultural “baggage”. Moreover, from the Ethnomathematics perspective, the concept of culture moves away also from a conservative view that expresses culture as “ a heritage of humankind”. Considering that this cultural heritage is a social production

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resulting from all our efforts, the expression thus supports the argument that humanity as a whole has the right to access and use knowledge created by humans. Nevertheless, the expression cultural heritage of humankind is very often identified only with academic mathematics. It is precisely this identification that masks power relations that, in turn, legitimise a very specific way of producing meaning – the Western, white, male urban and heterosexual one – as the cultural heritage of humankind. By providing visibility to other Mathematics besides the academic one, Ethnomathematics discusses precisely this apparent “consensus” as what counts as cultural heritage of humankind. It is, in fact, a part of a broad, heterogeneous production, precisely the part produced by hegemonic groups. (Knijnik, 2002a) In summary, Knijnik considered that Ethnomathematics had as one of its main concerns precisely to highlight those mathematical ways of giving meaning to life different from the hegemonic one, the one which is called “the” mathematics.

In this second phase of her trajectory, as before, Knijnik’s ethnomathematics research involved carrying out fieldwork in which ethnographic techniques such as participant observation, audio recording, field diary and interview were used. She was aware that even considering that she was not doing ethnography in the strictest sense of the word, in using some of its elements it was necessary to take into account questions and challenges which have been asked contemporarily by Anthropology an area strongly marked by its links with the colonial area, with the description of the “Other”.

In the last years, Knijnik’s Ethnomathematics perspective was extended (Knijnik and Wanderer 2008, Knijnik 2007, Wanderer 2007). Still considering the relevance of the sociological analysis, new theoretical lens are used. Here, Post-Structuralist theorizations, specially Michel Foucault thinking, and the Ludwig Wittgenstein’s late work assume a central role. Based on these theorizations it was reconceptualized the Ethnomathematics perspective which is giving support to the studies developed by the Unisinos Research Group (Wanderer, 2007, Giongo 2008, Duarte 2009, Knijnik and Bocasanta 2010, Knijnik and Wanderer 2010). They have considered an ethnomathematical perspective that consists of a toolbox which allows analyzing the Eurocentric discourses of academic and school mathematics; the effects of truth produced by such discourses; issues of difference in mathematics education, considering the centrality of culture and the power relations that institute it; and the language games that constitutes different mathematics and its family resemblances.

From a methodological point of view, the empirical settings of Knijnik’s work were increased. She continues doing fieldwork with the Landless Movement. After almost 20 years since it started, each new study conceived by the Research Group she coordinates includes this peasant social movement. But currently other cultural groups have been involved, like rural German-descended communities (Wanderer, 2007), Technical Agricultural School (Giongo, 2008). The diversified data and the use of the theoretical toolbox mentioned above have being productive to Knijnik’s

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Research Group since they are bringing other interrogations about mathematics education, other questions to think about.

References

Acesso Giongo, I. & Knijnik. (2009). Educação matemática e currículo escolar: um estudo das matemáticas da escola estadual agrícola de Guaporé. Zetetiké, Campinas, v.17 n.32

Knijnik, G. & Bocasanta, D. (2010). About the craft of research: Neoaphoristic pills. In: Alro, H. & Ravn, O. & Valero, P. (ed).Critical Mathematics Education: Past, Present and Future. Rotterdam: Sense Publishers, p. 101-120.

Knijnik, G. & Duarte, C. (2009). Entrelaçamentos e dispersões de enunciados no discurso da Educação Matemática: um estudo sobre a importância de trazer a ‘realidade’ para as aulas de matemática. Paper presented at the ANPEd 32nd Annual Meeting.

Knijnik, G. &Wanderer, F. (forthcoming). Mathematics Education and Differential Inclusion: A Study about Two Brazilian Time-Space Forms of Life. Berlin, Zentralblatt für Didaktik der Mathematik. v.42, n.4

Knijnik, G. (1998). Ethnomathematics and Political Struggles. Berlin, Zentralblatt für Didaktik der Mathematik.v. 30, n.6.

Knijnik, G. (2007). Mathematics education and the Brazilian Landless Movement: three different mathematics in the context of the struggle for social justice. Philosophy of Mathematics Education Journal. , v.21, p.1 - 18.

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YOUTH AND ADULT MATHEMATICS EDUCATION: RESEARCH AND CHALLENGES TO A SOCIAL INCLUSION PROJECT

Maria da Conceição Ferreira Reis Fonseca

Federal University of Minas Gerais

In Brazil, and in most Latin American and African countries we use the expression “Youth and Adult Education” mainly to refer to Basic Education2 initiatives for persons over 18 who did not have access to formal school education during childhood or youth. The inclusion of the word “Youth” in the traditional expression “Adult Education” makes official the long-term demand for policies, and the necessity of pedagogical alternatives, which emphasize the needs and the rights of youth, with their own expectations towards schooling.

However, the characterization of Youth and Adult Education can not be based only on age of students. The Youth and Adults at whom the initiatives are aimed bring to the discussion of the educational project of the country, and of each school, the demands of a substantial part of the population which has been excluded from the educational system. Such exclusion is both a reflection and a causal factor of a much broader mechanism of the denial of rights and opportunities by a society marked by inequalities of many kinds.

This restriction of rights and the limitations of the opportunities available to those people dictate their ways of fitting into the world and of inserting themselves into social practices, including scholarly ones. Confronted with the novelty presented by the various ways that these students position themselves in relation to knowledge (their intention, production, transmission and use), the theories on which educators have traditionally based their pedagogical projects, actions and evaluations were revealed to be incomplete, inefficient and even inadequate to address, understand and answer the demands of a new public not identified with school culture.

Therefore, it is not possible to talk about Youth and Adult Mathematics Education research in Brazil without also taking into consideration the seriousness and diversity of the challenges of Social Inclusion, as well as the defining character of these challenges to the constitution of this research field, such as it is today.

In a special way, these challenges are presented to those researchers whose works focus on literacy and numeracy practices (Adelino, 2009; Araújo, 2001; Azevedo, 2002; Bail, 2002; Barreto, 2005; Cabral, 2007; Castro, 2006; Carvalho, 2001; Conti, 2009; Fantinato, 2003; Faria, 2007; Ferreira, 2009; Fonseca, 2001; 2004; 2009; Franco, 2004; Giongo, 2008; Knijnik, 2006; Lima, 2007; Mendes, 2001; Monteiro,

2In Brazil, what is called “Basic Education” comprises nine years of “Fundamental School” and three years of “Medium School”. These levels were originally aimed at children and adolescents from 6 to 17 years old.

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1998; Santos, 2004; Silva, 2008; Souza, 2008; Wanderer, 2001; 2007). In fact, a new perspective on literacy began to request the work and the assumption of a new perspective by mathematical educators. This perspective demands investigations about skills, knowledge and principles that Young and Adult students, as readers, need to bring together in order to handle texts available in their social practices. Included among these skills, knowledge, and principles are also those related to quantification, ordering, spatial and metric relations and the ludic, esthetical or philosophical aspects, usually associated with mathematics.

In this way, challenges of Youth and Adult Education have induced studies in which popular mathematical practices begin to be interpreted and decoded as answers to the demands of a society that is marked by writing culture and quantitative arguments. These studies help to understand the internal coherence of numeracy practices and their close connection with the practical world.

Nevertheless, these persons that are now included in the school system, with their specific reactions, expectations, desires and reservations, expect, at some level, to comprehend the concepts or procedures of the academic mathematics traditionally considered as objectives of the teaching process, either because of their utility or their social value. Thus, to support Youth and Adult Education, the investigations must not treat academic and popular knowledge as a dichotomy. Their relationship must be permanently examined, and the power relations involved in the use of each of these forms of knowledge must be applied as the parameters for their analysis. It obliges us to include in the approach to mathematics, and in the studies about educational processes and their actors, the historical character of mathematical practices pointed out by the thematization of the confrontation or the solidarity – both involving power relations – between academic and popular knowledge.

However, the historical character of mathematical practices may be seen in more than just the power relations involved in these practices. Some Brazilian investigations into Mathematics Education (Adelino, 2009; Barreto, 2005; Cabral, 2007; Castro, 2006; Conti, 2009; Faria, 2007; Ferreira, 2009; Fonseca, 2001; Giongo, 2008; Knijnik, 2006; Lima, 2007; Silva, 2008; Souza, 2008; Wanderer, 2007) and their ways of proposing, realizing and analysing the pedagogic practices have helped Youth and Adult educators and researchers to understand the History permeating the production of meaning in mathematics practices by considering meaning as printed in the discursive materiality of social practices. Meaning is somehow treated as discursive and is defined by the enunciative occurrence insofar as creating, using, teaching and learning mathematics are activities performed in contexts of interaction, through different media, in which language and ideology, getting into contact with each other, produce meaning effects among interlocutors .

These Brazilian Youth and Adult Mathematics Education studies are being developed by considering, explicitly or implicitly, the teaching-learning situations as discursive processes of negotiating meanings. These studies has helped educators both to

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comprehend how the socio-cultural aspects of mathematical knowledge permeate their pedagogic practices in Youth and Adult Education and how these aspects persist even if, in the analyzed experience, the approach that is adopted tries to silence them or dissociate the produced knowledge from them.

And so, these studies helps us Youth and Adult educators to understand our own practices and to product new pedagogical practices engaged in a Social Inclusion Project.

References

Adelino, P.R. (2009). Práticas de Numeramento nos livros didáticos de matemática voltados para a Educação de Jovens e Adultos. Belo Horizonte: UFMG (Master’s in Education).

Araújo, D.A. (2001). O Ensino Médio na Educação de Jovens e Adultos: um material didático de matemática e o atendimento às necessidades básicas de aprendizagem. Belo Horizonte: UFMG, (Master’s in Education)

Azevedo, P.M.A.S. (2002) Um processo de ensino-aprendizagem de equações vivido por alunos jovens e adultos em sala de aula: transitando por registros de representação. Campinas: UNICAMP (Master’s in Education)

Bail, V.S. (2002) Educação Matemática de jovens e adultos, trabalho e inclusão. Campinas: UNICAMP (Master’s in Education)

Barreto, M.F.T. (2005) O tempo vivido pelo alfabetizando adulto nas aulas de matemática. Rio Claro: UNESP (PhD in Education).

Cabral, V.R.S. (2007) Relações entre conhecimentos matemáticos escolares e conhecimentos do cotidiano forjados na constituição de práticas de numeramento na sala de aula da EJA. Belo Horizonte: UFMG (Master’s in Education).

Carvalho, D.L. (2001) Diálogo Cultural, Negociação de Sentidos e Produção de Significados Matemáticos por Jovens e Adultos. Zetetiké (UNICAMP), Campinas, v. 9, p. 43-76

Castro, L.R.C. (2006) Narrativas sobre a educação matemática de jovens e adultos: um estudo etnomatemático. São Leopoldo, RS: UNISSINOS (Master’s in Education).

Conti, K.C. (2009) O papel da estatística na inclusão de alunos da educação de jovens e adultos em atividades letradas. Campinas: UNICAMP (Master’s in Education).

Fantinato, M.C.C.B. (2003). Identidade e sobrevivência no Morro de São Carlos: representações quantitativas e espaciais entre jovens e adultos. São Paulo: USP. (PhD in Education).

Faria, J.B. (2007). Relações entre práticas de numeramento mobilizadas e em constituição nas interações entre os sujeitos da Educação de Jovens e Adultos. Belo Horizonte: UFMG (Master’s in Education).

Ferreira, A.R. (2009). Práticas de numeramento, conhecimentos cotidianos e escolares em uma turma de Ensino Médio da Educação de Jovens e Adultos. Belo Horizonte: UFMG (Master’s in Education).

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Fonseca, M.C.F.R. (2001). Discurso, memória e inclusão: reminiscências da matemática escolar de alunos adultos do Ensino Fundamental. Campinas: UNICAMP (PhD dissertation).

Fonseca, M.C.F.R.(Ed.) (2004). Letramento no Brasil: habilidades matemáticas. São Paulo: Global: Ação Educativa: Instituto Paulo Montenegro.

Fonseca, M.C.F.R (2009) Conceito(s) de numeramento e relações com o letramento In: Nacarato & Lopes (ed). Educação matemática, leitura e escrita: armadilhas, utopias e realidade. Campinas : Mercado das Letras, p. 47-60.

Franco, I.C.A. (2004) Procedimentos multiplicativos: do cálculo mental à representação escolar na educação matemática de jovens e adultos. Campinas: UNICAMP (Master’s in Education).

Giongo, I.M. (2008). Disciplinamento e resistência dos corpos e dos saberes: um estudo sobre a educação matemática da Escola Estadual Técnica Agrícola Guaporé. São Leopoldo: UNISSINOS (PhD in Education).

Knijnik, G. (2006). Educação matemática, culturas e conhecimento na luta pela terra. Santa Cruz do Sul: EDUNISC.

Lima, P.C. (2007) Constituição de Práticas de Numeramento em eventos de Tratamento da Informação na Educação de Jovens e Adultos. Belo Horizonte: UFMG (Master’s in Education).

Mendes, J.R. (2001). Ler, escrever e contar: práticas de numeramento-letramento dos Kaiabi no contexto de formação de professores índios do Parque Indígena do Xingu. Campinas: UNICAMP (PhD dissertation).

Monteiro, A. (1998). Etnomatemática: as possibilidades pedagógicas um curso de alfabetização para trabalhadores rurais. Campinas: FE/UNICAMP. (PhD dissertation).

Santos, M.E.C. (2004). Posso fazer do meu jeito?: registros das estratégias de adultos desafiados a resolver problemas matemáticos aditivos. Itajaí, SC: Univali. (Master’s in Education).

Silva, F.B.S. (2008). A(prender) matemática é difícil: problematizando verdades do currículo escolar. São Leopoldo: UNISSINOS (Master’s in Education).

Souza, M.C.R.F. (2008). Gênero e Matemática(s) – Jogos de verdade nas práticas de numeramento de alunas e alunos da Educação de pessoas jovens e adultas. Belo Horizonte: UFMG (PhD. in Education).

Wanderer, F. (2001) Educação de jovens e adultos e produtos da mídia: possibilidades de um processo pedagógico etnomatemático. São Leopoldo, RS: UNISSINOS (Master’s in Education).

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