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In this paper we first explore how Wittgenstein’s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein’s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics (Wittgenstein 1973). We will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics. The central philosophical investigation concerns Wittgenstein’s shift to abandoning the essentialist concept of language and therefore denying the existence of a universal language. Languages—or ‘language games’ as Wittgenstein calls them—are immersed in a form of life, in a cultural or social formation and are embedded in the totality of communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figure-making and sand drawing, to illustrate Wittgenstein’s ideas. Based on an ethnomathematical comparative analysis, we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality, as expressed in both string figure-making and sand drawing practices, from one society to another. Finally, we suggest that a philosophical-anthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cultures. Philosophical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.

In this paper we first explore h ow Wittgenstein ’ s philosophy provides a conceptual tools to discuss the possibility of the simultaneous existence of culturally different mathematical practices. We will argue that Wittgenstein ’ s later work will be a fruitful framework to serve as a philosophical background to investigate ethnomathematics ( Wittgenstein 1973) . W e will give an overview of Wittgenstein’s later work which is referred to by many researchers in the field of ethnomathematics . The central philosophical investigation concerns Wittgenstein’s shift to abandon ing the essentialist concept of language and therefore deny ing the existence of a universal language. Languages — or ‘language games’ as Wittgenstein calls them — are immersed in a form of life, in a cultural or social formation and are embedded in the totality o f communal activities. This gives rise to the idea of rationality as an invention or as a construct that emerges in specific local contexts. In the second part of the paper we introduce, analyse and compare the mathematical aspects of two activities known as string figure - making and sand drawing, to illustrate Wittgenstein ’s ideas . Base d on an ethnomathematical comparative analysis , we will argue that there is evidence of invariant and distinguishing features of a mathematical rationality , as expressed in both string figure - making and sand drawing practices, from one society to another . Finally, w e suggest that a philosop hical - anthropological approach to mathematical practices may allow us to better understand the interrelations between mathematics and cul tures. Philoso phical investigations may help the reflection on the possibility of culturally determined ethnomathematics, while an anthropological approach, using ethnographical methods, may afford new materials for the analysis of ethnomathematics and its links to the cultural context. This combined approach will help us to better characterize mathematical practices in both sociological and epistemological terms.

Some mathematics educational reform policies indicate that mathematics education should be connected to learners‟ cultures. However, teaching in schools rarely brings the interconnection between mathematics and culture in pedagogically informed ways. Connections are often done superficially; the curriculum in schools lacks content and specific strategies that enable the making of the connections explicit in the context of teaching. The qualitative study from which this paper emerges worked with three mathematics teachers in an attempt to teach mathematics in ways that connect key concepts with culture. Through mathematizing culturally-based activities performed at a cultural village2, two Grade 9 mathematics topics in the South African curriculum were indigenised. A teaching unit on the indigenised topics was designed and implemented in five Grade 9 classes at the same school. The paper demonstrates that the experience of designing, implementing, and reflecting on the intervention study had some positive contribution to the participating teachers‟ pedagogical repertoire. Teachers saw the possibility of using cultural villages as instructional resources for connecting mathematics education to learners‟ cultures in the South African curriculum. I argue that cultural villages can be used as contexts for mediating culture and mathematics education.

Beginning from the reflections about a methodology used in a research project with an indigenous Colombian community, this paper outlines some possibilities for ethnomathematical research. Issues like intellectual property and social relevance are discussed in order to propose a broader concept of “academic instance”, trough the acknowledgement and legitimation of alternative scenarios of generation, transmission and transformation for mathematical knowledge. This paper has five sections: a) preliminaries about the indigenous community, b) description of the research process and its products (for their very nature, it will be written in a first-person plural voice), c) individual thoughts, treating the harmony between the ethnomathematical methodology and its theoretical, humanistic and political foundations, d) final remarks, sharing insights for further development, e)an epilogue or a review about the experience, to discuss the spirit which aims the analysis made.

Ethnomathematics is the mathematics of cultural groups, but often those cultural groups are “others” and “elsewhere”. However, it is also valuable to look inward to see the interaction of mathematics and one’s own culture. An assignment in a graduate course offers this opportunity to students. The assignment is to find an area of the student’s personal “culture” (sometimes defined rather broadly) and find its use of mathematics. Students are asked to write about (a) the cultural area; (b) how they are personally tied to it; and (c) how and where it uses mathematics. In addition to the paper, students make an oral presentation. Thus, all students learn (often surprising) aspects of their classmates’ non-professional life, and the presenter digs into areas of family and heritage that may not have been reviewed before. Since all students are classroom mathematics teachers, finding their own cultural mathematics is not only enlightening for themselves, but also offers opportunities to include new mathematics applications in their teaching. This paper includes much personal background that led to this assignment and is a report on more than a decade of using it, including brief student examples.

This paper aims to recover, develop, explore and expand the cultural practices of the ethnic group Nyanekankhumbi from southwestern Angola from an ethnomathematical perspective evidenced in the construction of houses of cob wall, keeping in mind the social multiculturalism of schools that exists, in part, almost everywhere, and in particular, in Portugal with a considerable and growing number of immigrants from Africa and other parts of the world. It becomes increasingly relevant to preserve the cultural memory of these people. This paper presents an introduction to the ethnomathematics studies in Africa by various authors usnig a methodology based on informal conversation with group members, participant observation, and interviews, supported by photographic records and written field notes. The results of the research developed in Huíla (Angola) registered much important mathematical knowledge about building the traditional houses of the Nyaneka-nkhumbi that can be understood in the creation of tasks in a context of formation, a topic of interest in the first author‟s doctoral project.

T his paper aims to recover, develop, explore and expand the cultural practices of the ethnic group Nyaneka - nkhumbi from southwestern Angola from an ethnomathematical perspective evidenced in the construction of houses of c ob wall, keeping in mind the social multiculturalism of schools that exists, in part, almost everywhere, and in particular, in Portugal with a considerable and growing number of immigrants from Africa and other parts of the world. It becomes increasingly r elevant to preserve the cultural memory of these people. This paper presents an introduction to the ethnomathematics studies in Africa by various authors usnig a methodology based on informal conversation with group members, participant observation, and in terviews, supported by photographic records and written field notes. The results of the research developed in Huíla (Angola ) registered much important mathematical knowledge about building the traditional houses of the Nyaneka - nkhumbi that can be understoo d in the creation of tasks in a context of formation, a topic of interest in the first author‟s doctoral project.

This paper starts from two statements based on a literature review. The first one concerns the learning process and states that learning is situated and socioculturally contextualized. Learning happens in the space of the background and the foreground of the learner in his or her particular environment of experience. This statement is based on the Vygotsky and the Cultural psychology approach (Cole, 1996) and on the work of Vithal & Skovsmose (1997).The second statement concerns the deficient theory of the learning process (instead of the deficiently of the learner). Based on the international comparative research on mathematical skills we claim that the drop out of school of many groups of children (OECD, 2010) has to do with the insufficient learning system at school that fail to fit with the daily background knowledge of the children.In the final part of the paper we will present three different ethnomathematical cases based on the educational practices that the authors developed in recent years.

This paper starts from two statements based on a literature review. The first one concerns the learning process and states that learning is situated and socioculturally contextualized. Learning happens in the space of the background and the foreground of the learner in his or her particular environment of experience. This statement is based on the Vygotsky and the Cultural psychology approach (Cole, 1996) and on the work of Vithal & Skovsmose (1997). The second statement concerns the deficient theory of the learning process (instead of the deficiently of the learner). Based on the international comparative research on mathematical skills we claim that the drop out of school of many groups of children (OECD, 2010) has to do with the insufficient learning system at school that fail to fit with the daily background knowledge of the children. In the final part of the paper we will present three different ethnomathematical cases based on the educational practices that the authors developed in recent years.

This paper, based on a PhD research, aims to discuss the conception of mathematics, and its developments in terms of methodology, of five preeminent ethnomathematics researchers: Bill Barton (University of Auckland, New Zealand), Eduardo Sebastiani Ferreira (Universidade Estadual de Campinas, Brazil), Gelsa Knijnik (Universidade do Vale do Rio Sinos, Brazil), Paulus Gerdes (Universidade Eduardo Mondlane, Mozambique) e Ubiratan D¿Ambrosio (Universidade Bandeirante de São Paulo, Brazil). Thresearch was carried out under a phenomenological perspective, and its methodology involved aninterview with each ofthe above mentioned researchers. Theses interviews were analyzed hermeneutically, and through phenomenological reductions, thematic categories were articulated. This paper brings the category which talks about the presence of mathematics within ethnomathematics.

The article aims to analyze the rationality employed in the cassava flour farmers' production in a mill located in the Ibiraquera neighborhood of the Imbituba city in Santa Catarina - Brazil. In this study we will emphasize the planting and harvesting of cassava, and procedures of its processing into flour, based on a qualitative methodology of ethnographic inspiration. We conducted a description from direct observations and semi-structured interviews, in addition to field diaries, and photographic and filmic records. We analyze the material from the theoretical framework of Ethnomathematics. Our main goal with this approach is to distance us radically from attempts to frame popular knowledge in terms of the hegemonic knowledge of mathematics. This paper discusses the insurrection of subjugated knowledges within their specific language games. In this sense, we use the theories of Michel Foucault on subjugated knowledges and of Ludwig Wittgenstein in his second phase, about the language limits. Finally, we suggest that research can enable a reflection on pedagogical practices, in relation to the curriculum and ways of dealing with mathematical school knowledge and the knowledge produced by other ways of life.

In this paper we account for the creation and analysis of an environment of mathematical modeling from the socio-critical perspective (Araújo, 2009). It was developed with a group of eleventh graders at a public school in Bogotá, Colombia, where this approach does not usually work. The main aim of the environment was to make evident opportunities and challenges that could be found incorporating: i) non mathematical problems in the math class, ii) Students' active participation in the model construction, iii) Students' active participation in the society and, iv) teacher's performance as mediator. To account for this, a critical methodological approach was adopted (Vithal, 2000; Skovsmose & Borba, 2004 and Valero, 2012), as well as an analysis of the discourses produced by participants in the environment created. It was found, as a possibility, that students work in groups, demonstrate critical positions and democratic attitudes in class. As challenges, it was established that students’ active participation in the society needs to be thought over and more deeply worked.

Systems of measurement were fundamental to the development of peoples, so that the most diverse oriental and western cultures developed their own tools parallel to one another, bringing man to the creation of norms and rules of universal measurement. In Brazil, the 6th-grade math books of elementary school students bring them into formal contact with the metric system and their multiples and sub-multiples, but the students are not induced to think about other forms of measuring present in their daily lives. This paper shows, through narratives of the Kanamari indigenous people of Maraã-AM, how their ancestors measured length, distances, lands and even the height of the houses. These narratives were introduced into the classroom and presented to the 6th-grade students of the Indigneous Municipal School of Kanamari in order to teach the units of measurement with an ethnomathematical approach. Beyond rescuing a small part of their culture, this activity favored a reflection on the part of teachers about their pedagogical practices, since, community leaders, indigenous teachers and non-indigenous have noted that an institutionalized education allied to indigenous knowledge favors student learning of content, allowing for the perpetuation of Kanamari culture.

We present findings from a research project aimed at developing instructional resources to support the learning with understanding of the numeration systems of Mexican indigenous languages. The project was conducted following the design research methodology. Products include instructional resources, as well as theoretical and methodological innovations developed to support the instructional design process. The paper centers on the process of developing an instructional sequence for teaching the Tu’un Savi (or Mixtecan) numeration system. This process is framed as a paradigmatic case of the kind of design research conducted in the project.

This paper sustains the affirmation: mathematic relations and art dialogue in the work of the Ecuadorian artist Estuardo Maldonado who motivates a poetic flight imagine inviting new times and new spaces .After the “S” structure, Maldonado goes in communion with Pre-Columbian peoples and from that experience he shows the ludic ways of making mathematics so to put into doubt the rational calculi from “modern reason”. The proposal is to make a trip with Maldonado through the constellations, but not through which we already know about them, yet towards what we could imagine “with” them in order to motivate the "creative act" that makes the human being a co -creator of the world in the development of the technique involves knowing how to think and know-how . The methodology used to develop this research is the observation, critical analysis and reflection. We conclude that primarily involve mathematical relations, beyond applying simple formulas, opening infinite field of thought.

This paper presents a preview of an investigation aimed at identifying and systematizing some mapuche [mathematical] knowledge. The conceptual references are taken from Ethnomathematics defined by D'Ambrosio (2008) and the cross-cultural activities defined by Bishop (1999). The methodological reference is given by the Collaborative Research (Boavida y Ponte, 2002) and some elements of the Inductive Intercultural Method (Gasché, 2008). The team of co-researchers is composed of the some members of Indigenous Peoples Program of a municipality of Santiago, and members of research group about Ethnomathematics of a particular university. This text presents the research project and some partial results in relation to the space-time location according to the mapuche worldview: the sun is the central reference for the space location and for the time location over one day and over the years; whereas the moon shows the time location over larger cycles within one year (such as the moon cycles by themselves). The results will be incorporated into the educational area of the Indigenous Peoples Program of the municipality El Bosque.

S ystem s of measurement were fundamental to the development of peoples, so that the most diverse oriental and western cultures developed their own tools parallel to one another , bringing man to the creation of norms and rules of universal m easurement. In Brazil, the 6th - grade math books of elementary school students bring them into formal contact with the metric system and their multip les and sub - multiples , but the student s are not induced to think a bout other forms of measuring present in their daily lives. This paper shows , through narrative s of the Kanamari indigenous people of Maraã - AM, how their ancestors measured length, distances, lands and even the height of the houses. These narratives were i ntroduced into the classroo m and presented to the 6th - grade students of the Indigneous Municipal School of Kanamari in order to teach the units of measurement with an ethnomathematical approach . Beyond rescuing a small part of their culture, this activity favored a reflection on the part of teachers about their pedagogical practices, since, community leaders, indigenous teachers and non - indigenous have noted that an institutionalized education allied to indigenous knowledge favors student learning of conten t , allowing for the perpetuation of Kanamari culture.

As cultures grow they must count and calculate, in turn creating their own systems of numeration and ways of representing numbers; it was not different in the case of Inka culture. In this paper we formulate a hypothesis about the Inka numeration system in the Yupana and Khipu, and its respective algebraic equivalence using summation theories, matrices, and vectors. We also present a software based on those algorithms and developed for Android “YupanaInka.apk”, which allows one to simulate the mathematical computation of the Yupana. This work was motivated by an attempt to recover the ancestral knowledge of Inka mathematics in order to strengthen and/or develop the identity process in Tawantinsuyo.

When thinking about mathematics, seldom does one think about culture, context, history, or diversity. Many teachers believe that there is no place for such constructs in their mathematics classrooms. As an ethnomathematician, my primary goal is to find meaningful ways to bring components of ethnomathematics into the mainstream mathematics curriculum and classrooms. In this paper, I describe key aspects of a mathematics curriculum that was designed to promote meaningful connections between ethnomathematics theory and practice and highlight how this curriculum might help address the key tenets of a critical ethnomathematics curriculum.

The Brazilian prison system can be understood metaphorically as a large pressure cooker. It is eminent danger of an explosion. In this context there is a constant tension in relations between its main subjects: the arrested and jailers. The school existing in this space assumes, among its many functions, the role of mediator of tensions and of space of freedom. In this paper we present an excerpt of ongoing research which aims to investigate and understand the knowledge built / acquired by young people and adults in the context of deprivation and restriction of freedom, in search of a dialogue between this feasibility knowledge with math classes in a transdisciplinary perspective. The field research was conducted in a state school located within a male Prison Unit of the State of Rio de Janeiro, motivated by the teaching practice of the author in this school. In this sense, we have adopted an ethnographic approach that has allowed us to immerse ourselves in the context of the arrested students. In this work the following aspects of the research will be presented along with a brief introduction: the definition of the theme and the objectives (general and specific), the methodology, the theoretical framework, the relevance and some notes for possible outcomes.

Math knowledge and skills are not found exclusively in schools. The Nyaneka-nkhumbi ethnic group is one of the linguistic groups in southern Angola, with rich and unknown ethnomatematics knowledge and know-how. With this study we intend to contribute to the research in ethnomathematics and to the development and dissemination of mathematics knowledge 'hidden' in the culture of Nyaneka-nkhumbi; in particular, we will identify this knowledge in the traps made by this ethnic group. In this context traps are the artifacts often used by hunters to catch prey such as birds and / or various small animals. This paper is based on studies by various authors such as Gerdes, D'Ambrosio, Palhares and many others who are apologists of ethnomathematical studies. The procedures used to collect the data were: direct observation, informal interviews with hunters and former hunters, photographic records and field notes.

Diverse forms of representation used in the past and in the present by American indigenous peoples of many cultures are pointed out and analyzed in this paper. Some representations commonly analyzed as mere decorative motifs, are considered now as parts of a very precise and logical system of expressing and storing cosmological knowledge. Following that aim, multiple indigenous peoples turn to structures and mathematical forms of representation that could allow them to save data related to astronomy, calendars, harvest knowledge, censuses, among others. The resulting myths were integrated as part of their cosmogonic space. Outstanding representations of Stepped fretworks have passed from one culture to another, from Alaska to the Patagonia, in models of representation apparently widespread during centuries before the arrival of the Spaniards to America. The reason why these models are generalized through very long distances and very diverse peoples, is explained by the relation of all their knowledge to the creation of weaved textiles. As far as we know, a majority of the indigenous peoples weaved cotton or other fibers, which created a simple and solid base to save information precisely once the data collection process had started. The examples for computations made using cords, as in the Quipus in South America, are multiple. The art of weaving related to mathematical knowledge has allowed for the saving of varied information related to calendars, astronomy, culture, and others, that were later expressed in a vision of the world and, finally, in a cosmogony.

Comparing and networking theoretical frameworks is a topic treated by various authors and discussion forums, as a source of enrichment and starting point for advancing theory development. In this paper we present a study about this topic, establishing a relationship between ethnomathematics and the onto-semiotic approach (OSA) to mathematical knowledge and instruction. Ethnomathematics is a research program with a consolidated international presence that proposes an expanded vision of mathematics and mathematics education. The OSA is a theoretical framework, which is widespread and valued in the community of scholars in mathematics education. A reciprocal glance of both theoretical frameworks can provide mutual enrichment. Specifically the basic features and the paradigmatic issues addressed by both theoretical frameworks are presented and compared. Some commonalities and complementarities are also identified.

Wheatley and Frieze‟s book, Walk Out Walk On, provides the conceptual framework for an examination of Project MACIMISE, a National Science Foundation funded project that focused on the languages and cultural practices of nine Pacific islands and the state of Hawai„i. MACIMISE, pronounced as if spelled „maximize‟, is a 5-year Project. The Project‟s task is the development of elementary school mathematics curriculum units sensitive to local mathematical thought and experience. There were twenty-one participants (who call themselves the Macimisers) in the Project. The participants were educated in ethnographic and anthropological research strategies to enable them to retrieve/uncover cultural practices extant in the communities where they lived. This academics work was accomplished partially via distance learning when the participants were registered in advanced degree programs at the University of Hawai„i—Mānoa. In this paper, the Project is analyzed in terms of the concepts (scaling across, start anywhere—follow it everywhere, intervention to friendship, the art of hosting and the use of circle) advanced by Wheatley and Frieze.

This paper shows the development and results of a research developed with primary teachers working in indigenous environments in Costa Rica. It has a double object of study: the Ethnomathematics of three ethnic groups and the teacher training within a model intercultural based on Ethnomathematics. The theoretical foundations of the study are based on Ethnomathematics, enculturation and learning by micro-projects. Also has some empirical foundations that consist of four sub-studies: an ethnological diagnose made with three ethnic groups in Costa Rica, as well as three diagnoses in which participated 30 professionals that work on issues of education and culture, 69 elementary school teachers working in indigenous contexts, and 11 professionals specialist on pedagogical issues. With the results of the four sub-studies, has been proposed a course of teacher training based on Indigenous Ethnomathematics. The experience of implementation of the course was evaluated through a participatory ethnographic method, and also by the content analysis generated by the micro-project and the portfolio developed and based on Indigenous Ethnomathematics. Its evaluation led to propose a teacher training model based on Indigenous Ethnomathematics. It is proposed a discussion about cultural relevance and about the ways to provide teacher training pertinent to contexts, as well as about the relationship between cultural knowledge and school mathematical knowledge.

This paper narrates facts and consequences of a course focused on Interculturality and Ethnomathematics developed in a Venezuelan Pedagogical University. The reflection is centered on Ethnomathematics and the development of "curricular ethnomathematics micro-projects". The form of the group work was cooperative, with each team chosing a culture and an important cultural symbol in it, to allow for the reconstruction of this cultural practice in the classroom. This was materialized through a forum and supplemented with visits to schools in indigenous populations, which have been developing the Intercultural Bilingual Education Program (PEIB) of that University. During the course various aspects associated with the idea of mathematics in different cultures were developed, which allowed us to initiate research on Teacher Education and Ethnomathematics, emphasizing the identification, evaluation and description of mathematical practices that take place outside school (mathematics in the design and construction of: blowguns, indigenous baskets, musical instruments, houses). In the medium-term (5 years), the products emerging because of the impulse of the course are: Ethnomathematics training courses, MA and PhD postgraduate work, research projects and numerous publications in conferences and scientific journals.

Māori mathematical practices were excluded from schooling in Aotearoa/New Zealand for over 150 years as a result of explicit policies precluding the use of the Indigenous language and culture. As a consequence of the range of assimilationist policies, by the 1970s, the Māori language was considered endangered. In response to the perilous state of the language, Māori communities set up their own schools, initially outside of the state system, to support the revitalisation of the language and culture. However, the reintroduction of the cultural knowledge in areas such as mathematics has not matched language revitalisation efforts. Many original ethnomathematical practices are no longer in general use in the Indigenous community and the practices valued by the European majority remain the norm in the state-mandated school curricula. Spatial orientation is an example of the mathematics curricula content that is based on Western mathematical perspectives. To provide a Māori perspective, this paper draws on interviews with elders and historical data to examine Māori spatial orientation terms and the spatial frames of references that they are derived from. Students in a Māori-medium school were tested on their understanding of this traditional knowledge. As a consequence, a series of learning activities based on Māori spatial orientation concepts were developed and trialled. The outcomes of these learning activities showed some increased understandings about Māori spatial orientation concepts.

Math knowledge and skills are not found exclusively in schools. The Nyaneka-nkhumbi ethnic group is one of the linguistic groups in southern Angola, with rich and unknown ethnomatematics knowledge and know-how. With this study we intend to contribute to the research in ethnomathematics and to the development and dissemination of mathematics knowledge 'hidden' in the culture of Nyaneka-nkhumbi; in particular, we will identify this knowledge in the traps made by this ethnic group. In this context traps are the artifacts often used by hunters to catch prey such as birds and / or various small animals. This paper is based on studies by various authors such as Gerdes, D'Ambrosio, Palhares and many others who are apologists of ethnomathematical studies. The procedures used to collect the data were: direct observation, informal interviews with hunters and former hunters, photographic records and field notes.

The Brazilian prison system can be understood metaphorically as a large pressure cooker. It is eminent danger of an explosion. In this context there is a constant tension in relations between its main subjects: the arrested and jailers. The school existing in this space assumes, among its many functions, the role of mediator of tensions and of space of freedom. In this paper we present an excerpt of ongoing research which aims to investigate and understand the knowledge built / acquired by young people and adults in the context of deprivation and restriction of freedom, in search of a dialogue between this feasibility knowledge with math classes in a transdisciplinary perspective. The field research was conducted in a state school located within a male Prison Unit of the State of Rio de Janeiro, motivated by the teaching practice of the author in this school. In this sense, we have adopted an ethnographic approach that has allowed us to immerse ourselves in the context of the arrested students. In this work the following aspects of the research will be presented along with a brief introduction: the definition of the theme and the objectives (general and specific), the methodology, the theoretical framework, the relevance and some notes for possible outcomes.

This position paper argues that ethnomathematics, while not a curriculum (nor a discipline, that is why D’Ambrosio call it a program), is a confluence of ideas and concepts that may lead to different mathematics engagement of our students. Just as water swirls together from multiple sources, so does our cognition when many ideas are brought together, leading to a cognitive confluence. The definition of ethnomathematics, as coined by org.siir.client.entities.Xref@64f8a745sets the foundation. Scholarship from other disciplines is included, as these areas motivate new thinking, new questions, and new perspectives.

This paper aims to recover, develop, explore and expand the geometric practices of women of the ethnic group Nyaneka-nkhumbi from southwestern Angola as evidenced in the construction of baskets of different dimensions. We show the mathematics ‘hidden’ in the process by identifying several of these mathematics concepts and reflecting on their use in the classroom. We present the results of the research developed in connection with the ethnic group using participant observation and interviews, supported by photographic records and written notes. Women of this ethnic group continue to build and use baskets which involve many interesting geometry concepts for use in classroom context. We refer, for example, to the notion of volume, conic (truncated) and cylindrical forms, spirals, proportions, geometric figures, patterns, plane transformations and friezes. The remarkable mathematical practices in the baskets are characterized by much unknown ethnomathematical knowledge that constitutes a challenge for future studies.

The majority of research in mathematics education in Brazil focuses on aspects related to urban schools. Although there has been a discussion for decades about the need for a curriculum in schools located in rural areas that considers the importance of the local culture and the concept of context-based knowledge, there is still precarious teaching of mathematics in these schools. This paper discusses research data that addresses the teaching of mathematics in rural schools located in Pernambuco, Brazil. The method used is a qualitative research method based on interviews and discourse analysis. We focus on the discourse of teachers who participated in the empirical study by analyzing their views on rural education, the conceptualization of resources, and on their performance in mathematics teaching. Although most teachers positively evaluate their performance when they teach mathematics, they do not identify differences between teaching in the city and in rural areas. Generally, the teachers were unaware of the field schools' specificities. The results lead us to reflect on the possibilities of teaching mathematics to empower rural communities.

Mathematical thoughts are in interaction with culture and they, together, form an indivisible whole. Therefore, Ethnomathematics both helps us understanding the nature of mathematics and contributes to understanding of one’s self as well as the other people sharing the same planet. Ethnomathematics reflects the studies of mathematical thoughts of multicultural or traditional societies. Mathematical thought is an approach in which people try to find quick and systematic solutions to a problem in many ways. The role of Ethnomathematics, which studies mathematical thoughts, cannot be ignored in a historical-cultural context. This paper examines some reflections of Ethnomathematics in Anatolian culture through geometry perception in engineering field, carpet, rug motifs and intelligence games.

The majority of research in mathematics education in Brazil focuses on aspects related to urban schools. Although there has been a discussion for decades about the need for a curriculum in schools located in rural areas that considers the importance of the local culture and the concept of context-based knowledge, there is still precarious teaching of mathematics in these schools. This paper discusses research data that addresses the teaching of mathematics in rural schools located in Pernambuco, Brazil. The method used is a qualitative research method based on interviews and discourse analysis. We focus on the discourse of teachers who participated in the empirical study by analyzing their views on rural education, the conceptualization of resources, and on their performance in mathematics teaching. Although most teachers positively evaluate their performance when they teach mathematics, they do not identify differences between teaching in the city and in rural areas. Generally, the teachers were unaware of the field schools' specificities. The results lead us to reflect on the possibilities of teaching mathematics to empower rural communities.

In this paper we give special emphasis to the mathematic concepts involved in the building of the oiler and we seek to contribute to the development, dissemination and preservation of one of the oldest traditional arts and crafts - the coppersmith – of one region of northern Portugal - the region of Trás-os-Montes e Alto Douro. We developed a study on knowledge (ethno) mathematics involved in building an oiler as elements that can promote a possible relationship with mathematics and the possible contribution to elementary mathematics education. As a result, we identified geometric properties (in Portugal) and that will lead students to discover mathematics through the analysis of this cultural artifact. The study was developed through a literature review and fieldwork with participant observation through the interviews to coppersmiths in this northeastern region of Portugal.

This paper presents a research which empirical data were derived from a carnival floats creator – the carnivalesque. Was aimed to develop a comparative analysis between the creation process of floats, mathematical modelling and ethnomathematics. Modelling is the art of elaborating of a model; and ethnomathematics, of knowing, explaining and understanding the doings and knowledges of people in their cultural contexts. Methodological procedures were divided in two steps: empirical apprehension and data signification, in the light of the theory. Apprehension, ethnography of the place and the person who creates floats to a carnival parade. The dataset was collected in loco: Documents, recordings, pictures, videos and the interview with de carnivalesque. Signification in organizing, classifying and analysis of data, comparing to the processes of modelling under a etnomathematical perspective. The result showed that the carnivalesque creates models of the floats in his mind, derived from perceptions and apprehensions from his environment. That, from comprehension and understanding, trespasses in an external model – ensemble of sub models represented in drawings and sketches that, when produced, shall illustrate the carnival parade. Analysis of the carvalesque’s processes involved in the creation of floats suggests that people who create something, in the diverse areas of knowledge, pervades the same steps prescribed in modelling under a ethnomathematics perspective: a problem-situation catalyzes the endeavor to a solution, attempting to perceive and apprehend data derived from diverse sources, comprehend and explain, formulating a model, and signify and explicitate, solving and validating the model, engendering new knowledges forward the needs implied by the environment, whether economical, social, historic or cultural needs.

This paper is part of a research that focuses on Mathematics education in the context of Education in Solidarity Economy as a possibility of teaching and learning Mathematics to meet specific demands of mathematical knowledge inherent to the implementation of Entrepreneurships in Solidarity Economy (ESE). Within this context, it investigates, from a theoretical point of view, possible approaches between Solidarity Economy and Mathematics Education. The research has pointed out the Ethnomathematics program as a possibility of approaching Mathematics education in the context of Solidarity Economy, defending the idea of an Ethnomathematics for Enterprises in Solidarity Economy.

Our intention in this paper is to articulate some of the characteristics of Ethnomathematics Education and of Inclusive Education. For this, we present a myth known as The Bed of Procrustes, as well as perpetrate a possible approach to this metaphor with the practices adopted by the current school system of basic education. In methodological terms, we anchored our considerations in a theoretical/bibliographical study, where we highlight the concepts of integration and inclusion, which when reviewed, appear irreconcilable due the prominence of the inclusion how much its capacity to propose substantial changes in the society/school, so that all people with different biological-physical-sensorial conditions may have, well as the others people, their differences respected and valued by all in these environments. Thence, using precepts of Ethnomathematics Education, we suggest look at the Education/Mathematics Education/InclusiveEducation starting at an ethic which have as fundamental principles the respect, the solidarity, the cooperation, the symmetrical dialogue etc., what notably remits to rethink the teachers formation.

This paper aims to discuss the meaning of the description of a practice within Ethnomathematics. For that, we start revealing what a description is by using a discussion by Ostrower (1983) on different perspectives about the relation between a painter and his painting. In sequence, we take into consideration a conception of description that does not separate describer from a described practice. Within this idea, we point out what the Ethnomathematics researcher’s role within this vision would be, as well as a methodological path based on Alangui’s Mutual Interrogation methodology (2006).

The paper presents the results of a research Project, which aims to analyzing the truths about mathematics education that circulate in Teacher Education courses (of the Southernmost state of Brazil) and how cultural difference operates in the constitution of such truths. The empirical data are composed by interviews done with teachers who work in the first years of schooling and have completed their Teacher Education courses. Using Ethnomathematics theoretical tools, linked to Foucault and Later Wittgenstein‟s theorizations, data analysis showed that Teacher Education courses enable students to reflect on their past experiences with school mathematics; the Teacher Education courses allow the appropriation of language games associated with diverse mathematics, instituted by different forms of life; and the boys are positioned as those who perform better in school mathematics and the girls as those who are obedient and able to follow rules in mathematics classes.

This work aims to present a discussion, consequential from a masters degree research, of how the use of Ethnomathematics in continuing education programs for mathematics' teachers implies inthe production osubjectivities. Therefore, two Masters dissertations that studied the Ethnomathematics in teachers educationline were analyzed: one from a private institution and one produced at a publicinstitution,disclosed in theyears 2006 and 2009, respectively, in Porto Alegre-RS. The analysis has focused on investigating, according to Foucault's discourse theories, through the discursive regularities and power/knowledgerelations, how different modes of subjectivity are produced in the researching subjects. By using twoguiding principles of visibility: Ethnomathematics as art and technique to explain and understand within thedifferent cultures andthe relation of it with the teacher`s training and the Ethnomathematics as a mechanism of government, itfollows that the discussions of the analyzed papers operate as governance mechanisms able to qualify and lead conducts, producing this way, unique modes of subjectivity.

The paper presents the results of a research Project, which aims to analyzing the truths about mathematics education that circulate in Teacher Education courses (of the Southernmost state of Brazil) and how cultural difference operates in the constitution of such truths. The empirical data are composed by interviews done with teachers who work in the first years of schooling and have completed their Teacher Education courses. Using Ethnomathematics theoretical tools, linked to Foucault and Later Wittgenstein"s theorizations, data analysis showed that Teacher Education courses enable students to reflect on their past experiences with school mathematics; the Teacher Education courses allow the appropriation of language games associated with diverse mathematics, instituted by different forms of life; and the boys are positioned as those who perform better in school mathematics and the girls as those who are obedient and able to follow rules in mathematics classes