ࡱ>  @ 5bjbj)) .KzKz &$ $ $ :^ ,  9998>99\ G{2:2:"T:T:T:/;/;/;zzzzzzz$|R!>z q/;/;qqz T:T:{xxxq T: T:zxqzxx xT:&: `9xtxz{0G{x_w_x _ x/;=N x[ f5 /;/;/;zz $"Dx^ "MATHEMATICS EDUCATION AND THE BRAZILIAN LANDLESS MOVEMENT: THREE DIFFERENT MATHEMATICS IN THE CONTEXT OF THE STRUGGLE FOR SOCIAL JUSTICE Gelsa Knijnik Gelsak(at)unisinos.br ABSTRACT This paper aims to discuss issues related to mathematics education and social justice in our Empire times taking as an empirical base for the discussion the work developed by the author in the last 16 years with the Brazilian Landless Movement. The paper analyzes this peasant social movement, focusing on the political role it is assuming in what Hardt and Negri (2001) called Empire, and more specifically, the educational work it is improving in the country. It also presents the theoretical background that informs the authors ethnomathematics thinking based on a Post-Modern perspective in its connections with Post-Structuralist theorizations, more specifically, those associated with the work of Michel Foucault. Moreover, using the work of the Second Wittgenstein (which corresponds to his book Philosophical Investigations) three different mathematics: are shown: a mathematics produced by a form of life associated to MST peasants, another one produced by a form of life of the urban sawmill men and a third, produced by a form of life found in the Western Eurocentric school, even considering that all of them have family resemblances. Introduction The well-known Brazilian educator Paulo Freire, in the language of his time, stated, in his book Pedagogy of the Oppressed, that education had a political dimension. Later, he himself reformulated that first statement he had made, saying that education is political. Freirian thinking, in particular its emphasis on the politicity of education and the central role given to culture in the constitution of educational processes, in its time had a major impact on the peripheral countries and also on the central ones, an impact that, possibly, gradually also reached the area of Mathematics Education. Decades after Freires initial ideas, we are examining the politicity of mathematics education from other theoretical perspectives, opening the possibility of assigning new meanings to this key issue. Such theories were aligned with the new world economic, social and political designs that shape these times of Empire, marked by large numbers of the poor crossing borders, producing culturally plural and socially even more unequal scenarios. Today it seems that the aims of Enlightenment thinkers have failed. In fact, as pointed out by twentieth-century authors like Sarup (1996:94), the ideas of linear progress, absolute truths, the rational planning of ideal social orders and the standardizations of knowledge and production embraced by Modernity, the extraordinary intellectual effort produced by its project in developing objective science, universal morality and autonomous art and its beliefs in justice and possibility of happiness of human have been cruelly shattered. More than ever, the majority of the worlds population is living in subhuman conditions, wars are being waged everywhere and nature is being destroyed all over the planet. Even so, or precisely because of this, our hopes of a more just and egalitarian world still remain. The old project of Modernity is gone but new ones are being built. It can be considered that at least punctually educators can contribute to their implementation. In fact, our curricular practices as well as our work as researchers are not neutral, since they have the potentialities to favor or not the inclusion of those who are the other, the different: unequally different, since their cultural differences such as gender, social class, race/ethnicity, sexuality and ageness are those which have less value in our society. There are many questions that we have to ask ourselves as mathematics educators committed to understanding and resisting the social injustice of these Empire times. This critical exercise cannot avoid problematizing the position assigned to mathematics as well as to mathematics education by Western society. Following Dunne and Johnston, our analytic exercise must consider (and deconstruct) the privilege that a access to mathematics confers on its chosen few, to understand its gate-keeping role in relation to further education and future careers and consider this in the production and reproduction of hierarchical gender (and class and race) relations. What constitutes mathematics, what counts as valued mathematical knowledge, how things came to be this way and how they are sustained are critical questions (Dunne and Johnston, 1994: 227). This paper aims to problematize some of these questions, taking as an empirical base for the discussion the work I have been doing in the last 16 years with the Brazilian Landless Movement. An analysis of this peasant social movement, focused on the political role it is assuming in what Hardt and Negri (2001) called Empire, and more specifically, the educational work it is improving in the country are the subject of the next section of the paper. The following one will outline the theoretical background that informs the Ethnomathematics thought which I have been developing with the Landless peasants. The third section presents empirical data that show three different mathematics: a mathematics produced by a form of life associated to MST peasants, another one produced by a form of life of the urban sawmill men and a third, produced by a form of life found in the Western Eurocentric school, even considering that all of them have family resemblances. Brazilian Landless Movement in the time of the Empire Michel Hardt and Antonio Negri (2001) begin their well-known book Empire, saying that it is materializing before our very eyes (...) [since] we have witnessed an irresistible and irreversible globalization of economic and cultural exchanges (Ibidem:11) which instituted a global order, a new logic and structure of rule in short, a new form of sovereignty. Empire is the political object that effectively regulates these global exchanges, the sovereign power that governs the world. (Ibidem:11). Sovereign power, according to the authors, ultimately scrambled the spatial divisions between the First, Second and Third World, since (w)e continually find the First World in the Third, the Third in the First, and the Second almost nowhere at all(...). In the postmodernization of the global economy, the creation of wealth tends even more toward what we will call biopolitical production, the production of social life itself, in which the economic, the political and the cultural increasingly overlap and invest one another (Ibidem:13). This new imperial order is taken as a background to this paper, considering the importance of attempting to understand adult education as a field of knowledge as well as the contemporary social movements and their educational processes within this new world configuration characterized by the absence of boundaries, in which the rule of the Empire operates on all registers of the social order, extending down to the depths of the social world (Ibidem:15). Hardt and Negri, on examining the potentials for constructing alternatives that counter the imperial power, highlight the role taken on by the struggles of the proletariat a social subject that they consider beyond the industrial working class constituted, in fact, by all those exploited and subject to capitalist domination (Ibidem:72) but who do not make up a homogeneous or undifferentiated unit: They further consider that there are new forms of struggle through which this new proletariat expresses its desires and need (Ibidem:72), struggles which must be identified, not as the appearance of a new cycle of internationalist struggles, but rather [as] the emergence of a new quality of social movements (Ibidem:74). This new quality is expressed by fundamentally new characteristics of the struggles of the social movements: first, each struggle, though firmly rooted local conditions, leaps immediately to the global level and attacks the imperial constitution in its generality. Second, all the struggles destroy the traditional distinction between economic and political struggles. The struggles are at once economic, political and cultural and hence they are biopolitical struggles, struggles over the form of life. They are constituent struggles, creating new public spaces and new forms of community. (Ibidem: 56). Among the many struggles of social movements that could be analyzed in their relationship with education, especially mathematics education, we can consider the struggles for land reform carried out by the Brazilian Landless Movement, which is well known on the international scene, mainly but not only because of new aspects that have been instituted in the sphere of education. In fact, as written in the official MSTs website (http://www.mstbrazil.org ): Landless Movement, in Portuguese, Movimento Sem Terra (MST) is the largest social movement in Latin America with an estimated 1.5 million landless members organized in 23 out of 27 states. The Landless movement carries out long-overdue land reform in a country where less than 3% of the population owns two-thirds of the land on which crops could be grown. Since 1985, the MST has occupied unused land where they have established cooperative farms, constructed houses, schools for children and adults and clinics, promoted indigenous cultures and a healthy and sustainable environment and gender equality. The MST has won land titles for more than 250,000 families in 1,600 settlements as a result of MST actions, and 200,000 encamped families currently await government recognition. Land occupations are rooted in the Brazilian Constitution, which says land that remains unproductive should be used for a larger social function. The paths followed by the Brazilian Landless Movement in the 22 years of its history, and especially the educational processes that are being produced there, have been the subject of discussion in many academic forums and in publications involving this theme. Thus, in this paper, I am interested in highlighting a few more recent strategies that the MST has implemented, and which may be creating fissures in the smooth space of imperial sovereignty, in which there is no place of power it is both everywhere and nowhere (Hardt e Negri, 2001:210) and there is progressively less distinction between inside and outside (ibidem:206). It is in this space crisscrossed by so many fault lines that it only appears as a continuous, uniform space (ibidem: 210) that one can conjecture on the potentials of social movement struggles to subvert the imperial order. The MST strategy of occupying large unproductive rural properties as a way of pressuring the State to carry out Land Reform, which marked the initial years of struggle, was gradually expanded to include the occupation of other spaces, such as public buildings, organization of regional and national marches, and setting up camps at the side of main highways. Thus, MST has aimed at occupying many and all possible territories. Moving continuously along the roads, staying for short periods in small towns and large cities constitute a strategy which counters the Empires need to restrict and isolate the spatial movements of the multitude to stop them from gaining political legitimacy (ibidem: 422). While the Empire seeks to isolate, divide and segregate, MST defined fighting strategies that, in a sense, undermine this segregationist operation. Its slogan: Land Reform, everybodys struggle indicates that if it is everybodys struggle, divisions must be smoothed down, more and more joint actions are needed. It is in this sense that one may understand two of the strategies developed by MST in recent years. The first concerns the implementation of MST actions organized with other popular social movements which are attuned with their position of repudiating the different social repercussions of neoliberal policies, as well as its participation in large national and international demonstrations of opposition to the imperial order. The second strategy that has been acted upon by MST, with a less occasional character, refers to its integration to the Via Campesina, an organization that brings together the main rural social movements in the world, to fight against neoliberalism and defend the peasant life and culture.(JST, 2004). In brief, what is presented here points at new strategies that have been more recently implemented by the Movement, strategies that can be considered as having the potential to undermine the Imperial order, contributing to the constitution of a society in which the basis of power is defined by the expression of the needs of all (Hardt e Negri, 2001:434). But there are other kinds of strategies which were implemented since the beginning of Landless Movement struggle and can be seen as reinforcing those above mentioned. I am referring to the work coordinated by its Educational Sector, in providing education to their members. First of all it is important to highlight that the educational process, which has been developed by the MST over its 22-year history must be understood beyond schooling, since each Landless subject educates her/himself through her/his participation in the everyday life of their communities and also through the wide range of political activities developed by the Movement. This means that the children, youth and adult peasants are educated by the multiple facets of the struggle for land, which produce very specific social identities. Nevertheless, these social identities do not form something compact, uniform, in which hundreds of family from different social strata would ultimately become a unified whole, homogenized by the struggle for land. To look at this social movement with such lenses implies considering that if there is some kind of intention of establishing a Landless identity, this intention is never completely fulfilled. Summing up, the Landless educate themselves in the struggle in the occupations, the marches, in their ways of organizing the settlements, through their cultural artefacts learning the many possible meanings of being landless. But in this educational process there is a sort of rebellion against fixing one social identity. There are many axes such as those of gender, sexuality, race/ethnicity, ageness ( which in their crossovers ultimately shape multiple Landless identities, multiple ways of giving meaning to the struggle for land. This position allows us to say that the peasant culture of the Brazilian Landless Movement in Wittgensteins words, its form of life is marked by difference. The schooling activities developed by the Landless Movement cover Child Education, Elementary and High School Education, Teacher Training Courses and projects of Education of Youths and Adults. As shown on the MST website, the Landless Movement Schooling project involves 1800 schools in camps or settlements (grade 1 to 8), with 160 thousand students and 3900 teachers; 250 educators who work with children up to 6 years; 3000 educators working with 30 thousand peasants of literacy and numeracy projects of Adult Education; and Teacher Training Courses implemented in partnership with public and private universities around the country. This schooling project, according to one of the Landless Movement official documents, sees the need for two articulated struggles: to extend the right to education and schooling in the rural area; and to construct a school that is in the rural area, but that also belongs to the rural area: a school that is politically and pedagogically connected to the history, culture, social and human causes of the subjects of the rural area (...) (Kolling et al., 2002:19). The movement has dedicated itself to conceiving the schooling of its children, youths and adults paying attention to these two struggles. In particular, such struggles are providing the guidelines for its adult mathematics education. This means that landless educators considered peasant culture a key issue also for those teaching and learnings processes related to mathematics. But they explicitly mention that this valorization cannot deny the relevance of acquiring mathematical tools connected to academic mathematics that can improve the use of new technologies for managing the production in rural areas and can allow the learners to go further in their schooling trajectory. As will be shown in the next section these ideas are strongly connected to the field of Ethnomathematics, more specifically to the ethnomathematics thinking I have been developing, which is rooted in the Landless culture and in my experience in working with the peasants living in southern Brazil. The theoretical basis of an ethnomathematics thought The basis of the ethnomathematics thought I have been elaborating is based on a Post-Modern perspective in its connections with Post-Structuralist theorizations, more specifically, those associated with the work of Michel Foucault. According to such theorizations, I have considered that Ethnomathematics may consist of a toolbox which allows analyzing: a) the Eurocentric discourses that institute academic mathematics and school mathematics; b) the effects of truth produced by the discourses of academic mathematics and school mathematics; c) issues of difference in mathematics education, considering the centrality of culture and the power relations that institute it (Knijnik, 2006). Operating with a toolbox that has this configuration is attuned with the positions of authors like Santos (1995), who argue about the need to cast suspicion on the education practiced today in the Western world, which he described as centrally Eurocentric: The cultural map underlying the modern educational systems is, cartographically speaking, a map with a Mercator projection. The central characteristic of this projection is that it places the European continent at the center of the map, inflating its size to the detriment of the other continents. In symbolical terms, the modern educational map is a Mercator map. The Eurocentric culture occupies almost all of the size of the map and, only marginally and always taking the central space into account, are the other cultures drawn () This is the map of the Imperial culturalism of the West. In this map the conflict between cultures either does not appear completely, or it appears as a conflict solved by the superiority of Western culture in relation to the other cultures.(ibidem:26). We are facing an issue that concerns the politics of knowledge, in the dispute around the definition of which knowledges are included and which excluded in the schooling processes. This dispute is marked by power-knowledge relations, which ultimately legitimate and are the legitimizer of some discourses, which interdict others, precisely those that are about the knowledges, the rationalities, the values, the beliefs of cultural groups we place in the position of the others. One should then ask how a single rationality among other rationalities the rules by which individuals and cultures deal with space, time and quantification processes all that which Western civilization associates with the notion of mathematics became a truth, the only truth that could be accepted as mathematics in the school curriculum. What is at stake here is to problematize the sovereignty of the Modern rationality, which scorns all other rationalities associated to other forms of life; the existence of a single mathematics the official one with its Eurocentric bias and its rules marked by abstraction and formalism. To be more precise, we must say that this official mathematics the academic one is composed by a set of branches, including all those associated with so-called pure mathematics and applied mathematics. The so-called school mathematics the traditional set of knowledges taught at school inherits at least part of the formal and abstract grammar that constitutes academic mathematics, through pedagogical recontextualized processes, in Bernsteins words. In summary, it can be said that all these different mathematics offers a dream of order, regularity, repeatability and control () and with it the idea of a pure, disembodied reason (Rotman, 1993:194). These issues lead us to Ludwig Wittgensteins ideas presented in his book Philosophical Investigations (2004) in which he criticized not only his earlier work (presented in Tractatus) but also the whole tradition to which it belongs (Glock, 1996:25), the foundationist schools, and dwells at length upon knowing as a process in mathematics (Ernest, 1991:31). In his remarkable book The Philosophy of Mathematics Education Paul Ernest (1991) examines philosophical schools and their contributions to the conceptualization of (academic) mathematics. In his analysis about the conventionalist view of mathematics, which considered that mathematical knowledge and truth are based on linguistic convention (ibidem: 30) the author refers to the work of Wittgenstein, saying that the philosopher proposes that the logical necessity of mathematical (and logical) knowledge rests on linguistic conventions, embedded in our social linguistic practices (ibidem: 32). In shaping a new philosophy of mathematics social constructivism Ernest refers to Wittgenstein as one of the philosophers who considers knowledge not only as a product, giving great weight to knowing and the development of knowledge (ibidem: 90). He considers that social constructivism employs a conventionalist justification for mathematical knowledge (ibidem: 64), assuming that the basis of mathematical knowledge is linguistic knowledge, conventions and rules, and language is a social construction (ibidem:42). Ernest argues that his philosophical perspective assumes a unique natural language showing that an alternative (i.e. different) mathematics could result (ibidem:64) as a consequence of this position. Mentioning the work of Alan Bishop as an evidence of different mathematics (ibidem: 67), Ernest will say that such evidence of cultural relativism strengthens rather than weakens the case in favour of social constructivism (ibidem: 64). These ideas are strongly connected to the ethnomathematics thinking presented in this paper based on the work of the Second Wittgenstein. In fact, viewing mathematics not as a body of truths about abstract entities, but as part of human practice (Glock, 1996: 24), the philosophers work gives us tools for thinking about rationality as forged from social practices of a form of life, which implies to consider it as invention, as construction (Cond, 2004: 29). Moreover, with the support of the philosophers ideas and using the expressions that he coined one can admit the existence of distinct mathematics distinct ethnomathematics, in DAmbrosio words. The basis of this statement can be found in the argument that these different mathematics in Wittgensteins words, different language games are produced by different forms of life, a term conceived by the Second Wittgenstein as stress[ing] the intertwining of culture, world-view and language (Glock, 1996:124), as patterns in the weave of our life Glock (1996: 129)). In Wittgensteins late work, especially in the new conception of language presented by the philosopher, Cond (1998, 2004) argues about the crucial role of the notion of use: In such work, use is directly connected to the concept of meaning () the meaning is determined by the use we make of the words in our ordinary language. () The meaning of a word is given based on the use we make of it in different situations and contexts. () the meaning is determined by the use. (ibidem: 47) It is in this sense that this notion of use, according to Wittgenstein, is considered pragmatic, no essentialist. Meaning is determined by the use of words and such a use respects rules, which are themselves produced in social practices, constituting language games. As pointed out by Cond (1998:91) the notion of language games involves not only expressions, but also the activities with which these expressions are linked. Language games are produced based on sets of rules (that are rooted in social practices), each of them constituting a specific grammar. So, the grammar that marks each language game is itself a social institution. Moreover, authors like Spaniol (apud Cond, 1998:110) argue that the grammar constitutes the logic itself, the grammar is the logic. () It is impossible to analyze the logic without considering the language. From what was briefly explained here based on the work of the Second Wittgenstein and some of his interpreters (like Cond and Glock before mentioned), it follows that different forms of life produce different language games, each of them marked by a specific grammar and such grammar, as a set of rules, constitutes the specific logic. This rationale drives us to admit that there is more than a single language game: there are different language games. Is there some kind of relationship between them? If the answer is positive, how does it operate? The response to these questions is given by the Second Wittgenstein through the notion of family resemblances. The philosopher would say (as shown in aphorisms 66 and 67 of Philosophical Investigations) that language games form a complicated network of similarities overlapping and criss-crossing: sometimes overall similarities, sometimes similarities of detail (Wittgenstein, 2004: 320) and adds: I can think of no better expression to characterize these similarities than family resemblances; for the various resemblances between member of a family? Build, features, color of eyes, gait, temperament, etc. etc overlap and criss-cross in the same way and I shall say: games form a family. Operating with the ideas of the Second Wittgenstein in the context of the struggle for land in the south of Brazil, leads us to assume the existence of three different mathematics: a mathematics produced by a form of life associated to MST peasants, another one produced by a form of life of the urban sawmill men and a third, produced by a form of life found in the Western Eurocentric school, even considering that all of them have family resemblances. Landless language game of cubagem of wood and two others mathematics Cubagem of wood (in Portuguese Cubagem da madeira) to calculate how many cubics there are in a truck load is a common practice in the Landless culture. The peasants perform it when it is necessary to build houses or animal shelters in camps and settlements and to purchase or sell planks, i.e., in our negotiations with the sawmill men, as said one MST member. Throughout my work with MST groups I have realized the importance they give to such language games, produced by their form of life. In teacher education courses and at settlement schools, particularly, I have found great interest in discussing that practice, constituted by a specific grammar, a specific set of rules. On one of the occasions when I was working with cubagem of wood in a Teacher Education Course class with lay teachers, Edinei, a student who was then living in a camp, explained about how important it was for his community to learn about cubagem of wood: My sponsor took an interest in this question of cubagem of wood. So I also didnt know about it. So, I said that Id be coming here and bring him the answer. Because he was working like this, as a workman, they [he and his fellows] logged and sold the wood, and sold it to the sawmill, in this case to the sawmill owner () When one took those meters there to sell, one got so much. When the fellow who worked as a workman himself bought the wood that had been sawed, there were many more cubic meters. So, how could this be? During my work with that group of students I found that Edineis sponsors question was shared by many in the class. They were expecting that I help them to go further in learning about the grammar that marks cubagem of wood language game. But it was expected that I also assume another role. In consonance with the Sector of Education pedagogical guidelines (as discussed in the previous section) they aimed to acquire the school mathematics knowledge the one called by them book mathematics. Avoiding a nave perspective, they were aware of the social importance of such a language game and the need to learn its specific grammar as part of their struggle to undermine the Empire sovereignty. The starting point of the pedagogical work was a students narrative. Roseli, a municipal teacher at the time, still without a degree like her colleagues, told what she had learned from her father about cubagem of wood. Going to a place where tree trunks lay the group of students took up position around one of them, helped by Helena (who had an electronic calculator), Antonia (who took notes on a sheet of paper) and Nelci and Cleci, (who contributed in issues concerning measurement). With their assistance, Roseli described what would henceforth be called by the group Roselis method of cubagem of wood. Roseli: First one takes hold of this by the middle [of the log] because there it is thicker and here it is thinner [pointing to the ends of the trunk]. Then,around the middle, one has more or less the average,it is the average. Now I take this string and pass it around it. Done. Now I fold it in four, then after I fold it in four I will measure it to see how many centimeters one will have.. Cleci: 37. Roseli: There, the result is 37 centimeters. Now I take these 37 and multiply by itself, multiply by 37. Helena: [using a calculator to perform the multiplication] 37 by 37 gives 1369. Roseli: [talking to her colleague] Write it down, Antonia, so what we will not forget it. Now Im going to measure the length. After this, now, I know that there is 37, so now is when I measure the length. The result was 1meter and 46. Now then, I multiply the length by the number I had before, which came from the small piece of string, which had given 37 times 37:1369. Helena: I make this 1369 by 1 and 46. Roseli: It is all centimeters. One has to do 1369 by 146. Helena: [with the calculator] 1360 times 146 gives ... 199874. Roseli: That is the number one gets. Rosane: 199874 what? Roseli: 199874 cubic [centimeters] of wood. Rosane: It is the same as doing side time side times length. Juarez: She did almost the same thing that Jorge did. She went and measured the trunk diameter,  then she made a square and multiplied by the length. Even considering the theoretical difficulties involved in translating language games, I found it important to express Roselis method using words and the syntax of the school mathematics language game, the one we are more familiar with. I am aware that in doing so some (or maybe most of the) specificities that constitute the Landless form of life which produced the cubagem of wood language game are suppressed. So, it can be said that Roselis method became a hostage of school mathematics language game when it is said that her method basically involves two steps: the first, to identify, by modeling, a tree trunk with a cylinder whose circumference coincides with that of the middle part of the trunk, and the second, identification, also by modeling the cylinder in a quadrangular prism, whose measure on the side is one fourth of the perimeter of the cylinder base. Thus, Roselis Method for cubagem of wood finds, as trunk volume, the volume of the quadrangular prism whose side of the base was obtained by determining the fourth part of a circumference. This, in turn, corresponds to the cylinder base, obtained by modeling from the initially given tree trunk. Roseli explained her method step by step, as she pointed to the different parts of the trunk involved in the process. This narrative triggered the study on cubagem of wood which we developed from then on. During the discussion of Roselis method there were students who immediately related its grammar to the one that constitutes the land cubao language game called by the group Jorges method, which was studied before (Knijnik, 1997). In effect, both grammars have one rule in common: the identification process which associates a cylinder base (in the case of cubagem of wood) or a quadrilateral (in the case of land cubao) with a square. The relationship established by the group between both language games was an interesting pedagogical issue linked to what Wittgenstein called family resemblances. This notion of Wittgenstein can be helpful in understanding another language game which emerged in the pedagogical process. At some point in the discussions, a two-student dialogue produced a shift in the debate about Roselis method. Jorge: The measurement process that I know is almost the same [as Roselis method], except that we measure at the narrow end of the wood. Ildemar: The point is that the right thing would be to do it in the middle. But the purchasers do not want to buy a piece that will fall away, if they want it for square wood or things like that. They will want a piece that goes from here to there [which goes from one end to the other of the log]. Those chips that are produced will only be for burning. According to these students, there were urban sawmill men who did not use the middle of the log as reference, considering only its narrower end, since they were interested in obtaining whole planks. For this purpose a different rule of calculation was introduced, conforming a specific grammar, which leads to a new language game, different from Roselis. The sawmill mens method was mentioned by most of the group as being practiced at sawmills in the urban areas close to their communities. We found that we were dealing with a language game which is produced by a specific form of life, different from that of the Landless peasant. But the pedagogical process was not circumscribed to Roseli method and to the sawmill men one. The book mathematics language game with the specific rules that shape its grammar was also analyzed. Moreover, the family resemblances of these three language games were emphasized. The work involved studying the modeling process of Roselis Method and learning mathematical tools such as relations between a cubic meter and its multiples. In different situations the results of calculating the amount of wood obtained by Roselis method were compared empirically to the volume of the cylinder produced by her method, which would correspond to a better approach to the total quantity of wood of the trunk, reckoning not only the part useful to obtain whole planks. The group also found that the results of Roselis method minimize those obtained using the cylinder volume. The group showed particular interest in learning the formulas of book mathematics connected to the discussion we were holding. In learning how to calculate volumes of the cylinder and rectangular prisms (which also requires knowing how to determine the length of the circumference, the area of a circle ) the group was dealing with the specific grammar which constitutes the language game of Western Eurocentric school mathematics. Bringing those three language games into the mathematics class enabled the group to go further in the appropriation of rules that shape the grammars produced by each form of life. They learned more about the Landless peasant cubagem of wood practiced in their communities. When the family resemblances of those language games were analyzed, the students were able to identify the remnants of wood that were produced by Roselis method, which were even greater when the initial measure of the log circumference was determined at the narrow end, as considered by the sawmill mens method. So, in this case, the wood not used for making planks could be useful for other purposes and therefore, in given situations, it should also be included in the accountancy of their calculations. Summing up, it can be said that learning about different mathematics and their family resemblances allowed the peasant students to broaden not only their mathematical world, but also their ways of seeing the complex social relations involved in different forms of life that produce such different language games. Some closing words I would like to end saying that the issues I attempted to discuss here are no more than provisional, unmarked by hopes for certainty, in the sense given by Stronach and Maclure (1997). I follow them when they say that we must recognize and try to work within the necessary failure of methodologys hope for certainty, and its dream of finding an innocent language in which to represent, without exploiting or distorting, the voices and ways of knowing of its subaltern subjects (ibidem: 4). The ideas I brought to this paper are inspired by this position. In fact, throughout my trajectory as a researcher I have always tried to mobilize all my efforts in order to never forget to problematize my own discourse, since it is necessarily marked by my privileged voice as an intellectual working with subaltern subjects like the Brazilian landless people. As all discourses, it is marked by power-knowledge relations. In my attempts I have been favoured by this social movement, which is very much aware of the risk of exposing themselves to academic research and of being narrated by the others, of being represented by us. The Brazilian Landless Movement peasants chose to take this risk not only because of my good intentions of being vigilant about my role in the work I have been developing with them for all these years. It is also because they see education as one of the central issues of their struggle to undermine the Empire and consider the importance ( at least at this point of their trajectory as a social movement ( to have academics contributing to the construction of their educational schooling processes, which, for them, is a key element for the social justice project they are attempting to build. References: Dunne, M. & Johnston, J. (1994) Research in Gender and Mathematics Education: The Production of Difference. In: Ernest, P. (ed.) Mathematics Education and Philosophy: An International Perspective. London: The Falmer Press. Cond, M. (1998). Wittgenstein: linguagem e mundo. So Paulo: Annablume. Cond, M. (2004). As teias da Razo: Wittgenstein e a crise da racionalidade moderna. Belo Horizonte: Argvmentvm. DAmbrosio, U. (2001). Etnomatemtica: elo entre a tradio e a modernidade. Belo Horizonte: Autntica. Ernest, P. (1991). The Philosophy of Mathematics Education. London: The Falmer Press. Glock, H. (1996). A Wittgenstein Dictionary. Oxford: Blackwell Publishers. Hardt, M. & Negri, A. (2001). Empire. London: Harvard University Press. JST Jornal Sem Terra (2004). Landless Movement Newspaper. Knijnik, G. (2006). Educao matemtica, culturas e conhecimento na luta pela terra. Santa Cruz do Sul: EDUNISC. Knijnik, G. (1997). Politics of Knowledge, Mathematics Education and the peasants' struggle for land. Educational Action Research Journal, v. 5, n. 3. Kolling, E.; Cerioli, P.; Caldart, R. (org.). (2002). Educao do Campo: Identidade e Polticas Pblicas. Braslia, DF: Articulao Nacional por uma Educao do Campo. Coleo Por Uma Educao do Campo, n. 4. Klusener, R. & Knijnik, G. (1986). A prtica de cubao de madeira no meio rural: uma pesquisa na pespectiva da Etnomatemtica. (Porto Alegre, UFRGS, Instituto de Matemtica). Texto datilografado Mattos, M. Nepstad, D. & Vieira, I. (1992). Cartilha sobre mapeamento de rea, cubagem de madeira e inventrio florestal. (Belm do Par, EMBRAPA/ Woods Hole Research Center). Rotman, B. (1993). Ad Infinitum: The ghost in Turings Machine. Stanford: Stanford University Press. Santos, B. (1995). Para uma pedagogia do conflito. In: Silva, L. (org). Novos mapas culturais, novas perspectivas educacionais. Porto Alegre: Sulina. p. 15-34. Sarup, M. (1996). Identity, culture and the postmodern world. Edinburgh: Edinburgh University Press. Stronach, I. & Maclure, M. (1997). Educational research undone. The Postmodern Embrace. Buckingham, Philadelphia: Open University Press. Wittgenstein, L. (2004). Philosophical Investigations. Oxford: Publishers.  Here a point should be highlighted. In their formulations about social movements, which are considered central by Hardt and Negri, as I mentioned previously, it is the new proletariat, defined based on the domination of work by capital and by the exploitation processes associated with it. In this sense, possibly one could say that social movements that articulate around other axes of submission (such as that of ethnicity, gender, sexuality), remain outside the discussions by the authors.  In fact, DAmbrosio (2001) considers that each branch of academic mathematics shapes an ethnomathematics; school mathematics is an ethnomathematics and also the ways in which specific cultural groups like the Brazilian Landless peasant deal with numbers, space, measurement, etc are considered different ethnomathematics.  The terms cbicos and cbicos de madeira are used in the Brazilian rural areas to mean cubic meters of wood. The term metros de madeira, in English, meters of wood, is also used.  It is interesting to observe that on this occasion different from other pedagogical situations involving Landless peasant practices the girl students were conducting the explanation. At that time I considered that this gender issue could be connected to the following fact: The whole group was (previously) asked to interview members of their communities about the cubagem of wood. As I had observed, it was a male bias practice in the Brazilian countryside but the disciplined women did their homework in such a detailed way that they felt more confident than the male students who know it only by practice about it.  The student used the expression trunk diameter to refer to what is considered in the language game of school mathematics the trunk circumference.  Several students referred to the use of Roselis Method in their communities. The so called Roselis method had already been identified in fieldwork previously performed in the south of Brazil (Klsener & Knijnik, 1986) and it was also practiced in the state of Acre, in the north of the country (Mattos, Nepstod & Vieira, 1992).  At that time, some students used the expression square wood to refer to a wooden plank.  Taking into account the remarks made before, concerning the translation issues from one language game to another, it could be said that the sawmill mens method consists of calculating the volume of a quadrangular prism whose height is given by the tree trunk. The quadrangular base, however, different from Roselis method, is obtained by the inscription of a square with a maximum side at the log base, considered as a circle.  The group questioned the possibility of applying what they had studied in the context of both Roselis method and the sawmill mens method to other peasant practices. One of the students mentioned that the rules he learned in those mathematics classes could be used in planning the construction of silos for crop storage, at the time one of the main goals of his comrades to render the settlement economically feasible.     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