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- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices Milton Rosa and Daniel Clark Orey Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ethnomathematics and Modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Exploring Ethnomodelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Ethnomodelling and its Three Approaches of Viewing Cultures. . . . . . . . . . . . . . . . . . . . . . . . 9 Etic: The Global/Outsider Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Emic: The Local/Insider Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Dialogic: The Glocal/Emic-Etic Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Characterizing Ethnomodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Emic and Etic Ethnomodels of the Mangbetu Ivory Sculpture. . . . . . . . . . . . . . . . . . . . . . . . 16 An Etic Ethnomodel of Brazilian Roller Carts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 A Dialogic Ethnomodel of a Local Farmer-Vendor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Relevance of Ethnomodelling in a Mathematics Curriculum. . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Abstract One of the major dilemmas in mathematics education in contemporary society is its hidden bias towards a western orientation in its scholarly and research paradigms. While being mindful of emerging glocalization of science, math- ematics, religion, art, music, and other aspects of a given culture, the use of innovative or what may seem alternative approaches and methodologies is necessary to record historically diverse forms of mathematical knowledge that occur in distinct cultural contexts. With 500 years of colonization of science and mathematics, it seems important at this critical stage of human development M. Rosa () · D. C. Orey Departamento de Educação Matemática, Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brazil e-mail: milrosa@hotmail.com; milton.rosa@ufop.edu.br; oreydeema@gmail.com © Springer Nature Switzerland AG 2019 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_70-1 1
- 2 M. Rosa and D. C. Orey that people look at diverse traditions in the field. So, it is that the authors have come to apply fundamentally different philosophies, modelling techniques, and an ethnomathematical perspective to the mathematics curriculum. It is the linking of mathematics and culture that the authors find appropriate and necessary for a deeper understanding of the development of mathematical knowledge aimed at providing a holistic comprehension of human behavior. It is important to develop an understanding of the role of ethnomathematics and modelling processes in the development of an innovative theoretical basis for ethnomodelling, which uses emic, etic, and dialogic approaches in its investigation process. In this theoretical chapter, the authors demonstrate how ethnomodelling is a pedagogical action for the process of teaching and learning mathematics that challenges the prevailing way of the universality of mathematics and the thinking involved therein. Keywords Cultural groups · Ethnomathematics · Ethnomodelling · Modelling · Pedagogical action Introduction Throughout history, traders, navigators, and explorers studied members from other cultures and shared knowledge often hidden or embedded in religious traditions that often times were mixed with mathematical and scientific practices, behaviors, and customs. This exchange of cultural capital (Cultural capital is the knowledge, experiences, and connections that members of distinct cultural groups acquired through the course of their lives, which enabled them to succeed more than individuals from a less experienced background. It also functions as a social relation within a system of exchange that includes the accumulated sociocultural knowledge that confers power and status to the individuals who possess it (Rosa 2010).) enriched all cultures when their members were engaged in a constant, dynamic, and natural process of evolution and growth through the process of cultural dynamism (Cultural dynamism refers to the exchange of systems of knowledge that facilitate members of distinct cultures to exploit or adapt to the world around them. This cultural dynamic facilitates the incorporation of human invention, which is related to changing the world to create new abilities and institutionalizing these changes that serve as the basis for developing more competencies (Rosa and Orey 2016).). For example, the Greek foundations of European civilization were themselves developed through interaction with the Egyptian civilization (Powell and Franken- stein 1997). One consequence of this recognition is a widespread consensus towards the supremacy of Western scientific and logical systems at the exclusion of many other traditions developed in diverse contexts. In mathematics, as in many other academic subjects, methods of problem-solving and teaching materials are based on the traditions of the written sciences and, with very few exceptions, are defined by Western academia and science. Most examples used in the teaching of mathematics are derived from non-Latino, North American,
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 3 and European contexts. These problem-solving methods mainly rely on the Greek- based European view of mathematics. There is certainly nothing wrong with this, but the authors have found that it is important to highlight how cultures and societies considerably affect the way individuals come to understand and comprehend concepts of their own mathematical ideas, procedures, and practices. According to D’Ambrosio (1999), this interaction is in danger of leaving out a significant amount of knowledge and supports forms of colonization that are subtle and often go unnoticed. By observing this context, D’Ambrosio (2006) demonstrates how the culture of a group results from the fraction of reality that is reachable by its members. However, the multiplicity of and constant interactions between members of distinct cultural groups and their unique cultural contexts, each one with a system of shared experience, history, and knowledge and an equally compatible set of behavior and values, facilitates the development of unique set of cultural dynamics by enabling an expanding familiarity with a rich diversity of humanity. This has created an important need for a field of research that studies phenomena and the application of modelling techniques developed in diverse cultural settings. This cultural perspective is applied to the development of problem-solving techniques, conceptual categories, and structural methods used to elaborate models that represent data to translate mathematical practices by using modelling processes. The authors refer to this process as ethnomodelling (Bassanezi 2002; Rosa and Orey 2010) that is one way in which they can recognize, through their lens of Western mathematical experience, how its foundations differ from the traditional modelling methodologies. The authors’ sources are firmly grounded and rooted in the theoretical basis of ethnomathematics (D’Ambrosio 1985), and they have found that the culturally bound views of mathematical modelling support the assumption that research of culturally bound modelling processes addresses issues of mathematics education by bringing the diverse backgrounds of learners into the mathematics curriculum by connecting it to the local and cultural aspects of the school community to the process of teaching and learning of mathematics. Ethnomathematics and Modelling The authors have seen that many models arising from reality have become the first paths that have provided numerous abstractions of deeper mathematical concepts. Ethnomathematics can use these models taken from reality and modelling as a translation to incorporate the codifications provided by the members of distinct cultural groups in order to understand mathematical ideas and procedures developed in other mathematical systems (D’Ambrosio 1993; Rosa and Orey 2003). For us, mathematical modelling becomes a concrete methodology closer to an ethnomathematics program (D’Ambrosio 1990; Rosa and Orey 2006), which is defined as the intersection between cultural anthropology and mathematics that uti- lizes mathematical modelling to explain, analyze, interpret, and solve real-world and
- 4 M. Rosa and D. C. Orey Mathematical Modelling Ethnomathematics Mathematics Practices Contexts Explain Analyze Interpret Cultural Anthropology Fig. 1 Ethnomathematics as an intersection between three research fields. (Source: Rosa and Orey 2010) daily problems (D’Ambrosio 2000; Rosa 2000). Figure 1 shows ethnomathematics as an intersection between cultural anthropology, mathematics, and mathematical modelling. Investigations in modelling have been found to be useful in the translation (Translation is an important transfer takes place when two cultures meet and interact, as the language, scientific, and mathematical knowledge of one cultural group pass into the interpretative realm of another. In this process, the translation of mathematical ideas, procedures, and practices of the studied culture is understood and comprehended through dialogic terms that are different in temporal and special frames and is transformed (Rosa and Orey 2017).) of ethnomathematical contexts by numerous scholars in Latin America (Bassanezi 2002; Biembengut 2000; D’Ambrosio 1995; Ferreira 2004; Rosa and Orey 2016). In order to document and study widely diverse mathematical practices and ideas found in many traditions, modelling is an important tool used to translate, describe, and solve problems arising from cultural, economical, political, social, and environmental contexts. It brings with it numerous advantages to the learning of contextualized mathematics (Barbosa 1997; Bassanezi 2002; Biembengut and Hein 2000; Hodgson and Harpster 1997; Orey 2000). For example, outside of the community of ethnomathematics researchers, it is known that many scientists search for mathematical models that translate their deepening understanding of both real-world situations and diverse cultural contexts. This approach enables them to take cultural, social, economic, political, and
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 5 environmental positions in relationship to the objects under study (Bassanezi 2002; D’Ambrosio 1993; Rosa and Orey 2006). Ethnomodelling is a process that allows for the elaboration of problems and questions that grow from real situations (systems) and forms an image or sense of an idealized version of the mathema. (According to D’Ambrosio (1985), mathema is considered as the actions taken by the members of distinct cultural groups to explain and understand the world around them. Thus, they must manage and cope with their own reality in order to survive and transcend. Throughout the history of mankind, technes (or tics) of mathema have been developed in very different and diversified cultural environments, that is, in the diverse ethnos. Thus, in order to satisfy the drive towards survival and transcendence, human beings have developed and continue to develop, in every new experience and in diverse cultural environments, their own ethnomathematics.) According to Rosa and Orey (2010), this perspective essentially forms a critical analysis for the generation and production of knowledge (creativity) and develops the intellectual process for its production, the social mechanisms of institutionalization of knowledge, and its transmission through generations. For example, D’Ambrosio (2000) affirmed that “this process is modelling” (p. 142) because it gives us the tools to analyze its role in reality as a whole. In this holistic context, modellers study systems taken from reality in which there is an equal effort made to create an understanding of their components as well as their interrelationships (Bassanezi 2002; Rosa 2000). By having started with a social or reality-based context, the use of modelling as a tool begins with the knowledge of the student by developing their capacity to assess the process of elaborating a mathematical model in its different applications and contexts (D’Ambrosio 2000). This uses the reality and interests of students versus the traditional model of instruction, which makes use of external values and curriculum without context or meaning. In this context, Bassanezi (2002) characterized this process as “ethno/modelling” (p. 208) and defined ethnomathematics as “the mathematics practiced and elabo- rated by different cultural groups and involves the mathematical practices that are present in diverse situations in the daily lives of members of these diverse groups” (p. 208). This interpretation is based on D’Ambrosio’s (1990) trinomial: Reality – Individual – Action (Fig. 2). For example, D’Ambrosio (2006) affirmed that the “discourse above was about one individual. But there are many other individuals ( . . . ) from the most varied species, going through a similar process. For living individuals, the cycle is the same: → reality → individual → action → reality → individual → action →” (p. 5). In this context, “individual agents are permanently receiving information and processing it and performing action. But although immersed in a same global reality, the mechanisms to receive information of individual agents are different” (D’Ambrosio 2006, p. 5). According to this assertion, reality is defined in a very broad sense including natural, material, social, and psycho-emotional characteristics. This context enables the development of linkages among these three elements of the cycle through the mechanism of information, which includes both sensory and memory capabilities that produce stimuli in the members of distinct cultural groups (D’Ambrosio 1985).
- 6 M. Rosa and D. C. Orey INFORMATION INDIVIDUALS REIFICATION STRATEGIES FACTS REALITY ACTION Fig. 2 D’Ambrosio’s trinomial. (Source: D’Ambrosio 1985) Through reification (Reification is considered as a fallacy of ambiguity, when an abstraction is treated as if it is a concrete physical entity or real event. It is the error of treating as a concrete thing something which is not concrete but merely an ideal. It is also the mental activity in which hazily perceived and relatively intangible phenomena such as complex arrays of objects or activities are given a factitiously concrete form, simplified and labelled with words or other symbols (Lumsden and Wilson 1981).) these stimuli help the development of strategies based on codes and models that require action in many contexts. Therefore, action impacts reality by introducing facts into it, both artifacts and mentifacts. (Mentifacts are related to the analytical tools such as thoughts, reflections, concepts, and theories that represent the ideas and beliefs of the members of a distinct cultural group, for example, religion, language, and laws. They are also shared ideas, values, and behaviors developed by the members of a culture. Examples of mentifacts include viewpoints, worldviews, and notions about right or wrong behavior (D’Ambrosio 2006).) These facts are added into reality in order to modify it. This action produces additional information that, through this reificative process, modifies or generates new strategies for action. In this regard, it is valuable to highlight how members of distinct cultural groups capture and process information in diverse ways and, consequently, develop differ- ent actions encouraging the transformation of their own surroundings. According to this perspective, it is important to document and translate alternative interpretations and contributions of ethnomathematical knowledge as students learn to construct their own connections between both traditional and nontraditional learning settings through ethnomodelling. Exploring Ethnomodelling The etymology of the prefix ethno traces back to the Greek word ethnos meaning a people, nation, or foreign people. In the context of ethnomodelling, though, ethno does not refer only to specific races or peoples but also to the diversity and differences between cultural groups in general.
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 7 These differences may include those based on racial oppression or nationality but are mainly based on language, history, religion, customs, and institutions and on the subjective self-identification of a people. In so doing, ethno represents particularity and modelling universality and the combination of the specific and universal leads to all mathematical activity that takes place within a culture through the dynamic of the encounters. The goddess of practical knowledge in ancient Greece was techne, whose name relates to technique and technology. The Greek word for art is techne, and the Greek word tikein, which means to create, is also derived from techne. Techne is a form of practical knowledge that results in productive action. These mythic modes of knowledge are considered as practical knowledge that results in productive action. This etymology reveals a deep connection between technology and the practices of living and creating. It represents the relationship among humanity, sociocultural contexts, and the creation of all forms of technology and guides scientists and educators to develop a moral and cultural standard for the teaching and learning mathematics. This is one of the most important purposes of ethnomodelling. Ethnomodelling binds contemporary views in ethnomathematics. It recognizes the need for culturally based views on modelling processes. Studying the unique cul- tural differences in mathematics encourages the development of new perspectives on the scientific questioning methods. Research involving culturally bound modelling ideas may address the problem of mathematics education in non-Western societies by bringing local and cultural aspects into mathematical teaching and learning processes (Eglash 1999). This perspective is needed in mathematics education. Therefore, Rosa and Orey (2010) argue that ethnomodelling involves examining ways in which individuals or groups draw on traditional or curricular mathematical ideas in the course of their problem-solving experiences, not to idealize these as correct or appropriate ways of thinking but rather to highlight the relationship between cultural groups and the deeply embedded mathematics in their daily activities. In this context, Rosa and Orey (2013) affirm that the purpose of ethnomodelling is to invite students to explore others’ cultural practices (emic) and transit them into other mathematical systems, such as school or academic mathematics (etic). For example, students should compare how a particular problem is solved in different cultural contexts. Thus, ethnomodelling is “a practical application of ethnomathematics, and which adds the cultural perspective to modelling concepts” (p. 78). This presents us with a cultural perspective that broadens views of modelling because it recognizes it as a pedagogical bridge for students in the acquisition of mathematical knowledge (Bassanezi 2002). Hence, ethnomodelling brings an inclusion of a diversity of ideas brought by students from other cultural groups, which can give them confidence and dignity, while allowing them to discuss the inclusion of cultural perspectives into the modelling process (Rosa and Orey 2013). Ethnomodelling is a tool that responds to its surroundings and is culturally dependent (D’Ambrosio 2002; Bassanezi 2002; Rosa and Orey 2007). The goal of recognizing ethnomodelling is not to give mathematical ideas and practices of other cultures a Western stamp of approval but to recognize that they are, and always
- 8 M. Rosa and D. C. Orey have been, just as valid in the overall development of mathematics and sciences. According to this context, Rosa and Orey (2010) affirm that ethnomodelling is considered as the intersection of cultural anthropology, ethnomathematics, and mathematical modelling (Fig. 3). It is important to reiterate here that ethnomodelling studies mathematical ideas, procedures, and practices developed by the members of culturally different groups. Hence, it is necessary to understand how mathematical concepts were born, conceptualized, and adapted into the practices of a society (Huntington 1993; Eglash 1997; Rosa and Orey 2007). In this context, ethnomodelling does not follow the linear modelling approach that is prevalent in modernity. Previously, for example, Bassanezi (2002) stated that ethno/modelling process starts with the social context, reality, and interests of students and not by enforcing a set of external values and decontextualized activities without meaning for the students. This process is defined as the mathematics practiced and elaborated by different cultural groups, which involves the mathematical practices present in diverse situations in the daily lives of diverse group members. For example, the introduction of the term mathematization by D’Ambrosio (2000) set the stage for early scholarship in ethnomodelling. This context has allowed us to see that mathematization “is a process in which individuals from different cultural groups come up with different mathematical tools that help them organize, analyze, comprehend, understand, and solve specific problems located in the context of their real-life situation” (Rosa and Orey 2013, p. 118). This approach shows, indeed respects, that people of different cultures have different views of the relation between the nature of spirit and humankind, the individual and the group, and the citizen and the state, as well as differing views on the relative importance of rights and responsibilities, liberty and authority, and equality and hierarchy. Ignoring these cultural elements is a form of subtle Mathematical Modelling Cultural Anthropology Ethnomathematics Ethnomodelling Valuing and Respecting Validation Dialogue Fig. 3 Ethnomodelling as an intersection of three research fields. (Source: Rosa and Orey 2010)
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 9 colonialization and the authors stand firmly against it. In addition to these categories, the idea of culture is expanded to include differing professional groups, ages, classes, and functions (D’Ambrosio 1995) as well as sexual orientation and gender. The authors prefer a definition of culture as defined as the ideations, (Ideation means to come up with a more innovative bright idea that makes a difference in society. It involves both divergent thinking, which starts with the known and moving outwards, and convergent thinking, which starts with the known and moving inwards. Hence, ideation is the creative process of generating, developing, and communicating innovative ideas and transforming them into valuable outcomes for the well-being of the members of distinct cultural groups. In this context, ideas are understood as a basic element of thought that can be either visual, concrete, or abstract (Jonson 2005). It is important to emphasize that ideation also comprises all stages of a thought cycle, from innovation, to development, to actualization (Graham and Bachmann 2004).) that is, the symbols, behaviors, values, knowledge, and beliefs that are shared by a community (Banks and Banks 1993). The essence of a culture is not only its artifacts, tools, or other tangible cultural elements but the way members of distinct cultural groups interpret, use, and perceive them. An artifact may be used in different cultures in very diverse ways and for very distinct purposes. Mathematical ideas, procedures, and practices are good examples of this. Different cultures can contribute to the development of mathematical ideas, procedures, and practices that help to enrich the traditional mathematics curriculum. Traditional Eurocentric epistemologies and conceptions of mathematics have been imposed globally as the patterns of rational human behavior and are often closed to new ideas that originate in their former colonies. It is important to state here that the control of Western powers and the results of the globalization process are far from acceptable (D’Ambrosio 1997). Hence, the study of ethnomodelling, while being mindful of aspects of colonialization, and the importance of modern science, has encouraged the development of ethics of respect, solidarity, dignity, and cooperation across cultures. Consequently, it becomes necessary to discuss the development of mathematical ideas, procedures, and practices from three approaches of viewing cultures such as emics (local/insiders) and etics (global/outsiders) in order to develop and understand the dialogic (emic-etic/glocal) approach that is necessary for the development of ethnomodelling investigations. Ethnomodelling and its Three Approaches of Viewing Cultures The challenge both researchers and educators have in dealing with the connection between mathematics and culture is to develop forms of pedagogical action that helps us to understand culturally bound mathematical ideas, procedures, and practices developed by members of distinct cultural groups without letting their own (often dominant) culture interfere in the curricular process. In accordance with this context, the members of distinct cultural groups have developed their own interpretation of local culture (emic approach) opposed to
- 10 M. Rosa and D. C. Orey its global interpretation from the outsiders (etic approach) (Orey and Rosa 2014). The use of emics and etics for the interpretation of cultural systems includes cognitive, perceptual, and conceptual knowledge, which is influenced through a unique cultural dynamism. (Cultural dynamism refers to the exchange of systems of knowledge that enable members of distinct cultures to exploit or adapt to the world around them. Thus, this cultural dynamic facilitates the incorporation of human invention, which is related to changing the world to create new abilities and institutionalizing these changes that serve as the basis for developing more competencies (Rosa and Orey 2015).) Both emic and etic approaches provide ways of discriminating between various types of knowledge for the study of cultural phenomena such as the development of mathematical practices. Thus, Pike (1967) affirmed that: ( . . . ) it proves convenient – though partially arbitrary – to describe behavior from two different standpoints, which lead to results which shade into one another. The etic viewpoint studies behavior as from outside of a particular system, and as an essential initial approach to an alien system. The emic viewpoint results from studying behavior as from inside the system. (p. 37) The emic approach examines local principles of classification and conceptualization from within each cultural system (Berry 1989) in which distinctions made by the members of distinct cultural groups are emphasized. According to Lett (1990), the emic approach is essential for an intuitive and empathic understanding of a culture, while the etic approach is essential for cross-cultural comparison and indispensable for ethnology because such comparisons necessarily demand the application of standard units and categories. It is necessary to deconstruct the notion that mathematical ideas, procedures, and practices are uniquely modern or European in origin as they are based on certain philosophical assumptions and values that are strongly endorsed by Western civilizations. For example, Rosa and Orey (2017) assert that there are beliefs that mathematical procedures are unique and that the sociocultural unit of operation is the individual. On the other hand, there are beliefs that mathematical practices are the same and that its goals and techniques are equally applicable across all cultural groups. An important challenge for many educators is to strengthen existing mathematics curricula by minimizing the power of mathematical universality and their claims of descriptive, predictive, and explanatory adequacy (Rosa 2010). A second goal is to assist and support educators to understand and explain both existing and historical variations of mathematical ideas, procedures, and practices that have varied across time, place, cultures of origin, race, ethnicity, gender, and other sociocultural characteristics (Rosa and Orey 2015). Consequently, when researching ethnomodelling, it is possible to identify at least three cultural views or approaches that help us to investigate mathematical ideas, procedures, and practices developed by the members of distinct cultures: etic, emic, and dialogic approaches.
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 11 Etic: The Global/Outsider Approach This approach is related to the outsiders’ view on beliefs, customs, and scientific and mathematical knowledge of the members of distinct cultural groups. In this context, global analyses have a cross-cultural design because outsider observers develop global worldviews that seek objectivity across cultures. Thus, Helfrich (1999) examines the question of a cross-cultural perception in which observations are often taken according to externally derived criteria and frequently without the intentionality of learning the perspectives of others. Globalization has reinforced the utilitarian mechanization, indeed automatization of mathematics approach to school mathematics curricula. As well, it has helped to globalize pervasive western academic mathematical ideologies. Particularly, school mathematics is criticized as a cultural homogenizing force, a critical filter for status, a perpetuator of mistaken illusions of certainty, and an instrument of power (Skovsmose 2000). In this approach, comparativist researchers and educators attempt to describe differences among cultures. These individuals are considered as culturally universal (Sue and Sue 2003). In this context, Pike (1967) refers etic categories as culture-free features of the real world. Emic: The Local/Insider Approach This approach is related to the insiders’ view on their own culture, customs, beliefs, and scientific and mathematical knowledge. Local knowledge is important because it has been tested and validated within the local context. It creates a framework from which members of distinct cultural groups can understand and interpret the world around them. Local worldviews clarify intrinsic cultural distinctions that examine local principles of classification and conceptualization from within each cultural system. Currently, there is a recognition about the importance of local contributions to the development of scientific and mathematical knowledge. For example, local mathematical knowledge and interpretations are essential to emic analyses in the mathematics curriculum that cultivates values and fosters the conscientization of the students. An emic analysis is culturally specific regarding to the insiders’ beliefs, thoughts, behaviors, knowledges, and attitudes. It is from their viewpoint that mathematical knowledge is conveyed for the understanding of their cultural context. In this approach, these members describe their culture in its own terms. These individuals are considered as culturally specific (Sue and Sue 2003). In this context, Helfrich (1999) stated that what is emphasized in this approach is the self- determination and self-reflection of these members about the development of their mathematical ideas, procedures, and practices.
- 12 M. Rosa and D. C. Orey Dialogic: The Glocal/Emic-Etic Approach This approach represents a continuous interaction between etic (globalization) and emic (localization) approaches, which offers a perspective that they are both elements of the same phenomenon (Kloos 2000). It involves blending, mixing, and adapting two processes in which one component must address the local culture and/or a system of values and practices (Khondker 2004). In a glocalized society, (According to Rosa and Orey (2017), glocalization is the acceleration and intensification of interaction and integration among members of distinct cultural groups. Glocalization has emerged as the new standard in rein- forcing positive aspects of worldwide interaction in textual translations, localized marketing communication, sociopolitical considerations, and in the development of scientific and mathematical knowledges.) members of distinct cultural groups must be “empowered to act globally in its local environment” (D’Ambrosio 2006, p. 76). It is also necessary to work with different cultural environments and, acting as ethnographers, to describe mathematical ideas, procedures, and practices of other peoples in order to give meaning to these findings (D’Ambrosio 2006). Therefore, Rosa and Orey (2017) argued that glocalization has emerged as the new standard in reinforcing positive aspects of worldwide interaction in textual translations, localized marketing communication, sociocultural-political consider- ations, and in the development of scientific and mathematical knowledge. In this context, Eglash et al. (2006) stated that, in some cases, the translation between distinct mathematical knowledge systems is direct and simple such as counting and calendars. However, there are cases in which mathematical ideas, procedures, and practices are embedded in processes related to the iteration (repetition of techniques or procedures) in beadwork and/or in Eulerian paths found in African sand drawings. For example, Eglash (1997) argued that Gerdes (1991) used the sona sand drawings developed by the members of the Tchokwe cultural group, in Northeastern Angola, to demonstrate the value of indigenous mathematical knowledge by show- ing that the constraints necessary to define complex Eulerian paths and recursive generation systems are created by successive iterations through the application of the same geometric algorithm. The construction of these complex cultural artifacts indicates the conscious use of iterative constructions as a visualization of analogous iterations in cultural knowledge. Figure 4 shows the similarity between the Eulerian path and the sona sand drawing produced by the Tchokwe people in Angola. In this context, Eglash et al. (2006) developed a computational modelling process on traditional African architecture using fractal geometry, which are patterns that repeat themselves at many scales as they are usually used to model natural phenomena such as trees (branches of branches) and mountains (peaks within peaks). The results of their project showed that both computer simulations and mea- surement of fractal dimensions of these traditional village architectures are formed
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 13 by several repetitions (iterations) in regard to the same pattern at different scales: circular houses arranged in circles of circles and rectangular houses in rectangles of rectangles (Eglash et al. 2006). In this context, Eglash and Odumosu (2005) argue that “in the African case many villages were constructed over many generations with no one in charge – yet there is a cohesive fractal pattern for the village as a whole” (p. 102). Figure 5 shows a Ba-ila settlement in southern Zambia that has a fractal shape. Figure 6 shows that this architecture can be modelled with fractals by applying the principle of iteration. It is possible to observe, in Fig. 6, the fractal generation of Ba-ila, in which the first iteration is similar to a single house, the second iteration is similar to a family ring, (At the back end of the interior of the settlement, there is smaller detached ring of houses, which is like a settlement with a settlement. This is the chief’s extended family ring (Eglash and Odumosu 2005).) and the third iteration is similar to the whole village. Fig. 4 The similarity between the Eulerian path and the sona sand drawing. (Source: Rosa and Orey 2014, p. 144) a b d f c e Eulerian Path Sona sand drawing produced by the Tchokwe People Fig. 5 Ba-ila settlement with a fractal shape. Source: Eglash and Odumosu (2005)
- 14 M. Rosa and D. C. Orey Fig. 6 Iterations used in the Ba-ila architectural structure. (Source: Eglash and Odumosu 2005) These examples show that the act of translation applied in these processes arises from emic rather than etic origins. Hence, ethnomodelling establishes relations between the local (emic) conceptual framework and the mathematical knowledge embedded in relation to the global designs (etic). Through focusing on local knowledge first and then integrating global influences, people can create individuals and collective groups rooted in their local cultural traditions and contexts, but they are also equipped with a global knowledge by creating a sort of localized globalization (Cheng 2005). For example, emic-oriented researchers and educators focus on the investigations of the intrinsic cultural distinctions meaningful to members of distinct cultural groups, especially when the natural world is distinguished from the supernatural realm in the worldview of those specific cultures (Rosa and Orey 2017). On the other hand, etic-oriented researchers and educators examine cross-cultural perspectives so that their observations are taken according to externally derived criteria. This context allows for the comparison of multiple cultural groups in which “both the objects and the standards of comparison must be equivalent across cultures” (Helfrich 1999, p. 132). According to this context, researchers and educators should find points of agreement between the imposed cultural universality (Cultural universality refers to the belief that the origin, process, and manifestation of disorders are equally applicable across cultures (Bonnett 2000).) (global) of mathematical knowledge or take on techniques, procedures, and practices of its cultural relativism. (Cultural relativism is related to the assertion that human values, far from being universal,
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 15 vary according to different cultural perspectives in distinct cultures. Individuals’ beliefs, values, and practices are understood based on their own culture, rather than be judged against the criteria of another (Todorov 1993).) In this context, the use of both emic and etic approaches deepens their understanding of important issues in scientific research and investigations about ethnomathematics because they are complementary worldviews (Rosa and Orey 2013). Since these two approaches are complementary, it is possible to delineate forms of synergy between local and global aspects of mathematical knowledge. A suggestion for dealing with this dilemma is to use a combined emic-etic (local-global) approach, rather than simply applying local or global dimensions of one culture to other cultures. This combined approach requires researchers and educators to attain local knowledge developed by the members of distinct cultural groups, which allows us to become familiar with the relevant cultural differences in diverse sociocultural contexts (Rosa and Orey 2015). In the authors’ point of view, both local (emic) and global (etic) approaches are important to develop a clearer idea of what is needed for mathematics education in a given context, mainly, to the conduction of ethnomodelling research. In this context, local knowledge and its interpretations (emic) are essential to the conduction of these studies as well as the promotion of debates related to the comparisons between mathematical knowledge developed in distinct cultural contexts (etic) which are also necessary to the development of ethnomodelling investigations. In this regard, Pike (1967) stated that: Through the etic ‘lens’ the analyst views the data in tacit reference to a perspective oriented to all comparable events (whether sounds, ceremonies, activities), of all peoples, of all parts of the earth; through the other lens, the emic one, he views the same events in that particular culture, as it and it alone is structured. The result is a kind of ‘tri-dimensional understanding’ of human behavior instead of a ‘flat’ etic one. (p. 41) It is important to understand Pike’s (1967) view of the relation between the emic and etic approaches as a symbiotic process between two different mathematical knowledge systems. Similarly, the resurgence of debates regarding cultural diversity in the mathematics curriculum has also renewed the classic emic-etic debate since there is a need to comprehend how to build scientific generalizations while understanding and making use of sociocultural diversity. Yet, attending to unique mathematical interpretations developed by members of each cultural group often challenges fundamental goals of mathematics in which the main objective is to build a theoretical basis that can truly describe the development of mathematical practices in distinct cultures. Characterizing Ethnomodels Ethnomodelling privileges the organization and presentation of mathematical ideas, notions, procedures, and practices that describe systems (Systems are part of reality that are considered integrally as well a set of items taken from students’
- 16 M. Rosa and D. C. Orey sociocultural contexts. The study of systems seeks to understand all its components and the relationship between them, including sociocultural variables (Rosa and Orey 2013).) taken from the sociocultural context of the members of distinct cultural groups in order to enable its communication and transmission across generations. The representation of this mathematical knowledge helps these members to understand, comprehend, and describe their world by using small units of infor- mation, named ethnomodels, which links their cultural heritage to diverse contexts such as social, political, economic, environmental, and educational (Rosa and Orey 2010). This approach helps them to develop techniques, processes, and methods to solve problems they face daily. This context allows ethnomodels to be defined as cultural artifacts that can be considered as the pedagogical tools used to facilitate the understanding and comprehension of systems taken from reality of the members of distinct cultural groups (Rosa and Orey 2010). Hence, ethnomodels serve as external representations of local phenomena that are both precise and consistent with the scientific and mathematical knowledge socially constructed and shared by the members of specific cultural groups. In the ethnomodelling process, ethnomodels can be emic, etic, and dialogic. Emic ethnomodels are grounded in the mathematical features and characteristics that are important and valuable for members of distinct cultural groups since their models are built and based on the information obtained from the insiders’ viewpoint. Many ethnomodels are etic in the sense that they are built on data gleaned from the outsiders’ viewpoint. For example, etic ethnomodels represent how modellers think the world works through systems taken from reality, while emic ethno- models represent how people who live in such world think these systems work in their own. The dialogic ethnomodels enable a translational process between emic and etic knowledge systems. In this cultural dynamism, these systems are used to describe, explain, understand, and comprehend knowledge generated, accumulated, transmit- ted, diffused, and internationalized by people from other cultures. According to Rosa and Orey (2017), this process involves a process of negotiating mathematical meanings expressed between local and global contexts through translation in the ethnomodelling process. Emic and Etic Ethnomodels of the Mangbetu Ivory Sculpture It is useful to examine mathematical ideas found in an ivory hatpin from the Mangbetu people, who occupy the Uele River area in the northeastern part of the Democratic Republic of Congo, and the geometric algorithm involved in its production, which “gives explicit instructions for generating a particular set of spatial patterns” (Eglash 1999, p. 61). The creation of a Mangbetu design may reflect the artisans’ desire to “make it beautiful and show the intelligence of the creator” (Schildkrout and Keim 1990, p. 100) by adhering to angles that are multiples of 45 degrees. This emic ethnomodel
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 17 is only one part of an elaborated geometric esthetic based on these angles that are used in many Mangbetu designs. The combination of the 45-degree angle construction technique with the scaling properties of the ivory carving may reveal its underlying structure, which has three interesting geometric features (Eglash 1999). However, this also suggests that if there were no rules to follow, then it would have been difficult to compare designs. First, each head is larger than the one above it and faces in the opposite direction. Second, each head is framed by two lines that intersect at approximately 90 degrees: one formed by the jaw and one formed by the hair. Third, there is an asymmetry in which the left side shows a distinct angle about 20 degrees from the vertical. The decorative end of this ivory hatpin is composed of four scaled similar heads that shows a scaling design (Fig. 7). Figure 8 shows the geometric analysis of this sculpture in which the sequence of shrinking squares can be constructed by an iterative process that bisects one square to create the length of the side for the next square. However, Eglash (1999) stated that it is not possible “to know if these iterative squares construction was the concept underlying the sculpture’s design, but it does match the features identified in this process” (p. 68). The mathematical idea implicit in this emic knowledge was passed to the members of the Mangbetu people across generations, who were responsible for the construction and upkeep of this unique ivory cultural artifact. Fig. 7 Mangbetu ivory sculpture. (Source Eglash 1999)
- 18 M. Rosa and D. C. Orey Fig. 8 Geometric analysis of a Mangbetu ivory sculpture. (Source: Eglash 1999) Fig. 9 Geometric relations in the Mangbetu ivory sculpture (Source: Adapted from Eglash 1999) On the other hand, Fig. 9 (This figure is not to scale.) shows the geometric relations in the sculpture iterative square structure. In this regard, it is possible to elaborate an etic ethnomodel to show that since α1 and α2 are alternate interior angles of a transversal intersecting two parallel lines,
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 19 then α1 = α2. Thus, the equation shows that: tan α1 = √ 2 2 3 √ 2 2 = √ 2 3 √ 2 = 1 3 and α1 = arctan 1 3 ∼ = 18 ◦ The left side of the ivory sculpture is about 20 degrees from the vertical, while in the iterative squares structure, the left side is about 18 degrees from the vertical (Eglash 1999). The construction algorithm of this etic ethnomodel can be continued indefinitely, and the resulting structure can be applied to a wide variety of mathematics teaching applications, from simple procedural construction to formal trigonometry. In this regard, D’Ambrosio (1993) affirmed that mathematical practices are socially learned and transmitted to the members of cultural groups. In this example, an emic observation sought to understand this mathematical practice of making this sculpture from the perspective of the internal dynamics and relations within the Mangbetu culture by clarifying intrinsic cultural distinctions to the external observers and its contributions to the development of mathematics. An Etic Ethnomodel of Brazilian Roller Carts An investigation was conducted by Soares (2018) with 34 students in a public night school, ages ranging from 18 to 33 years old, in the second year of high school in the Youth and Adult Education Program in the Belo Horizonte metropolitan region, the state capital of Minas Gerais, Brazil. Figure 10 shows one of the most common types of Brazilian roller carts built with a wooden frame and steel bearings that are discarded in automotive repair shops. Etic ethnomodels enable students to analyze and interpret their data, to formulate and test their own hypotheses, and to verify the effectiveness of their elaborated mathematical models taken from their own reality. In this approach, students in their groups designed a model and constructed their roller carts by learning how mathematical concepts were used in the preparation, analysis, and resolution of their models. Fig. 10 Brazilian roller cart. (Source: Soares 2018)
- 20 M. Rosa and D. C. Orey For example, one of the concerns of the students was to determine the dimensions of the roller carts that were suitable for them, regardless of their height and size so that all of them could participate in the race. Figure 11 shows an example of an etic ethnomodel of the roller carts developed by the students in each group by choosing a standardized model of the cart to be used in the race competition. This approach helps students to move away from emotional arguments and to focus on and then apply data-based tools to build a model of a standardized roller cart for a race competition. These students applied their etic ethnomodels to develop a standardized roller cart for a competition by using mathematical content to accomplish the proposed activities related to the design of the model and the construction of their carts. During the process of elaborating their etic ethnomodels, students described, analyzed, and interpreted data collected in relation to the dimension of the parts of the cart roller in order to standardize its dimensions. Then, they sent their notes to the woodworker for the validation of their results. For example, the majority of the students affirmed that the standardization of procedures enables the roller cart competitions to be fairer. It is important to state that, according to Barbosa (2006), the results obtained in this process are linked to the students’ perceptions and reality. In this ethnomodelling process, students elaborated and developed their projects related to the design and construction of roller carts in which they could participate in a race competition under equal conditions for all competitors. Figure 12 shows a roller cart built by the students in the classroom. This etic ethnomodel provided a cross-cultural contrast and comparative perspec- tive by translating mathematical knowledge involved in this cultural phenomenon related to the construction of the roller carts for understanding individuals from different cultural backgrounds so as to holistically comprehend and explain this mathematical practice from the viewpoint of the outsiders by seeking objectivity across cultures. Therefore, the focus of this ethnomodelling process was to apply data in a specific sport competition related to roller carts that have been initially created by the social, cultural, climatic, and economic influences in Brazil in which popular and diverse forms of competition arose and are still practiced. Fig. 11 Models of the roller carts. (Source: Soares 2018)
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 21 Fig. 12 Roller cart built by the students. (Source: Soares 2018) A Dialogic Ethnomodel of a Local Farmer-Vendor A study that was conducted by Cortes (2017) in a public school in the metropolitan region of Belo Horizonte and in a local farmers’ market, in the state of Minas Gerais, Brazil, is an example of the application of a dialogic ethnomodel. The main objective of this study was to show how dialogic approaches of ethnomodelling can contribute to the process of re-signification of the function concept. The data gleaned from this study came from 38 students, aged from 15 to 17 years old, in the second year of high school, during their interaction with a local farmer- vendor and his daily labor practices. Figure 13 shows a farmer-vendor and a group of students in a local farmer market. A contribution of the ethnomodelling process to the development of re- signification of the scholarly function concepts was to provide an analytical way to examine local (emic) strategies applied by the farmer-vendor to his labor practices in the farmer market, as well as the academic techniques (etic) employed by the students in their school context. These contexts constituted ambiences of effective exchange of local and academic mathematical knowledge reciprocally through the elaboration of emic, etic, and dialogic ethnomodels. For example, the results of the study conducted by Cortes (2017) showed that the farmer-vendor developed through his observations and experiences an emic ethnomodel by mathematizing the calculation of the sale price of his products: Let’s assume that you buy a 10 kg box of tomatoes for 40 reais, (The Brazilian real or reais (R$) is the official currency of Brazil, which is subdivided into 100 cents.) and the kilogram is sold at 4 reais, thus each 100 grams cost 40 cents, then you cannot sell it at that price because we have expenses like gas, transportation, employees, packaging, etc. Thus, I sell
- 22 M. Rosa and D. C. Orey Fig. 13 A farmer-vendor and a group of students in a local farmer market. (Source: Cortes 2017) each kilogram of tomatoes by 5 or 6 reais because it should be more expensive since you do not go to the market to buy the products and sell them at the same price. In this case, I increased the price by 25 or 50 percent. Sometimes, I need to sell my products, for example, at 100 or 60 percent more, depending on the price I buy them and the expenses I have. This system is used to determine the price of any of the products I sell. For example, if I buy a product for 80 or 100 reais each box, then the price of the kilogram should be 16 or 12.80 reais [60% Mark up] or 20 or 16 reais [100% Mark up]. It is important to state that this emic ethnomodel was in accordance with the percep- tions, notions, and understandings deemed appropriate by the farmer-vendor and his cultural context. Thus, Rosa and Orey (2013) affirmed that the main objective of an emic approach is a descriptive idiographic orientation of mathematical phenomena because of the strength of the particularities of mathematical ideas, procedures, and practices developed by the members of distinct cultural groups. In this context, an etic ethnomodel provided cross-cultural contrasts and com- parative perspectives by using aspects of mathematical knowledge of the farmer- vendor’s practices to translate and guide the creation of connections and new understandings related to how individuals from a different cultural background use their own mathematical thinking. This etic approach is necessary to the holistic comprehension and explanation of this specific mathematical practice from the point of view of the students (outsiders). For example, Cortes (2017) affirmed that students developed an etic ethnomodel that is an approximation of the emic ethnomodel developed by the farmer-vendor: A product, whose cost price is 40 reais, has a sale price between R$ 5.00 and R$ 6.00. Another product, whose cost price is 80 reais, has a sale price between R$ 12.00 and R$ 16.00. And a third product whose cost price is 100 reais, has a sale price between R$ 16.00 and R$ 20.00. However, it is important to note that these sales prices may be increased by other costs related to the market’s expenses.
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 23 Table 1 Possible dialogic ethnomodel. (Source: Cortes 2017) If CP(m) = 40, then SP(m) = v. m, where 5 ≤ v ≤ 6 If CP(m) = 80, then SP(m) = v. m, where 12 ≤ v ≤ 16 If CP(m) = 100, then SP(m) = v. m, where 16 ≤ v ≤ 20 CP = cost price SP = Sale price m = mass (kg) of the product v = variation of price including expenses and charges The interpretation of these results shows that the determination of these prices, besides being related to the quantity of products purchased, is also bounded to the emic constructs developed by the daily labor experiences of the farmer-vendor. Table 1 shows the elaboration of a dialogic ethnomodel by the students, which represents the sale process developed by the farmer-vendor. This example shows that ethnomodelling allowed for the reconceptualization and application of the function concept through the elaboration of mathematical activities originating in the sociocultural context of the farmer-vendor and school community by applying the ethnomodelling process in the mathematics curriculum as an encounter of two complementary cultures. This approach enabled the dialogic development between the ideas, procedures, and mathematical practices intrinsic to the labor practices of farmer-vendor (emic approach) and school mathematical contents (etic approach) with the use of problem situations that emerged from the context of a farmer market (dialogic approach). By using ethnomodels, students try to understand and comprehend their own surroundings through the development of explanations that are organized as proce- dures, techniques, methods, and theories in order to explain and deal with daily facts and phenomena (Rosa and Orey 2015). These strategies are historically organized in every culture as knowledge systems, including mathematics. Relevance of Ethnomodelling in a Mathematics Curriculum In considering ethnomodelling as tool to study ethnomathematics, teaching is more than the transference of knowledge because it becomes an activity that introduces the creation of mathematical knowledge. For example, Freire (1970) argued that this approach is the antithesis of turning students into containers to be filled with information and that it is necessary for a mathematics curriculum to translate the interpretations and contributions of ethnomathematical knowledge because students need to be able to analyze the connection between both traditional and nontraditional learning settings. According to Rosa and Orey (2016), ethnomodelling applies mathematics as a language for understanding, simplification, and resolution of problems and activities linked to the students’ reality. Conversely, traditional mathematical modelling devel- oped in the academic mathematics curriculum aims at transmitting mathematical content by applying it to artificial situations presented as problems. In this context,
- 24 M. Rosa and D. C. Orey D’Ambrosio (1995) argued that these problems are artificially formulated in such ways that they can only help memorization skills. These techniques and problems are, for most students, boring, uninteresting, obsolete, and unrelated to their own reality. The characteristics of this kind of curriculum are responsible for the downgrading of school satisfaction and achievement in many countries. In this regard, Rosa and Orey (2016) affirm that ethnomathematics and modelling through ethnomodelling may restore a sense of pleasure in doing mathematics in the classrooms. One reason for this curricular failure is to ignore emic perspectives in the elabora- tion of mathematical activities that includes the recognition of other epistemologies, as well as the comprehension of a holistic and integrated nature of the mathematical knowledge of members of diverse cultural groups found in many cultures (Rosa and Orey 2010). In this regard, an ethnomodelling curriculum provides an ideological basis for learning with and from the diverse cultural and linguistic backgrounds of the members of distinct cultural groups. There are three reasons for the application of ethnomodelling into the mathematics curriculum (Rosa and Orey 2013): 1. Ethnomodelling is an effective path that can be developed to translate ideas, procedures, and practices between distinct mathematical systems, such as school or academic mathematics. 2. Ethnomodelling can be used to develop intercultural classroom activities. 3. Ethnomodelling is a pedagogical action that can be used to transform the relation between mathematics, culture, and society. This paradigm suggests that developing an ethnomodelling curricular praxis is to value the contributions of other mathematical knowledge traditions. Thus, in order to achieve this goal, it is recommended that teachers interpret alternative mathemat- ical ideas, procedures, and practices by starting with the outside sociocultural reality of students. It is important to emphasize here how ethnomodelling is not considered as a way to reach academic mathematical concepts because an emic approach does not work as a kind of scaffolding to school mathematics. Admittedly, this is a possible perspective, but in our investigations, this is different from perceiving emic and etic descriptions as parallel in such a way that ethnomodelling is much more as an encounter of distinct cultures than a scaffolding educational process. Consequently, it is beneficial to apply an ethnomathematical ethnographic perspective in order to come to an understanding of, and respect for, mathematical knowledge of the members of a given cultural group and having a clear purpose of this educational activity. Thus, the implementation of an ethnomodelling perspective must be preceded by an inventory of students’ tacit knowledge (Rosa and Orey 2013). In this regard, coming to comprehend students’ contexts, ethnomodelling pro- vides a deeper appreciation of mathematical beauty and utility. Thus, it is useful to develop mindfulness and an understanding of what kind of mathematical ideas,
- Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 25 procedures, and practices are important to particular cultural environments and historical contexts. Conclusion This chapter sought to outline ongoing research related to cultural perspectives in mathematical modelling. Contemporary academic mathematics is predominantly Eurocentric. This Eurocentrism, which is not necessarily bad, is insufficient to connect this kind of mathematics to the local realities of learners and educators as it facilitates an ongoing divide that has hindered the mathematics coming from non-Western traditions. The motivation towards a cultural approach presents us with an accompanied assumption that makes use of cultural perspectives through ethnomathematics and uses mathematical modelling to bring local issues into global discussion. The authors have suggested that mathematics education is an active and partici- patory social product in which there is a development of a dialogic relation between mathematical knowledge and society. Moreover, they have presented modern or westernized mathematics as primarily dominated by the preferences of science and capitalism of the West (European-North American) and that this Eurocentrism poses many problems in mathematics education in non-Western cultures. Ethnomodelling stands for the development of mathematical ideas, procedures, and practices that originated by members of diverse cultural groups. It is defined as the study of mathematical phenomena within a culture. In this context, ethnomod- elling differs from traditional definitions of modelling that considers the foundations of mathematics education as constant, universal, and applicable everywhere. Hence, it is necessary to point out that the study of ethnomodelling takes the position that mathematics curriculum, and the many unique problems that it may model, is a social construct and thus culturally bound. In order to keep up with modern Western developmental models, other cultures have been forced to adapt or perish. Relying primarily on constructivist theories, the authors argue that universal theories of mathematics take different forms in different cultures and that Western views on abstract ideas of modelling are culturally bound. The study of ethnomodelling is considered a powerful tool used in the translation of a problem-situation of mathematical ideas and practices within a culture. These new-found ethnomathematical lenses lead to new findings in the development of an inclusive model of the connection between mathematics and culture. Ethnomodelling is also a pedagogical action that enables students to link school mathematics and the mathematics as used by the members of other cultural groups. It has to do with developing an understanding or creating a sense of mathematical knowing of and between our own cultures and translating them into school perspectives and vice versa. In order to do so, it is important to develop the concepts of emic, etic, and dialogic approaches, which refer to describing the mathematical ideas, procedures, and practices in terms of local culture or school lens. The main contribution of this
- 26 M. Rosa and D. C. Orey chapter is to conceptualize the notion of ethnomodelling and to exemplify how it happens in mathematics education practices in the schools. Thus, three examples illustrate the use of emic, etic, and dialogic approaches as part of the pedagogical action of ethnomodelling. In the conduction of the ethnomodelling process, the promotion of dialogue (re- signification of function concept) between emerging knowledge (farmer-vendor) and existing (function concept) is important to enable the approximation of this knowledge through the proposition of contextualized mathematical activities. In this cultural dynamism, local knowledge has interacted dialogically with the knowledge consolidated by the academy, developing a reciprocal relationship between these two approaches. In an increasingly glocalized world, educators must consider the cultural and philosophical backgrounds of a society and, most importantly, their learners. Distinct cultures have very different perceptions of time and space, logic, problem- solving methods, society, and values. Learning to comprehend and appreciate these differences enriches the curriculum and increases understanding between peoples, which can only be a good thing! The adoption of an ethnomodelling perspective in a mathematics curriculum recognizes the importance of local cultures to the development of mathematics. This pedagogical aspect produces student-researchers who are active participants in their own mathematics education as they learn that they themselves can contribute to the development of mathematics. References Banks JA (1993) Multicultural education: characteristics and goals. In: Banks JA, CAM B (eds) Multicultural education: issues and perspectives. Allyn Bacon, Boston, pp 3–28 Barbosa JC (1997) O que pensam os professores sobre a modelagem matemática? [What do teachers think on mathematical modelling?]. Zetetiké 7(11):67–85 Barbosa JC (2006) Mathematical modelling in classroom: a critical and discursive perspective. ZDM 38(3):293–301 Bassanezi RC (2002) Ensino-aprendizagem com modelagem matemática [Teaching and learning with mathematical modelling]. Editora Contexto, São Paulo Berry J (1989) Imposed etics-emics-derived etics: the operationalization of a compelling idea. Int J Psychol 24:721–735 Biembengut MS (2000) Modelagem etnomatemática: pontos (in)comuns [Modelling ethno- mathematics: (un)common points]. In: Domite MC (ed) Anais do Primeiro Congresso Brasileiro de Etnomatemática – CBEm-1. FE-USP, São Paulo, pp 132–141 Biembengut MS, Hein N (2000) Modelagem matemática no ensino [Teaching mathematical modelling]. Editora Contexto, São Paulo Bonnett A (2000) White identities: an historical and international introduction. Pearson Longman, Harlow Cheng YC (2005) New paradigm for re-engineering education. Springer, New York Cortes DPO (2017) Re-significando os conceitos de função: um estudo misto para entender as contribuições da abordagem dialógica da Etnomodelagem [Re-signifying function concepts: a mixed method study to understand the contributions of the ethnomodelling dialogic approach].
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