Revisiting the Theoretical Foundations of Ethnomodelling

OPEN ACCESS EDUCAÇÃO POR ESCRITO Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 e-ISSN: 2179-8435 http://dx.doi.org/10.15448/2179-8435.2023.1.45054 SEÇÃO: ARTIGOS REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING REVISITANDO AS FUNDAMENTAÇÕES TEÓRICAS DA ETNOMODELAGEM Daniel Clark Orey1 orcid.org/0000-0002-8567-034X [email protected] Milton Rosa1 orcid.org/0000-0002-5190-3862 [email protected] Recebido em: 20 ago. 2023. Aprovado em: 31 out. 2023. Publicado em: 19 dez. 2023. Abstract: By including mathematical activities from outside of the school environment, the process of modelling shows us that mathematics is more than the manipulation of mathematical symbols, procedures, and practices. The application of ethnomathematical techniques along with the tools of modelling allows us to see a holistic reality to mathematics. From this perspective, one pedagogical approach that connects the cultural features of mathematics with its school/ academic aspects is named ethnomodelling, which is a process of translation and elaboration of problems and questions taken from systems that are part of the reality of the members of any cultural group. In this article we offer an alternative goal for educational research, which is the acquisition of both emic (local) and etic (global) approaches for the implementation of ethnomodelling in the classrooms. We also discuss a third approach on ethnomodelling, which is the dialogical (glocal) approach, which combines both emic and etic approach bases. Finally, we define ethnomodelling as the study of mathematical phenomena within a culture because it is a social construct and is culturally bound, which adds the cultural characteristics of mathematics into the modelling process. Keywords: ethnomathematics; ethnomodelling; mathematical modelling; sociocultural perspective. Resumo: Ao incluir as atividades matemáticas fora do ambiente escolar, o pro- cesso de modelagem nos mostra que a Matemática é mais do que a manipulação de símbolos, procedimentos e práticas matemáticas. A aplicação de técnicas etnomatemáticas juntamente com as ferramentas de modelagem permitem ver uma realidade holística para a Matemática. Nessa perspectiva, uma abordagem pedagógica que conecta os aspectos culturais da Matemática com os seus aspectos escolares/acadêmicos é denominada etnomodelagem, que é um processo de tradução e elaboração de problemas e questões retirados de sistemas que fazem parte da realidade dos membros de qualquer grupo cultural. Neste artigo, oferecemos um objetivo alternativo para a pesquisa educacional, que é a aquisição de abordagens êmicas (locais) e éticas (globais) para a implementação da etnomodelagem em salas de aula. Discutimos também uma terceira abordagem para a etnomodelagem, que é a abordagem dialógica (glocal), que combina as abordagenes êmica (local) e ética (global). Finalmente, definimos a etnomodelagem como o estudo de fenômenos matemáticos dentro de uma cultura porque é um construto social e culturalmente vinculado que adiciona as características culturais da Matemática ao processo de modelagem. Palavras-chave: etnomatemática; etnomodelagem; modelagem matemática; perspectiva sociocultural. INITIAL CONSIDERATIONS Ethnomathematics arises when researchers investigate the knowledge possessed by members of distinct cultural groups in the context of their Artigo está licenciado sob forma de uma licença Creative Commons Atribuição 4.0 Internacional. mathematical ideas, techniques, procedures, and practices. However, an outsiders’ understanding of cultural traits remains an interpretation that 1 Universidade Federal de Ouro Preto, Ouro Preto, Minas Gerais, Brasil. 2/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 may come to emphasize inessential features of ments (ROSA, 2010). the culture and/or is in danger of creating a mi- Furthermore, members of distinct cultural sinterpretation of a unique and culturally specific groups have developed unique and often very mathematical paradigm. distinct ways to mathematize their own realities The challenge that arises from this approach is (D’AMBROSIO, 1990). In this context, mathema- how culturally bound mathematical ideas can be tization is the process in which individuals from extracted or understood without letting the cultu- distinct cultural groups come up with different re of the researchers (outsiders) interfere with their mathematical tools that can help them to organi- findings, interpretation, and discussions related ze, analyze, comprehend, understand, solve, and to the mathematical knowledge of the members model specific problems located in the context of of the cultural group under study. This fact may their own real-life situation (ROSA; OREY, 2006). happen when the members of distinct cultural These tools allow them to both identify and groups develop the interpretation of their own describe specific mathematical ideas, procedures, culture, which is named an emic (local) approach or practices in a general context by schematizing, as opposed to an outsiders’ interpretation, which formulating, and visualizing a problem in different is named the etic (global) approach. ways, discovering relations and regularities, and The concepts of emic (local) and etic (global) transferring a real world problem to academic were first introduced by the linguist Pike (1954) mathematics through mathematization. In this who drew upon an analogy regarding two lin- context, ethnomodelling is considered one alter- guistic terms: native methodological approach that enables us to record historical forms of mathematical ideas, a) Phonemic, which is considered as the specific sounds used in a particular language. procedures, techniques, and practices that occur b) Phonetic, which is considered as the general aspects of vocal sounds and sound production in a particular language. ethnomodelling as the practical application of In this context, we consider that all the possible bound view on modelling, our sources are rooted sounds human beings can make constitute the on the theorical basis of ethnomathematics and phonetics of a given language. However, when on the sociocultural perspective of mathematical people speak a particular language, they do not modelling. in distinct cultural contexts. Consequently, Rosa and Orey (2010a) define ethnomathematics that adds a cultural perspective to the modelling process. Thus, when justifying the need to the development of a culturally hear all its possible sounds. In this regard, not all sounds make a difference when spoken because they are locally significant. This means that they are the phonemics of that language. Researchers and educators who take on an emic (local) approach consider that many factors such as cultural and linguistic backgrounds, social, moral values, and lifestyle come into play when mathematical ideas, procedures, and practices are developed by members of their own culture. Thus, these members have developed different ways of doing mathematics in order to understand and comprehend their own cultural, social, political, economic, and natural environ- ETHNOMATHEMATICS Ethnomathematics was introduced by the Brazilian educator and Mathematician Ubiratan D’ Ambrosio in the 1970s. Ethnomathematics uses the etymology of three Greek roots: ethno, mathema, and tics (ROSA; OREY, 2016). It is a program that incorporates mathematical ideas and procedures practiced by the members of distinct cultural groups, which are identified not only as indigenous societies but as groups of workers, professional classes, and groups of children of a certain age as well (D’AMBROSIO, 1990). Thus, ethno refers to members of a specific Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING 3/19 group within a cultural environment identified by relevant to the work of ethnomathematics in- their cultural traditions, codes, symbols, beha- clude the essential elements of culture such as viors, myths, and specific ways used to reason and language, economy, politics, religion, art, and the to infer. Mathema means to explain and unders- daily mathematical practices of diverse groups tand the world to transcend, manage, and cope of students. with reality so that the members of the cultural Since cultural anthropology gives us tools that groups can survive and thrive in their daily ende- increase our understanding of the internal logic of avors and tics refers to techniques, procedures, the members of a given cultural group; detailed and strategies such as counting, ordering, sor- anthropological studies of the mathematics of ting, measuring, weighing, ciphering, classifying, distinct cultures most certainly allows us to further inferring, and modelling (D’AMBROSIO, 1993). our understanding of the internal logical system Ethnomathematics as a research paradigm is and beliefs of diverse group of students. Thus, wider than traditional concepts of mathematics, we consider ethnomathematics as the intersec- ethnicity, or any current sense of multiculturalism. tion of cultural anthropology, mathematics, and Ethnomathematics is described as the arts and mathematical modelling, which is used to help us techniques (tics) developed by individuals from understand and connect diverse mathematical diverse cultural and linguistic backgrounds (ethno) ideas and practices found in our communities to to explain, to understand, and cope with their traditional and academic mathematics. own social, cultural, environmental, political, and Consequently, Rosa and Orey (2010b, p. 60) economic environments (mathema) (D’AMBROSIO, consider ethnomodelling as the intersection 1990). Ethno refers to distinct groups identified by of three research fields: cultural anthropology, cultural traditions, codes, symbols, myths, and ethnomathematics, and mathematical modelling, specific ways of reasoning and inferring. which can be used “as a tool towards pedagogical Detailed studies of mathematical procedures action of an ethnomathematics program, students and practices of distinct cultural groups most have been shown to learn how to find and work certainly allow us to further our understanding with authentic situations and real-life problems”. of the internal logic and mathematical ideas of Figure 1 shows ethnomodelling as the intersection diverse groups of people. For example, Rosa of 3 (three) research fields: cultural anthropology, (2010) affirmed that principles of anthropology ethnomathematics, and mathematical modelling. Figure 1 – Ethnomodelling as the intersection of three research fields Source: Adapted from Rosa and Orey (2010). 4/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 According to Rosa and Orey (2010a), cultural the mathematical ideas, thoughts, concepts, anthropology studies distinct cultures and how procedures, and practices as developed by the their members shape the world around them by members of all cultures. From this perspective, studying similarities and differences regarding a body of anthropological research has come to the development of mathematical procedures focus on both the intuitive mathematical thinking and techniques. The goal for education is to learn and the cognitive process that are largely deve- how to collect data on how political, economic, loped in distinct cultural groups. social, environmental, and cultural practices to Following the above definition, Ethnomathe- understand how mathematics is influenced by matics may also be considered as a program the cultures that are studied. that seeks to study how students have come to Ethnomathematics seeks to study how stu- understand, comprehend, articulate, process, and dents have come to understand, comprehend, ultimately use mathematical ideas, concepts, pro- articulate, process, and ultimately use mathe- cedures, and practices that may solve problems matical ideas, procedures, and practices that related to their daily activities. Ethnomathematics enable them to solve problems related to their program and its pedagogical action of teaching daily activities. This helps students to reflect, and learning mathematics that is built on students’ understand, and comprehend extant relations tacit knowledge2, background, the role his envi- among all components of systems under study. ronment plays in terms of content, methods, and In this regard, educators should be empowered his past and present experience of his immediate to analyze the role of students’ ethnoknowledge environment (ROSA, 2010). in the mathematics classroom, which is acquired Ethnomathematics as applied in a school con- by students in the process of pedagogical action text involves taking students to situations that of learning mathematics in culturally relevant allow them to actively participate in culturally educational systems (ROSA, 2010). influenced activities in the classroom, such as Ethnomathematics researchers investigate house roofing, traffic, prices in a supermarket or ways in which members of distinct cultural groups neighborhood outdoor vegetable market, and the comprehend, articulate, and apply ideas, proce- representation of models of real-life situations in dures, and techniques identified as mathematical educational environments for purposes of interac- practices. In this regard, Rosa and Orey (2003) tion between the learner and materials to achieve affirmed that ethnomathematics uses cultural educational objectives (ROSA; OREY, 2016). experiences as vehicle to make mathematics In this regard, the arguments often given for learning more meaningful and to provide stu- using ethnomathematical examples in classrooms dents with the insights of mathematical know- are: (a) to show students of underrepresented ledge as embedded in their social and cultural cultures that their own cultures do contribute to environments. mathematical thinking and (b) to expose students Ethnomathematics empowers students intel- of majority cultures to diverse cultures from lectually, socially, emotionally, and politically by around the world, building respect for others using their sociocultural and historical realities and and generally contributing to global education. contexts to convey knowledge, impart academic These are certainly laudable goals, however, oc- skills, and change students’ attitudes towards aca- casionally ethnomathematicians have expressed demic instruction (ROSA; OREY, 2012). According concern that too often Western field research to Barton (1996), ethnomathematics embraces tends to seek the others to the extent of exploiting 2 Tacit knowledge is the unwritten, unspoken, and hidden knowledge held by members of distinct cultural groups, which is based on their emotions, experiences, insights, intuition, observations, and internalized information developed through the resolution of phenomena they face in their daily life. It is integral to the development of the consciousness of these members because it is acquired through association with members of other cultural groups and requires joint or shared activities to be imparted from one to another. It constitutes a set of informally developed knowledge and forms the underlying framework that makes explicit knowledge possible (ROSA; OREY, 2012). Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING indigenous cultures (ROSA; OREY, 2016) without giving them the voice that allows them to share and explain how they actually do the math. 5/19 communication (ROSA, 2010). This context enables the evolution of ethnomathematics as a research field in which one of In this context, Gavarrete (2014) states that one the main goals is to link local (emic) knowledge possible way to avoid this problem and, notably, to the mathematics curriculum (global, etic) by bring the goals of ethnomathematics even more applying innovative approaches to mathematics directly to students, is to encourage students through dialogue (glocal, cultural dynamism) to develop ethnomathematical studies of their (ROSA; OREY, 2017). own individual cultures, heritage, and personal It is important to discuss interrelated innovative interests. Therefore, if students make presenta- approaches in ethnomathematics programs, such tions to each other, they all learn about all the as their relation to social justice, civil rights, indi- cultures represented in the classrooms, not just genous education, professional contexts, game the one expressed by the formal curriculum and playing, urban and rural contexts, ethnotransdis- the textbook. Students from underrepresented ciplinarity, ethnopedology, ethnomethodology, groups can demonstrate the contributions of ethnomodelling, ethnocomputing, and the Trivium their cultural group. Curriculum (ROSA; OREY, 2017). Mathematical knowledge is perceived in an It is important to emphasize that researchers ethnomathematical perspective because tea- conducting ethnomathematical invetigations chers build on the students’ informal mathematics study the cultural congruence between the back- and direct the lesson toward their culture and ex- grounds of students, communities, and schools, periences while developing their critical thinking which in turn form one of the main principles of skills. This environment enables us to reflect on an ethnomathematics program. An important the nature of mathematics, culture, education, and characteristic of ethnomathematics is its trans- society and the relationships among them in order formational power, and how it can help to rethink to include pedagogical practices in the teaching the nature of mathematics (THOMAS; HART, 2010). and learning of mathematics that address deeper One possible purpose for ethnomathematical notions of equality and equity (GAVARRETE, 2014). studies, and its many innovative approaches, Examples can come from family traditions, could be to foster the development or transforma- hobbies, religions, and occupations; geography- tion of mathematics itself (D’AMBROSIO, 1993). To -based activities; celebrations of holidays and life remain relevant, mathematics instruction needs to events; personal interests such as sports, music, accommodate continuous and ongoing changes art, dance, or crafts; and even child-related ac- in students’ demographics in mathematics clas- tivities, from playground games and computer srooms around the world. Since ethnomathema- games to skateboarding, jumping rope, and bir- tics proposes that educators contextualize their thday parties. All bring the students’ attention to mathematics teaching and learning by relating cultures, and all show applications of mathematics the content to sociocultural experiences of their in context (GAVARRETE, 2014). students (ROSA, 2010). It is necessary for teachers to be supported to Hence, it has become necessary to integrate allow them to emphasize the connections betwe- diverse ethnomathematical perspectives into en mathematics and other curricular disciplines existing teacher education programs and encou- and consider students’ cultural backgrounds rage students to examine mathematical activities in designing and selecting mathematical acti- in their own sociocultural contexts. To help them vities. Students learn in ways characterized by to see, indeed realize how mathematics proce- social and affective approaches, harmony with dures and practices are not trivial and help them the community, holistic perspectives, field de- to connect what they are learning to their daily pendence, expressive creativity, and non-verbal lives. In this perspective, students may succeed 6/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 in mathematics when their understanding of it is of relevance, whereby enhancing meaningful linked to real and meaningful cultural referents connections and deepening their understanding and when instruction assumes that all students in mathematics. Ethnomathematics as part of are capable of mastering mathematics (ROSA; the school curriculum must reinforce and value OREY, 2007). the cultural knowledge of students rather than Ethnomathematics presents possibilities for ignore or negate it. educational initiatives and innovative curricu- Every culture has its own way of mathematizing lum objectives based on an ethnomathematical concepts which are part of its inheritance and the perspectives. However, one dilemma regarding result of the struggle for its survival (D’AMBROSIO, this issue is related to how we can prepare te- 1990). Throughout history, Mathematics has been achers to create curriculum activities based on used by different people in many ways. Egyptians ethnomathematics. One important approach to used geometry to construct pyramids for burial solve this dilemma is to focus on the importance purposes. Therefore, ethnomathematics may of promoting the dissemination of heritage as- be defined as how people of various cultures pects of local (emic), cultural, and mathematical use mathematics in their everyday life (ROSA; knowledge to help students strengthen their OREY, 2017). own cultural identities in school environments (GAVARRETE, 2014). Thus, it is necessary to highlight that the Trivium Curriculum for mathematics proposed by D’ Am- In this regard, Rosa and Orey (2003) affirmed brosio (1993) as an important, indeed innovative, that ethnomathematics uses cultural experiences ethnomathematics approach that needs more as vehicle to make mathematics learning more investigation in order to address pedagogical meaningful and to provide students with the purposes, as well as it helps to reach the propo- insights of mathematical knowledge as embe- sed objectives of ethnomathematics. The overall dded in their social and cultural environments. goal of this curriculum is to empower students Ethnomathematics contributes to restoring cul- through learning activities that help them develop tural dignity and offers the intellectual tools for literacy, numeracy, material and technological, the exercise of a citizen. social, and political skills in order for them to be In accordance with D’Ambrosio (1993), it enhan- active participants in a democratic society. ces creativity, reinforces cultural self-respect, and offers a broad view of mankind. In everyday life, it is a system of knowledge that offers the possibility of more favorable and harmonious relation between humanity and nature. In this approach, ethnomathematics aims at drawing from the learners’ cultural experiences and practices of the individual learners, the communities, and the society at large. Ethnomathematics uses cultural experiences as vehicles to make mathematics learning more meaningful and to provide students with the insight of mathematical knowledge as embedded in their social and cultural environments (ROSA; OREY, 2012). Ethnomathematics presents mathematical concepts of the school curriculum in such a way that they are related to students’ cultural and daily experiences, it creates a sense SOCIOCULTURAL PERSPECTIVE OF MATHEMATICAL MODELLING Mathematical modelling constitutes one of the most important research trends for the development of teaching and learning processes in mathematics (ROSA, 2010). In this regard, it is important to point out that this pedagogical action is directed towards the resolution of everyday problems and situations, with the use of modelling to encourage educators and learners to value and enable the connection between mathematics and the daily experiences lived by members of distinct cultural groups. In this context, Bassanezi (2002) states that mathematical modelling is the art of transforming reality problems into mathematical problems and solving them by interpreting their solutions Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING 7/19 in the language of the real world. According of members of distinct cultural groups can be to this approach, Rosa (2010) affirms that mo- considered as pedagogical tools that are used delling techniques provide the contextualization for the abstraction of mathematical concepts, be- of academic school/mathematics by providing cause member of distinct cultural groups develop necessary conditions to the development of its own set of ideas and mathematical concepts, pedagogical actions through the elaboration of among which some basic tools that are used in mathematical models, so that the members of the development of the modelling process stand. these cultural groups can act satisfactorily in the These tools can be understood as the ways glocalized world . 3 that members of each cultural group develop Accordingly, mathematical modeling presu- methods that allow them to deal with their own pposes the use of multidisciplinary approaches realities and to mathematize and model their because it has confluences with other trends world through the use of measurement, compa- in mathematics education. Ethnomathematics, rison, quantification, classification, and inference which, such as point to the removal of boundaries often uniquely developed locally. between the various areas of research (BASSA- From the perspective of Cortes (2017), this con- NEZI, 2002). Thus, there is a need to consider ma- text allows the exploration of ideas, procedures, thematics education as a scientific field directed and local mathematical practices, which aims to towards to the teaching and learning process in value and respect diverse cultural values and the mathematics through its resignification, which knowledge acquired by students through their enables the understanding and perception of its own experiences in society. Therefore, modelling importance in the development of this pedago- is an important tool to help students to unders- gical action (ROSA, 2010). tand, comprehend, analyze, and reflect on their Similarly, ethnomodelling relates the diversity own sociocultural contexts. In this regard, being of concepts inherent to ethnomathematics with proficient in the use of modeling is of fundamental mathematical modelling (ROSA; OREY, 2010b). importance so that members of distinct cultural For example, Caldeira (2007) states that it is ne- groups, through their actions, modify their own cessary to consider mathematics constructed reality so that they can be included in the process and signified in the cultural practices developed of social transformation in a critical and reflective by the members of local communities, as well way (ROSA; OREY, 2017). as the many influences of these meanings in For example, Rosa and Orey (2009) state that, the pedagogical process in order to apply ma- through the modeling process, it is possible to thematical modelling as a means to achieve the show that a key aspect of this process is to help objectives proposed for the conduction of this students realize their mathematical potential pedagogical action through its complementarity through the recognition of the importance of with ethnomathematics. culture for the appreciation of their own identity However, for this objective to be achieved, because this aspect influences the way they think, Rosa and Orey (2012) argue that there is a need for learn, reflect, infer, and takes informed decisions. students to be inserted in a learning environment Thus, Rosa and Orey (2012) state that ma- that enables them to use mathematical knowle- thematical modelling is a learning environment dge that was previously acquired at school and that facilitates the construction and transfer of developed and accumulated in the community mathematical knowledge through the use of in which they are inserted. their mathematical knowledge: a) explicit4 and Historically, models that originate in the reality tacit, which interact in this environment. In this A glocalized world enables the development of active, interactional, and dialogical processes that requires an ongoing negotiation between the local and the global mathematical, scientific, technological, and engineering knowledge through the development of a cultural dynamism (ROSA; OREY, 2017). 4 Explicit knowledge is related to a concrete fact, which can be disseminated by teachers through the use of textbooks, academic 3 8/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 context, members of distinct cultural groups have through ethnomathematics is considered as the developed and are developing diverse and diffe- study of ideas and procedures used in mathemati- rent ways of doing mathematics. Thus, D’Ambrosio cal practices, which were developed by members (1990) states that members of these groups have of different cultural groups when considering developed, throughout history, distinct ways to the acquired mathematical knowledge and cul- mathematize their own reality by using elements tural practices in the community with the use of of the modelling process. problem-solving techniques developed locally Therefore, mathematization is the process (ROSA; OREY, 2010b). through which members of distinct groups to use These techniques are considered as the basic different mathematical tools to help them organi- tools used by ethnomodelling that help teachers ze, analyze, understand, understand, model, and and researchers in carrying out the translation solve the problems faced in their daily lives (ROSA; between the emic and etic approaches (ROSA; OREY, 2006). These tools enable, according to OREY, 2016). Thus, ethnomodelling is a tool that Rosa and Orey (2017), the identification of ideas in aims to mediate the cultural forms of mathema- order to describe procedures and mathematical tics with the school curriculum to facilitate the practices specific to a cultural context, which aim development of its teaching and learning process. to help these members to discover relationships According to this context, mathematical mo- and regularities. delling is a teaching trend in mathematics edu- Hence, highlight that this cultural approach to cation that aims to develop critical and reflective modeling allows these members to schematize, students who are aware of the different problems formulate, and visualize problems and situations that are faced in their everyday life. However, for in different ways, which help them transcend the this objective to be achieved, there is a need for solution of real-world phenomena to mathema- students to be inserted in a learning environment tical conceptualization through the mathemati- that allows them to use of mathematical know- zation process (ROSA; OREY, 2003). ledge previously acquired at school and tacitly In this direction, Rosa and Orey (2017) state that in the community in which they are inserted. This the sociocultural perspective of mathematical approach will help students to contextualize cur- modelling involves the study of mathematical ricular activities in the daily life of the students ideas, procedures, and practices that are found (ROSA; OREY, 2007). in different cultural contexts so that they can be From this perspective, Rosa and Orey (2017) used in their pedagogical action in classrooms comment that modelling techniques provide through the elaboration of ethnomodels. the contextualization of school/academic ma- Thus, mathematical modelling procedures can thematics by providing the necessary conditions be employed when ethnomathematics is actively through the elaboration of mathematical models, used as a system based on a theoretical basis so that members of distinct cultural groups can that helps members of distinct cultural groups in act in the glocalized world. For example, Rosa the development of solving everyday problems and Orey (2007) state that this contextualization related to the social, cultural, economic, political, is an important concept for the development of and environmental contexts (ROSA; OREY, 2017). citizens students, as it offers an opportunity for The connection between mathematical modelling and the cultural aspects of mathematics the teaching of sociocultural efficiency5. In this regard, teachers have the responsibility approach about the subject, knowledge of pedagogical instructional practices, and any other method of using materials and technological instruments that can to help them to absorb, internalize and, consequently, transfer and diffuse the applicability of this knowledge to other areas of human knowledge (ROSA; OREY, 2012). 5 The fundamental characteristic of sociocultural efficiency is the emphasis on students’ critical analysis of society’s power structures. Another important feature is the students’ personal reflection on the social elements that underpin the globalized world. Thus, the critical perspective of students in relation to the social conditions that affect their own experiences can help them to identify common problems and, collectively, develop different strategies to solve them (ROSA; OREY, 2007). Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING to favor the establishment of relationships betwe- 9/19 cultural context. en school/academic mathematics and students’ Therefore, the sociocultural dimension of ma- tacit knowledge, so that they can perceive the thematical modelling has as background the presence of mathematics in the activities they social and cultural knowledge theories, which experience daily (ROSA; OREY, 2009). Thus, it are related to the emancipatory perspective is necessary that, in the classrooms, teachers and transformative learning that apply the phi- discard the traditional passive and transmissive losophical ideals of the critical thinking theory pedagogical models and favor the transformative (ROSA; OREY, 2007). Therefore, sociocultural pedagogical model. theory is related to learning processes that are Thus, Rosa and Orey (2007) state that the tra- triggered through socialization, as knowledge ditional teaching method predominant in the is better constructed when students interact to educational system tends to focus on the tradi- socialize learning. tional objective of learning for the transmission Thus, students act cooperatively and collabo- of mathematical knowledge. So, it is necessary ratively to support and encourage each other, so to discard this traditional model so that socio- that they can reflect on the resolution of complex cultural efficiency in education is implemented problems rooted in authentic situations (ROSA; in the classrooms. OREY, 2007). As in the mathematical modelling In this context, agreeing with the point of view process, it is important that students actively par- of Rosa and Orey (2012) who present modelling ticipate in the construction of their mathematical as a learning environment, in which teachers and knowledge by connecting it interdisciplinary with students are responsible for the development of other areas of knowledge in an interdisciplinary mathematical knowledge and for the conversion fashion (ROSA; OREY, 2017). between its tacit and explicit dimensions, from In the sociocultural theory, Rosa and Orey situations arising, preferably, from their own re- (2007) affirm that the joint work between teachers alities. and students makes learning more effective, be- From the perspective of Rosa and Orey (2007), cause cultural tools, such as artifacts, language, the conception of the role of students in this traditions, behaviors, and institutions are shared. approach is that of active collaborators in the So, the meaning of learning is constructed in the learning process, which is a more stimulating social context, as members of different cultural task than the one related to the simple reception groups learn together and collaboratively through of mathematical knowledge and practices. In integrated experiences. sociocultural mathematical modelling, students Thus, it is necessary for students to develop can be considered as creators of mathematical their abilities to solve problems, make decisions, knowledge, as this process provides the condi- work in teams, and communicate effectively, all tions for them to get involved with mathematics, of which are important characteristics of socio- so that they can challenge it, understand it, and cultural mathematical modelling. For example, interpret it by making it into a product of human Bassanezi (2002) states that the analysis of data creation. through statistics and the interpretation of results In this direction, Cortes (2017) states that le- determined in studies have contributed to direct arning is triggered according to the students’ the use of action strategies in commercial, social, purpose, as it develops differentiated capabilities environmental, and political contexts. so that they can act, react, reflect, and change the Consequently, Rosa and Orey (2007) argue environment in which they live by transforming it, about the need to apply the notions of the eman- strategically. Thus, this environment influences cipatory approach to mathematical modelling, the development of students’ cognitive process whose educational objectives address issues of in different ways, as it is related to their socio- a sociopolitical nature and their consequences 10/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 in the pedagogical practices used in school sys- relations found in measuring, calculation, games, tems. For example, Rosa and Orey (2007) claim divination, navigation, astronomy, modelling, and that this emancipatory approach is based on the a wide variety of other mathematical procedures sociocultural competence of members of distinct and cultural activity (EGLASH et al., 2006). cultural groups in which its main objective is to Researchers such as Eglash et al. (2006) and help students to face and solve challenges im- Rosa and Orey (2006) use the term translation to posed by the globalized society. describe the process of modelling local cultural Thus, this approach must be directed to trans- systems (emic/local), which may have a Wes- form students into flexible, adaptable, reflective, tern school/academic mathematical represen- critical, and creative citizens using alternative tation (etic/global). This context allows for the pedagogical methodologies that aims to value translation of interpretations and contributions and record ideas, procedures, and mathematical of ethnomathematical knowledge into systemi- practices that are developed in distinct cultural zed mathematics as students learn to construct contexts (ALVES, 2014). Therefore, Rosa and Orey their own connections between both traditional (2007) state that the sociocultural aspect of mo- (global/etic) and non-traditional (local/emic) delling is based on the expansion of students’ learning settings through translations and sym- autonomy, as it aims to provide a critical reading metrical dialogues. of their worldview, as well as for the development In this regard, ethnomathematics makes use of their autonomous thinking, which aims to con- of modelling by attempting to use it to establish tribute to the full exercise of their citizenship. relations between the local conceptual framework (emic/local) and the mathematics embedded in DISCUSSING ETHNOMODELLING Ethnomodelling is the study of mathematical ideas and procedures elaborated by members of distinct cultural groups. It involves the mathematical practices developed, used, practiced, and presented in diverse situations in the daily life of the members of these groups (ROSA; OREY, 2010a). This context is holistic and allows those engaged in this process to study mathematics as a system taken from their own contextual reality in which there is an equal effort to create an understanding of all components of these systems as well as the interrelationship among them (D’AMBROSIO, 1993; BASSANEZI, 2002; ROSA; OREY, 2003). Investigators and educators such as Ascher (2002), Eglash (1999), Orey (2000), Urton (1997), and Rosa and Orey (2009) “have revealed [in their research] sophisticated mathematical ideas and practices that include geometric principles in craft work, architectural concepts, and practices in the activities and artifacts of many indigenous, local, and vernacular cultures” (EGLASH et al., 2006, p. 347). These concepts are related to the numeric relation to local designs. On the other hand, many indigenous designs, such as the applications of symmetry classifications from crystallography to indigenous textile patterns, have been analyzed from a Western view (etic/global). In some cases, Eglash et al. (2006, p. 347) state that “the translation to Western mathematics is direct and simple such as counting systems and calendars”. However, there are cases in which mathematical ideas and concepts are “embedded in a process such as iteration in bead work, and in Eulerian paths found in sand drawings” (EGLASH et al., 2006, p. 348). Thus, the act of translation applied in this process is best referred to as ethnomodelling. In this process “mathematics knowledge can be seen as arising from emic rather than etic origins” (EGLASH et al., 2006, p. 349). This means that the mathematics we use in academic/scientific contexts was not conceived as a universal language because its principles, concepts, and foundations were not the same everywhere (ROSA, 2010). In this regard, any choice among equivalent representational systems can only be founded on considerations of simplicity, for no other consideration can adjudi- Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING 11/19 cate between equivalent systems that univocally the conceptual schemes and categories that are designate reality (CRAIG, 1998). regarded as meaningful and appropriate by the In this context, Rosa and Orey (2006) affirm community of scientific observers, researchers, that the processes of production of mathematical and investigators (LETT, 1996). An etic construct ideas, procedures, and practices operate in the is precise, logical, comprehensive, replicable, and register of the interpretative singularities regar- observer-researcher independent. ding the possibilities for a symbolic construction Any validation of etic approach thus becomes of knowledge in different cultural groups. This a matter of logical and empirical analysis, in par- context allows for the translation of interpretations ticular, contains the logical analysis of whether and contributions of ethnomathematical knowled- the construct meets the standards of comprehen- ge into systemized mathematics as students learn siveness and logical consistency, and then the to construct their own connections between both empirical analysis of whether or not mathematical traditional and non-traditional learning settings concepts have been replicated (LETT, 1996). through translations and symmetrical dialogue. It is important to emphasize that particular research techniques used in the acquisition of The Emic and Etic Constructs of Ethnomodelling In the ethnomodelling approach, the emic constructs are the accounts, descriptions, and analyses expressed in terms of the conceptual schemes and categories that are regarded as meaningful and appropriate by the members of the cultural group under study. This means that an etic mathematical knowledge has no bearing on the nature of that knowledge. The etic approach may be obtained at times through elicitation as well as observation (LETT, 1996). According to D’Ambrosio (1990), investigators and educators must acknowledge and recognize that local people possess scientifically and mathematically valid knowledge. emic construct is in accordance with the perceptions and understandings deemed appropriate Dialogical Approach of Ethnomodelling by the insider’s culture. The validation of emic If we make an analogy in regard to ethnomo- approach comes with a matter of consensus of delling, it is possible to state that its emic (local) local people who must agree that these cons- approach is concerned about differences that tructs match the shared perceptions that portray make mathematical practices unique from an the characteristic of their culture (LETT, 1996). insider’s point of view. We argue that emic eth- In other words, the emic approach investigates nomodels are grounded in what matters in the mathematical phenomena and their interrela- mathematical world of those being modeled. tionships and structures through the eyes of On the other hand, many ethnomodels are etic the people themselves. It is important to note in the sense that they are built on an outsider’s that research techniques used in acquiring emic observation of the others about the world being mathematical knowledge have nothing to do modeled. with the nature of that knowledge. In this regard, In this context, etic (global) ethnomodels re- “emic mathematical knowledge may be obtained present how modelers themselves think the either through elicitation or observation because world works through systems taken from reality it is possible that objective observers may infer while emic ethnomodels represent how people local perceptions” (LETT, 1996, p. 382) about who live in such worlds think these systems work mathematical ideas, procedures, and practices. in their own reality. We also would like to point It is necessary to state that etic constructs out how etic approaches play important roles in include individual accounts, descriptions, and ethnomodelling research, yet the emic approach analyses of mathematical ideas, concepts, pro- should be also taken in consideration. cedures, and practices expressed in terms of In this perspective, the emic ethnomodels 12/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 sharpen the question of what an agent-based analysis focuses on a single culture and employs model should include to serve practical goals in descriptive and qualitative methods to study a modelling research. Thus, mathematical ideas mathematical idea, concept, procedure, or prac- and procedures are etic if they can be compared tice of interest. Its focus is on the study within the across cultures using common definitions and cultural group context in which the researcher metrics while the focus of the analysis of these tries to develop research criteria relative to inter- aspects are emic if the mathematical ideas, con- nal characteristics or logic of the cultural system. cepts, procedures, and practices are unique to a In this perspective, meaning is gained relative to subset of cultures that are rooted on the diverse the context and therefore not easily, or of at all ways in which etic activities are carried out in a transferable to other contextual settings. specific cultural setting. For example, it is not intended to compare the Currently, the debate between emic-etic is observed mathematical patterns in one setting one of the most intriguing questions in ethnoma- with mathematical patterns in other settings. This thematics and mathematical modelling research means that the primary goal of an emic approach since researchers continue to explore questions is a descriptive idiographic orientation of mathe- such as: matical phenomena because it puts emphasis on the uniqueness of each mathematical idea, 1. Are there mathematical patterns that are identifiable and/or similar across cultures? procedure, or practice developed by the mem- 2. Is it better to focus on these patterns particularly arising from the culture under investigation? highlight meanings of these generalizations in 3. How does data gained from outsiders differ from that of the insiders in relation to emic (local)-etic (global) research data? contrast, an etic analysis would be comparative, Emic (local) and etic (global) approaches are identify lawful relationships and causal explana- often thought of as creating conflicting dicho- tions valid across different cultures. Thus, if rese- tomies. For example, Pike (1954) originally con- archers and educators wish to make statements ceptualized them as complementary viewpoints. about universal or etic aspects of mathematical According to this context, rather than posing a knowledge, these statements need to be phrased dilemma, the use of both approaches deepens in abstract ways. bers of cultural groups. Thus, if researchers and educators wish to local or emic ways, then they need to refer to more precisely specified mathematical events. In examining many distinct mathematical cultural practices by using standardized methods (LETT, 1996). This means that the etic approach tries to our understanding of important issues in scien- On the other hand, an etic (global) approach tific research and investigations (BERRY, 1999). may be a way of getting at the emics of the mem- A suggestion for dealing with this dilemma is to bers of cultural groups because it may be useful use a combined emic-etic approach, rather than for penetrating, discovering, and elucidating emic simply applying emic or etic dimensions of one (local) systems that were developed by members culture to other cultures. of different cultural groups (PIKE, 1954). In so A combined emic-etic approach requires re- doing, while traditional emic and etic concepts searchers to first attain emic approach about the are important points of view for understanding cultural groups under study. This may allow them and comprehending cultural influences on ma- to become aware of and then put aside cultural thematical modelling, we would like to propose a biases, and to become familiar with the relevant distinctively different view on ethnomathematics cultural differences in each setting (BERRY, 1999). and modelling research, which is referred as a Usually, in ethnomodelling research, an emic dialogical approach (MARTIN; NAKAYAMA, 2007). Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING 13/19 In this approach, the etic approach claims However, we argue that traditional mathe- that the knowledge of any given cultural group matical modelling does not fully consider the has no priority over its competing emic claims. implications of cultural aspects of human social According to this point of view, it is necessary to systems. It also serves to introduce learners to depend “on acts of ‘translation’ between emic increasingly powerful and formal mathematical and etic approaches” (EGLASH et al., 2006, p. knowledge needed to resolve more and more 347). In other words, the cultural specificity may complex problems through formal modelling. be better understood with the background of The cultural component in this process is cri- communality and the universality of theories and tical because its accounts “emphasize the unity methods and vice versa. of culture, viewing culture as a coherent who- In this context, these insights must be verified le, a bundle of [mathematical] practices and with methods independent of the subjectivity of values” (POLLAK; WATKINS, 1993, p. 490) that the observer and researcher to achieve a scientific are incompatible with the rationality of the ela- character. In so doing, it is important to analyze boration of traditional mathematical modelling the insights that have been acquired through process. However, in the context of mathematical subjective and culturally contextualized methods. forms of knowledge, what is meant by the cul- The rationale behind the emic-etic dilemma is the tural component, varies widely and ranges from argument that mathematical phenomena in their viewing mathematical practices as learned and full complexity can only be understood within the transmitted to the members of cultural groups context of the culture in which they occur. to formal academic practices viewed as made up of abstract symbolic systems with an internal Mathematical Phenomena and Ethnomodels Through history, many researchers, and investigators made extensive use of mathematical procedures ranging the encountered as Europeans colonized and extended their empires through colonial trade connections. The diverse procedures they encounters originated from geometrical and statistical methods, methods for the elucidation of patterns in behavior, and to the mathematical representations and logic processes of indigenous conceptual systems they encountered for better or worse (ROSA; OREY, 2017). Mathematical modelling has been considered by some metaphorically as a pedagogical tool and by others as a way to understand anthropological and archaeological research. Yet, others have decried the use of mathematical, and in particular, statistical and quantitative modelling as fundamentally in opposition to a humanistic approach to understanding human behavior and knowledge that takes into account contingency and historical embeddedness and in turn, decries universality. logic giving a symbolic system its mathematical structure. If the former is considered, then it is the process by which knowledge transmission takes place from one person to another, which is central to elucidating the role of culture in the development of mathematical knowledge (D’AMBROSIO, 1993). If the latter is considered, then culture plays a far more reaching and constructive role with respect to mathematical practices that cannot be induced simply through observation of these practices (EGLASH et al., 2006). In this regard, mathematical knowledge developed by members of a specific cultural group consists of abstract symbol systems whose form is the consequence of a unique internal logic. Then, people learn specific instances of and definite usages of the symbol system as well as what is derived from those instances and forms of cognitively based understanding of the internal logic of mathematical symbolic systems. The cognitive aspects needed in this framework are primarily decision processes by which members of cultural groups either accept or reject an ethnomodel as part of their own repertoire of 14/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 mathematical knowledge. We believe that the practices found in sociocultural systems (ROSA; conjunction of these two scenarios appear to be OREY, 2010a), which link cultural heritage with the adequate to the depth needed to encompass the development of mathematical practice. It is our full range of cultural mathematical phenomena. In understanding that this approach may help the effect, there are two ways in which we recognize, organization of pedagogical action that occurs represent, and make sense of a mathematical in classrooms using the emic and etic aspects phenomenon that impinges upon our sensory of mathematical knowledge found in the school apparatus: community. a) First, there is a level of cognition that we share, to varying degrees, with members of our own and other cultural groups. This level would include cognitive models that we may elaborate on at a non-conscious level, which serves to provide an internal organization of external mathematical phenomena to provide the basis upon which a mathematical practice takes place. b) Second, there is a culturally constructed representation of external mathematical phenomena that also provides its internal organization. However, this representation arises through the formulation of abstract and conceptual structures that provide forms and organizations for external phenomena. In other words, cultural constructs provide re- Ethnomodels Culture is a lens that shapes reality, as well as it is a blueprint that specifies a plan of action or expectations. At the same time, there are aspects of a culture that are unique to the members of distinct cultural groups, who together have grown, learned, and act daily in diverse contexts, such as economic, social, cultural, political, and environmental in which they live (ROSA; OREY, 2017). According to Rosa and Orey (2010a), research in ethnomodelling is linked to mathematical practices developed by members of different cultural groups that tend to be of benefit to the presentation and organization of mathematical ideas and procedures that enable the development of communication, diffusion, and further transmission through generations (emic approach). presentations for systems taken from reality. The The representative idea of this approach gi- implications for mathematical modelling are that ves room for the development of mathematical models of cultural constructs are considered as knowledge through scientific methods that may symbolic systems organized by the internal logic help researchers and educators to build and of members of cultural groups. We agree with understand the world (etic approach) by using Eglash et al. (2006) and Rosa and Orey (2010b) small units of information called ethnomodels who argued that models built without a first-hand that make up the set of these representations sense for the world being modeled should be (ROSA; OREY, 2010b). viewed with suspicion. In this regard, ethnomodels are considered Investigators and educators, if not blinded by cultural constructs because one of the main their prior theory and ideology, should come out objectives of its elaboration is to comprehend with an informed sense of distinctions that make the way of thinking of these members, as well a difference from the point of view of the mathe- as to understand how they organize and model matical knowledge of the work being modeled. In their mathematical ideas and procedures from so doing, they should, in the end, be able to tell their own point of view in order to mathematize outsiders (etic) what matters to insiders (emic). their own reality (ROSA; OREY, 2012). On the other Ethnomodelling respects the organization and hand, a model built without a first-hand sense presentation of mathematical ideas and proce- for the world being modeled should be viewed dures developed by the members of distinct with suspicion. cultural groups by constructing ethnomodels of Researchers and educators, if not hindered by Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING 15/19 their prior ideology, paradigms, cosmologies, and ders the processes that help the construction and worldviews, should come out with an informed development of local mathematical knowledge sense of the distinctions that are effective from systems, which include collectivity, creativity, and the point of view of the phenomena being mo- inventively (ASCHER, 2002) through the elabora- deled. In so doing, they should be able to inform tion of ethnomodels. the outsiders (etic/global) what matters to the insiders (emic/local) (ROSA; OREY, 2017). According to this approach, it is impossible to imprison mathematical ideas, procedures, and Ethnomodelling emphasizes the organization practices in registers of univocal designation of and presentation of mathematical ideas and pro- reality because there are distinct systems that cedures developed by the members of distinct provide unambiguous representations of reality cultural groups to facilitate its communication as well as universal explanations (CRAIG, 1998). and transmission across generations, which adds This means that mathematics cannot necessarily cultural aspects to the modelling process. In this be conceived as a universal language because its regard, these members construct ethnomodels principles are not always the same everywhere of their mathematical practices found in their around the world (ROSA; OREY, 2007). sociocultural systems, which link their cultural Or at very least, it serves like different dialects heritage to the development of the greater ma- or accents in the greater mathematical language thematical practices (ROSA; OREY, 2017). of humanity. In accordance with this context, This approach helps the organization of pe- the production process of mathematical ideas, dagogical action in classrooms by using emic procedures, and practices operates within the (local) and etic (global) aspects of mathematical register of interpretative singularities regarding knowledge through the development and elabo- possibilities for symbolic construction of local ration of ethnomodels, which are described as mathematical knowledge (ROSA; OREY, 2012). cultural artifacts that are the pedagogical tools Ethnomodelling studies local mathematical used to enable the understanding of systems processes developed by the members of distinct taken from the reality of the members of distinct sociocultural groups. Many interesting ethnomo- cultural groups (ROSA; OREY, 2016). dels have been formulated by using data obtai- In this regard, ethnomodels may be considered ned from studies related to ethnomathematics, as external representations that are precise and and which propose a rediscovery of knowledge consistent with the 2014scientific and mathe- systems adopted by the members of diverse matical knowledge that is socially constructed, groups (BASSANEZI, 2002; ROSA; OREY, 2012). developed, and shared by members of specific When this knowledge applies mathematical cultural groups. One objective for the elaboration ideas and procedures through the elaboration of ethnomodels is to translate emic constructs of ethnomodels, we can understand the origin which highlight the unique mathematical ideas, of mathematical practices more efficiently. procedures, and practices in order to establish re- In ethnomodelling research, emic (local) cons- lations between local conceptual knowledge and tructs represent the accounts, descriptions, and the mathematics embedded in these constructs analyses of mathematical ideas, procedures, through dialogical relations (EGLASH et al., 2006). and practices expressed in terms of conceptual Thus, ethnomodels help to link the develo- schemes and categories that are regarded as pment of mathematical practices to the cultu- meaningful and appropriate by members of the ral heritage of the members of distinct cultural cultural group under study. This means that emic groups, who detain necessary information to solve constructs are in accordance with the percep- problems and situations described in systems tions and understandings deemed appropriate taken from their own reality (ROSA; OREY, 2016). by the insider’s culture. The validation of these The emphasis of ethnomodelling research consi- constructs comes with a matter of consensus 16/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 from those who do the mathematics under study, ral cross-cultural concepts (ROSA; OREY, 2010a). in which local people who must agree that emic On the other hand, Rosa and Orey (2012) state constructs match shared perceptions, behaviors, that etic ethnomodels are mathematical repre- and knowledge that portray characteristics of sentations elaborated through a descriptive and their culture (LETT, 1996). external observations. Thus, the representation Emic (local) mathematical knowledge can be of local mathematical knowledge is developed obtained through elicitation and observation be- when the members of different cultural groups cause observers infer local perceptions. In emic have their own interpretation of their culture (local) approaches, researchers and educators (emic approach) as opposed to the researchers’ must put aside their own bias, prior theories, and interpretation and researchers who develop re- assumptions to let those who do the activity un- presentations of mathematical knowledge place der study to explain, and allow for understanding from the perspective of their own conceptions mathematical themes, patterns, and concepts (etic approach). that emerge locally. Some of its strength lies in the appreciation of the uniqueness of the context being studied in its respect for local viewpoints, and its potential to uncover unexpected mathematical findings (ROSA; OREY, 2012). For example, Lett (1996) states that etic constructs are accounts, descriptions, and analyses of mathematical ideas, procedures, and practices expressed in terms of conceptual schemes and categories that are regarded as meaningful and appropriate by the community of scientific observers. An etic approach uses these concepts as starting point theories, hypothesis, perspectives, and concepts from outside of the cultural setting being studied, which are developed by researchers and educators. Etic constructs are precise, logical, comprehensive, replicable, and observer-researcher independent (ROSA; OREY, 2017). The validation of etic approaches becomes a matter of logical and empirical analysis, in particular, the logical analysis of whether the construct meets the standards of comprehensiveness and logical consistency of concepts (LETT, 1996). It is important to emphasize that the particular research technique used in the acquisition of scientific and mathematical knowledge has little to no bearing on the nature of that knowledge. Etic (global) approaches may be obtained at times through elicitation, as well as observation. One of the strengths of the etic (global) approach is that it allows for comparison across contexts and populations, and the development of more gene- THE DIALOGICAL APPROACH INTO A MATHEMATICS CURRICULUM Mathematical knowledge of the members of cultural groups combined with Western-mathematical knowledge systems may result in a dialogical approach to mathematics education. An emic analysis of a mathematical phenomenon is based on internal structural or functional elements of a particular cultural group while an etic analysis is based on predetermined general concepts external to that cultural group (LOVELACE, 1984). The emic (local) approach provides both internal conceptions and perceptions of mathematical ideas and concepts while the etic approach provides the framework for determining the effects of those beliefs on the development of the mathematical knowledge. In this perspective, the acquisition of mathematical knowledge is based on the applications of current mathematics curriculum, which may be assessed based on multiple instructional methodologies across various cultures. In this regard, it could indeed be that one of the reasons for failure in many educational systems is that curriculum developers by using a one size fits all program, have ignored unique emic (local) approach in the school cultures. A dialogical approach includes the recognition of other epistemologies, and of holistic and integrated natures of mathematical knowledge of members of the diverse cultural groups found in many schools and urban centers. In other words, an ethnomo- Daniel Clark Orey REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING delling curriculum provides an ideological basis 17/19 approach to etic approach and vice versa. for learning with and from the diverse cultural In a dialogical context such as outlined here, and linguistic elements presented by members an ethnomodelling curriculum provides the un- of distinct cultural groups (ROSA; OREY, 2010a). derlying philosophy for knowledge generation In this kind of curriculum, it would be crucial and exchange within and between all subsystems to understand that an etic construct is a mathe- of mathematics education. Key elements of an matical-theoretical idea that is assumed to apply ethnomodelling curriculum approach ensure the in all cultural groups while an emic construct is balanced integration of the affective domain of one that applies only to members of specific educational objectives that are essential to the cultural groups. This means that there is concern recognition and utilization of the students’ pre- for cultural bias occurring if educators and re- vious knowledge. searchers assume that an emic construct is etic (EGLASH et al., 2006), which results in an imposed etic approach in which a culture-specific idea is wrongly imposed on the members of another cultural group. An ethnomodelling curriculum that combines key elements of local and academic approach in a dialogical approach is likely to produce students who can manage knowledge and information systems taken from their own reality and creatively apply this knowledge to other situations. This means that ethnomodelling can be considered part of a critical mathematics education because it provides a learning process in which teachers encourage a critical examination of multiple sources of knowledge and theories found in diverse learning styles. In this approach, acquired knowledge is centered, located, oriented, and grounded on the cultural background/context of the students, which could be applied and translated appropriately by them and thus equip them to be fully productive locally and globally. According to Rosa and Orey (2010b), ethnomodelling is a pedagogical approach to reach this goal. The nature of the previous mathematical knowledge of the students lends themselves to the principle of sequencing in curriculum development. By giving educators the freedom to start with previous mathematical knowledge and experience of their students, we can move from the familiar to the unfamiliar and from the concrete to the abstract in the process of promoting the acquisition of mathematical knowledge (ROSA; OREY, 2006). As well we can move from emic FINAL CONSIDERATIONS As we discussed in this article, it is important to highlight that an emic (local) observation of mathematical practices sought to understand them from the perspective of the internal dynamics and relationships as influenced within the culture of its members. On the other hand, an etic approach provides a cross-cultural contrast and comparative perspectives by using some aspects of academic mathematics to translate this phenomenon in order to amplify the understanding of those from a different cultural background. This approach is necessary to comprehend and explain this mathematical practice as a whole from the point of view of that from the outside. In this context, the emic (local) approach clarifies the intrinsic cultural distinctions of mathematical procedures and techniques while the etic (global) approach seeks objectivity as an outside observer across cultures regarding the development of mathematical practices. This is the dialogical approach, which concerns the stability of relationships between two distinct cultural approaches. It is important to state here that both perspectives are essential to understanding and comprehending human behaviors (PIKE, 1954), which help members of distinct cultural groups to shape mathematical ideas, procedures, and practices they developed overtime through history. As well, it is necessary to state that one of the latest trends in mathematics education points out to the need to integrate the teaching of this science with other knowledge areas in an interdisciplinary fashion at all levels of education. For this 18/19 Educação por escrito, Porto Alegre, v. 14, n. 1, p. 1-19, jan.-dez. 2023 | e-45054 process to be successful as well as mathematics emic and etic approach and understandings to be valued as a discipline whose contents can through the processes of dialogue. be considered as a human creation, it is necessary Finally, we define ethnomodelling as the study to understand and modify the environment we of mathematical phenomena within a culture live. In this regard, we can use ethnomodelling because it is a social construction, it is culturally to link theory into practice by the inclusion of bound, and it adds the culturas features of mathe- the dialogical approach into the mathematics matics into the mathematical modelling process. curriculum. Where other than academic forms of mathematics are valued, protected, archived and most importantly shared equally. Defined in this manner, the usefulness of the emic (local) and etic (global) distinction seems to be evident. For example, like all human beings, researchers, and educators, have been enculturated to some particular cultural worldview. They therefore need a means of distinguishing between the answers they derive as enculturated members of distinct cultural groups and the answers they derive as observers. Defining emics (local) and etics (global) in epistemological terms provides a reliable means of making that distinction. In this perspective, culture is a lens that shapes our reality and it can be considered as a blueprint that specifies a plan of action. At the same time, a culture is unique to a specific group of people. Thus, by conducting investigations provided by both approaches: emic (local) and etic (global), we acquire a more complete understanding of the culture of the members of distinct cultural groups. According to the discussion provides n this article, we have offered an alternative goal for the conductions of research in mathematics education, which is the acquisition of both emic (global) and etic (global) approaches for the implementation of ethnomodelling. Emic (local) knowledge is essential for an intuitive and empathic understanding of mathematical ideas of a culture, and it is essential for conducting effective ethnographic fieldwork. On the other hand, etic (global) approach is essential for cross-cultural comparison, the essential components of ethnology, because such comparison necessarily demands standard units and categories. We also offered here a third approach on ethnomodelling research, which is the dialogical approach that makes use of both REFERENCES ALVES, G. M. 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Endereço para correspondência: DANIEL CLARK OREY MILTON ROSA Rua Diogo de Vasconcelos, 122 Pilar, Ouro Preto, 35402-163 Minas Gerais, MG, Brasil Os textos deste artigo foram revisados pela Texto Certo Assessoria Linguística e submetidos para validação dos autores antes da publicação.