Ethnomodelling as the Translation of DiverseCultural Mathematical Practices

Milton Rosa and Daniel Clark Orey

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Ethnomathematics and Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Exploring Ethnomodelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Ethnomodelling and its Three Approaches of Viewing Cultures . . . . . . . . . . . . . . . . . . . . . . . . 9

Etic: The Global/Outsider Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Emic: The Local/Insider Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Dialogic: The Glocal/Emic-Etic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Characterizing Ethnomodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Emic and Etic Ethnomodels of the Mangbetu Ivory Sculpture . . . . . . . . . . . . . . . . . . . . . . . . 16An Etic Ethnomodel of Brazilian Roller Carts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A Dialogic Ethnomodel of a Local Farmer-Vendor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Relevance of Ethnomodelling in a Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Abstract

One of the major dilemmas in mathematics education in contemporary societyis its hidden bias towards a western orientation in its scholarly and researchparadigms. While being mindful of emerging glocalization of science, math-ematics, religion, art, music, and other aspects of a given culture, the useof innovative or what may seem alternative approaches and methodologies isnecessary to record historically diverse forms of mathematical knowledge thatoccur in distinct cultural contexts. With 500 years of colonization of scienceand mathematics, it seems important at this critical stage of human development

M. Rosa (�) · D. C. OreyDepartamento de Educação Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MinasGerais, Brazile-mail: milrosa@hotmail.com; milton.rosa@ufop.edu.br; oreydeema@gmail.com

© Springer Nature Switzerland AG 2019B. 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 havecome to apply fundamentally different philosophies, modelling techniques, andan ethnomathematical perspective to the mathematics curriculum. It is the linkingof mathematics and culture that the authors find appropriate and necessary for adeeper understanding of the development of mathematical knowledge aimed atproviding a holistic comprehension of human behavior. It is important to developan understanding of the role of ethnomathematics and modelling processes in thedevelopment of an innovative theoretical basis for ethnomodelling, which usesemic, etic, and dialogic approaches in its investigation process. In this theoreticalchapter, the authors demonstrate how ethnomodelling is a pedagogical action forthe process of teaching and learning mathematics that challenges the prevailingway of the universality of mathematics and the thinking involved therein.

KeywordsCultural groups · Ethnomathematics · Ethnomodelling · Modelling ·Pedagogical action

Introduction

Throughout history, traders, navigators, and explorers studied members from othercultures and shared knowledge often hidden or embedded in religious traditionsthat 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 acquiredthrough the course of their lives, which enabled them to succeed more thanindividuals from a less experienced background. It also functions as a social relationwithin a system of exchange that includes the accumulated sociocultural knowledgethat confers power and status to the individuals who possess it (Rosa 2010).)enriched all cultures when their members were engaged in a constant, dynamic, andnatural process of evolution and growth through the process of cultural dynamism(Cultural dynamism refers to the exchange of systems of knowledge that facilitatemembers of distinct cultures to exploit or adapt to the world around them. Thiscultural dynamic facilitates the incorporation of human invention, which is relatedto changing the world to create new abilities and institutionalizing these changesthat serve as the basis for developing more competencies (Rosa and Orey 2016).).

For example, the Greek foundations of European civilization were themselvesdeveloped through interaction with the Egyptian civilization (Powell and Franken-stein 1997). One consequence of this recognition is a widespread consensus towardsthe supremacy of Western scientific and logical systems at the exclusion of manyother traditions developed in diverse contexts.

In mathematics, as in many other academic subjects, methods of problem-solvingand teaching materials are based on the traditions of the written sciences and, withvery few exceptions, are defined by Western academia and science. Most examplesused 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 itis important to highlight how cultures and societies considerably affect the wayindividuals come to understand and comprehend concepts of their own mathematicalideas, procedures, and practices. According to D’Ambrosio (1999), this interactionis in danger of leaving out a significant amount of knowledge and supports forms ofcolonization that are subtle and often go unnoticed.

By observing this context, D’Ambrosio (2006) demonstrates how the cultureof a group results from the fraction of reality that is reachable by its members.However, the multiplicity of and constant interactions between members of distinctcultural groups and their unique cultural contexts, each one with a system of sharedexperience, history, and knowledge and an equally compatible set of behavior andvalues, facilitates the development of unique set of cultural dynamics by enablingan expanding familiarity with a rich diversity of humanity. This has created animportant need for a field of research that studies phenomena and the applicationof modelling techniques developed in diverse cultural settings.

This cultural perspective is applied to the development of problem-solvingtechniques, conceptual categories, and structural methods used to elaborate modelsthat represent data to translate mathematical practices by using modelling processes.The authors refer to this process as ethnomodelling (Bassanezi 2002; Rosa and Orey2010) that is one way in which they can recognize, through their lens of Westernmathematical experience, how its foundations differ from the traditional modellingmethodologies.

The authors’ sources are firmly grounded and rooted in the theoretical basisof ethnomathematics (D’Ambrosio 1985), and they have found that the culturallybound views of mathematical modelling support the assumption that research ofculturally bound modelling processes addresses issues of mathematics educationby bringing the diverse backgrounds of learners into the mathematics curriculumby connecting it to the local and cultural aspects of the school community to theprocess of teaching and learning of mathematics.

Ethnomathematics and Modelling

The authors have seen that many models arising from reality have become the firstpaths that have provided numerous abstractions of deeper mathematical concepts.Ethnomathematics can use these models taken from reality and modelling as atranslation to incorporate the codifications provided by the members of distinctcultural groups in order to understand mathematical ideas and procedures developedin other mathematical systems (D’Ambrosio 1993; Rosa and Orey 2003).

For us, mathematical modelling becomes a concrete methodology closer to anethnomathematics program (D’Ambrosio 1990; Rosa and Orey 2006), which isdefined 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

MathematicalModelling

Ethnomathematics

Mathematics

PracticesContexts

ExplainAnalyzeInterpret

Cultural Anthropology

Fig. 1 Ethnomathematics as an intersection between three research fields. (Source: Rosa and Orey2010)

daily problems (D’Ambrosio 2000; Rosa 2000). Figure 1 shows ethnomathematicsas an intersection between cultural anthropology, mathematics, and mathematicalmodelling.

Investigations in modelling have been found to be useful in the translation(Translation is an important transfer takes place when two cultures meet andinteract, as the language, scientific, and mathematical knowledge of one culturalgroup pass into the interpretative realm of another. In this process, the translation ofmathematical ideas, procedures, and practices of the studied culture is understoodand comprehended through dialogic terms that are different in temporal and specialframes and is transformed (Rosa and Orey 2017).) of ethnomathematical contextsby 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 andideas 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 learningof contextualized mathematics (Barbosa 1997; Bassanezi 2002; Biembengut andHein 2000; Hodgson and Harpster 1997; Orey 2000).

For example, outside of the community of ethnomathematics researchers, it isknown that many scientists search for mathematical models that translate theirdeepening 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 andquestions that grow from real situations (systems) and forms an image or sense of anidealized version of the mathema. (According to D’Ambrosio (1985), mathema isconsidered as the actions taken by the members of distinct cultural groups to explainand understand the world around them. Thus, they must manage and cope with theirown reality in order to survive and transcend. Throughout the history of mankind,technes (or tics) of mathema have been developed in very different and diversifiedcultural environments, that is, in the diverse ethnos. Thus, in order to satisfy the drivetowards survival and transcendence, human beings have developed and continue todevelop, in every new experience and in diverse cultural environments, their ownethnomathematics.) According to Rosa and Orey (2010), this perspective essentiallyforms a critical analysis for the generation and production of knowledge (creativity)and develops the intellectual process for its production, the social mechanisms ofinstitutionalization 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. Inthis holistic context, modellers study systems taken from reality in which there isan equal effort made to create an understanding of their components as well as theirinterrelationships (Bassanezi 2002; Rosa 2000).

By having started with a social or reality-based context, the use of modellingas a tool begins with the knowledge of the student by developing their capacity toassess the process of elaborating a mathematical model in its different applicationsand contexts (D’Ambrosio 2000). This uses the reality and interests of studentsversus the traditional model of instruction, which makes use of external values andcurriculum 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 arepresent 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 aboutone individual. But there are many other individuals ( . . . ) from the most variedspecies, going through a similar process. For living individuals, the cycle is thesame: → reality → individual → action → reality → individual → action →”(p. 5). In this context, “individual agents are permanently receiving informationand processing it and performing action. But although immersed in a same globalreality, 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 includingnatural, material, social, and psycho-emotional characteristics. This context enablesthe development of linkages among these three elements of the cycle through themechanism of information, which includes both sensory and memory capabilitiesthat 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, whenan abstraction is treated as if it is a concrete physical entity or real event. It is theerror of treating as a concrete thing something which is not concrete but merelyan ideal. It is also the mental activity in which hazily perceived and relativelyintangible phenomena such as complex arrays of objects or activities are given afactitiously concrete form, simplified and labelled with words or other symbols(Lumsden and Wilson 1981).) these stimuli help the development of strategies basedon codes and models that require action in many contexts. Therefore, action impactsreality by introducing facts into it, both artifacts and mentifacts. (Mentifacts arerelated to the analytical tools such as thoughts, reflections, concepts, and theoriesthat 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, andbehaviors developed by the members of a culture. Examples of mentifacts includeviewpoints, worldviews, and notions about right or wrong behavior (D’Ambrosio2006).) These facts are added into reality in order to modify it. This action producesadditional information that, through this reificative process, modifies or generatesnew strategies for action.

In this regard, it is valuable to highlight how members of distinct cultural groupscapture and process information in diverse ways and, consequently, develop differ-ent actions encouraging the transformation of their own surroundings. According tothis perspective, it is important to document and translate alternative interpretationsand contributions of ethnomathematical knowledge as students learn to constructtheir own connections between both traditional and nontraditional learning settingsthrough ethnomodelling.

Exploring Ethnomodelling

The etymology of the prefix ethno traces back to the Greek word ethnos meaninga 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 anddifferences 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 butare mainly based on language, history, religion, customs, and institutions and on thesubjective self-identification of a people. In so doing, ethno represents particularityand modelling universality and the combination of the specific and universal leadsto all mathematical activity that takes place within a culture through the dynamic ofthe encounters.

The goddess of practical knowledge in ancient Greece was techne, whose namerelates to technique and technology. The Greek word for art is techne, and the Greekword tikein, which means to create, is also derived from techne. Techne is a formof practical knowledge that results in productive action. These mythic modes ofknowledge are considered as practical knowledge that results in productive action.

This etymology reveals a deep connection between technology and the practicesof living and creating. It represents the relationship among humanity, socioculturalcontexts, and the creation of all forms of technology and guides scientists andeducators to develop a moral and cultural standard for the teaching and learningmathematics. This is one of the most important purposes of ethnomodelling.

Ethnomodelling binds contemporary views in ethnomathematics. It recognizesthe need for culturally based views on modelling processes. Studying the unique cul-tural differences in mathematics encourages the development of new perspectives onthe scientific questioning methods. Research involving culturally bound modellingideas may address the problem of mathematics education in non-Western societiesby bringing local and cultural aspects into mathematical teaching and learningprocesses (Eglash 1999). This perspective is needed in mathematics education.

Therefore, Rosa and Orey (2010) argue that ethnomodelling involves examiningways in which individuals or groups draw on traditional or curricular mathematicalideas in the course of their problem-solving experiences, not to idealize these ascorrect or appropriate ways of thinking but rather to highlight the relationshipbetween cultural groups and the deeply embedded mathematics in their dailyactivities.

In this context, Rosa and Orey (2013) affirm that the purpose of ethnomodellingis to invite students to explore others’ cultural practices (emic) and transit theminto other mathematical systems, such as school or academic mathematics (etic).For example, students should compare how a particular problem is solved indifferent cultural contexts. Thus, ethnomodelling is “a practical application ofethnomathematics, and which adds the cultural perspective to modelling concepts”(p. 78).

This presents us with a cultural perspective that broadens views of modellingbecause it recognizes it as a pedagogical bridge for students in the acquisitionof mathematical knowledge (Bassanezi 2002). Hence, ethnomodelling brings aninclusion of a diversity of ideas brought by students from other cultural groups,which can give them confidence and dignity, while allowing them to discuss theinclusion of cultural perspectives into the modelling process (Rosa and Orey 2013).

Ethnomodelling is a tool that responds to its surroundings and is culturallydependent (D’Ambrosio 2002; Bassanezi 2002; Rosa and Orey 2007). The goal ofrecognizing ethnomodelling is not to give mathematical ideas and practices of othercultures 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 isconsidered as the intersection of cultural anthropology, ethnomathematics, andmathematical 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; Eglash1997; Rosa and Orey 2007). In this context, ethnomodelling does not follow thelinear modelling approach that is prevalent in modernity.

Previously, for example, Bassanezi (2002) stated that ethno/modelling processstarts with the social context, reality, and interests of students and not by enforcinga set of external values and decontextualized activities without meaning for thestudents. This process is defined as the mathematics practiced and elaborated bydifferent cultural groups, which involves the mathematical practices present indiverse 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 hasallowed us to see that mathematization “is a process in which individuals fromdifferent cultural groups come up with different mathematical tools that help themorganize, analyze, comprehend, understand, and solve specific problems located inthe context of their real-life situation” (Rosa and Orey 2013, p. 118).

This approach shows, indeed respects, that people of different cultures havedifferent views of the relation between the nature of spirit and humankind, theindividual and the group, and the citizen and the state, as well as differing viewson the relative importance of rights and responsibilities, liberty and authority,and equality and hierarchy. Ignoring these cultural elements is a form of subtle

MathematicalModelling

CulturalAnthropology

Ethnomathematics

EthnomodellingValuing

andRespecting

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, (Ideationmeans to come up with a more innovative bright idea that makes a differencein society. It involves both divergent thinking, which starts with the known andmoving outwards, and convergent thinking, which starts with the known and movinginwards. Hence, ideation is the creative process of generating, developing, andcommunicating innovative ideas and transforming them into valuable outcomes forthe well-being of the members of distinct cultural groups. In this context, ideasare understood as a basic element of thought that can be either visual, concrete, orabstract (Jonson 2005). It is important to emphasize that ideation also comprisesall 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 essenceof a culture is not only its artifacts, tools, or other tangible cultural elements butthe way members of distinct cultural groups interpret, use, and perceive them. Anartifact may be used in different cultures in very diverse ways and for very distinctpurposes. 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 beenimposed globally as the patterns of rational human behavior and are often closed tonew ideas that originate in their former colonies.

It is important to state here that the control of Western powers and the results ofthe globalization process are far from acceptable (D’Ambrosio 1997). Hence, thestudy of ethnomodelling, while being mindful of aspects of colonialization, and theimportance 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 mathematicalideas, procedures, and practices from three approaches of viewing cultures such asemics (local/insiders) and etics (global/outsiders) in order to develop and understandthe dialogic (emic-etic/glocal) approach that is necessary for the development ofethnomodelling investigations.

Ethnomodelling and its Three Approaches of Viewing Cultures

The challenge both researchers and educators have in dealing with the connectionbetween mathematics and culture is to develop forms of pedagogical action thathelps us to understand culturally bound mathematical ideas, procedures, andpractices 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 havedeveloped 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 includescognitive, perceptual, and conceptual knowledge, which is influenced through aunique cultural dynamism. (Cultural dynamism refers to the exchange of systemsof knowledge that enable members of distinct cultures to exploit or adapt to theworld around them. Thus, this cultural dynamic facilitates the incorporation ofhuman invention, which is related to changing the world to create new abilitiesand institutionalizing these changes that serve as the basis for developing morecompetencies (Rosa and Orey 2015).)

Both emic and etic approaches provide ways of discriminating between varioustypes of knowledge for the study of cultural phenomena such as the development ofmathematical practices. Thus, Pike (1967) affirmed that:

( . . . ) it proves convenient – though partially arbitrary – to describe behavior from twodifferent standpoints, which lead to results which shade into one another. The etic viewpointstudies behavior as from outside of a particular system, and as an essential initial approachto an alien system. The emic viewpoint results from studying behavior as from inside thesystem. (p. 37)

The emic approach examines local principles of classification and conceptualizationfrom within each cultural system (Berry 1989) in which distinctions made by themembers of distinct cultural groups are emphasized. According to Lett (1990), theemic approach is essential for an intuitive and empathic understanding of a culture,while the etic approach is essential for cross-cultural comparison and indispensablefor ethnology because such comparisons necessarily demand the application ofstandard 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 oncertain philosophical assumptions and values that are strongly endorsed by Westerncivilizations. For example, Rosa and Orey (2017) assert that there are beliefs thatmathematical procedures are unique and that the sociocultural unit of operation isthe individual. On the other hand, there are beliefs that mathematical practices arethe same and that its goals and techniques are equally applicable across all culturalgroups.

An important challenge for many educators is to strengthen existing mathematicscurricula by minimizing the power of mathematical universality and their claims ofdescriptive, predictive, and explanatory adequacy (Rosa 2010). A second goal is toassist and support educators to understand and explain both existing and historicalvariations of mathematical ideas, procedures, and practices that have varied acrosstime, place, cultures of origin, race, ethnicity, gender, and other socioculturalcharacteristics (Rosa and Orey 2015).

Consequently, when researching ethnomodelling, it is possible to identify at leastthree 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 scientificand mathematical knowledge of the members of distinct cultural groups. In thiscontext, global analyses have a cross-cultural design because outsider observersdevelop global worldviews that seek objectivity across cultures. Thus, Helfrich(1999) examines the question of a cross-cultural perception in which observationsare often taken according to externally derived criteria and frequently without theintentionality of learning the perspectives of others.

Globalization has reinforced the utilitarian mechanization, indeed automatizationof mathematics approach to school mathematics curricula. As well, it has helped toglobalize pervasive western academic mathematical ideologies. Particularly, schoolmathematics is criticized as a cultural homogenizing force, a critical filter forstatus, a perpetuator of mistaken illusions of certainty, and an instrument of power(Skovsmose 2000).

In this approach, comparativist researchers and educators attempt to describedifferences among cultures. These individuals are considered as culturally universal(Sue and Sue 2003). In this context, Pike (1967) refers etic categories as culture-freefeatures 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 becauseit has been tested and validated within the local context. It creates a framework fromwhich members of distinct cultural groups can understand and interpret the worldaround them. Local worldviews clarify intrinsic cultural distinctions that examinelocal principles of classification and conceptualization from within each culturalsystem.

Currently, there is a recognition about the importance of local contributionsto the development of scientific and mathematical knowledge. For example, localmathematical knowledge and interpretations are essential to emic analyses in themathematics curriculum that cultivates values and fosters the conscientization ofthe students. An emic analysis is culturally specific regarding to the insiders’beliefs, thoughts, behaviors, knowledges, and attitudes. It is from their viewpointthat mathematical knowledge is conveyed for the understanding of their culturalcontext.

In this approach, these members describe their culture in its own terms. Theseindividuals are considered as culturally specific (Sue and Sue 2003). In thiscontext, Helfrich (1999) stated that what is emphasized in this approach is the self-determination and self-reflection of these members about the development of theirmathematical 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 bothelements of the same phenomenon (Kloos 2000). It involves blending, mixing, andadapting two processes in which one component must address the local cultureand/or a system of values and practices (Khondker 2004).

In a glocalized society, (According to Rosa and Orey (2017), glocalization isthe acceleration and intensification of interaction and integration among membersof distinct cultural groups. Glocalization has emerged as the new standard in rein-forcing positive aspects of worldwide interaction in textual translations, localizedmarketing communication, sociopolitical considerations, and in the developmentof scientific and mathematical knowledges.) members of distinct cultural groupsmust 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, actingas ethnographers, to describe mathematical ideas, procedures, and practices of otherpeoples in order to give meaning to these findings (D’Ambrosio 2006).

Therefore, Rosa and Orey (2017) argued that glocalization has emerged as thenew standard in reinforcing positive aspects of worldwide interaction in textualtranslations, 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 translationbetween distinct mathematical knowledge systems is direct and simple such ascounting 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 foundin African sand drawings.

For example, Eglash (1997) argued that Gerdes (1991) used the sona sanddrawings developed by the members of the Tchokwe cultural group, in NortheasternAngola, to demonstrate the value of indigenous mathematical knowledge by show-ing that the constraints necessary to define complex Eulerian paths and recursivegeneration systems are created by successive iterations through the application ofthe same geometric algorithm.

The construction of these complex cultural artifacts indicates the conscioususe of iterative constructions as a visualization of analogous iterations in culturalknowledge. Figure 4 shows the similarity between the Eulerian path and the sonasand drawing produced by the Tchokwe people in Angola.

In this context, Eglash et al. (2006) developed a computational modelling processon traditional African architecture using fractal geometry, which are patternsthat repeat themselves at many scales as they are usually used to model naturalphenomena such as trees (branches of branches) and mountains (peaks withinpeaks).

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 ofrectangles (Eglash et al. 2006).

In this context, Eglash and Odumosu (2005) argue that “in the African case manyvillages were constructed over many generations with no one in charge – yet thereis a cohesive fractal pattern for the village as a whole” (p. 102). Figure 5 shows aBa-ila settlement in southern Zambia that has a fractal shape.

Figure 6 shows that this architecture can be modelled with fractals by applyingthe principle of iteration.

It is possible to observe, in Fig. 6, the fractal generation of Ba-ila, in which thefirst iteration is similar to a single house, the second iteration is similar to a familyring, (At the back end of the interior of the settlement, there is smaller detached ringof houses, which is like a settlement with a settlement. This is the chief’s extendedfamily ring (Eglash and Odumosu 2005).) and the third iteration is similar to thewhole village.

Fig. 4 The similaritybetween the Eulerian pathand the sona sand drawing.(Source: Rosa and Orey 2014,p. 144)

a

b

d

f

c

e

EulerianPath

Sona sand drawing producedby 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 arisesfrom emic rather than etic origins. Hence, ethnomodelling establishes relationsbetween the local (emic) conceptual framework and the mathematical knowledgeembedded 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 culturaltraditions and contexts, but they are also equipped with a global knowledge bycreating a sort of localized globalization (Cheng 2005).

For example, emic-oriented researchers and educators focus on the investigationsof the intrinsic cultural distinctions meaningful to members of distinct culturalgroups, especially when the natural world is distinguished from the supernaturalrealm in the worldview of those specific cultures (Rosa and Orey 2017).

On the other hand, etic-oriented researchers and educators examine cross-culturalperspectives so that their observations are taken according to externally derivedcriteria. This context allows for the comparison of multiple cultural groups inwhich “both the objects and the standards of comparison must be equivalent acrosscultures” (Helfrich 1999, p. 132).

According to this context, researchers and educators should find points ofagreement between the imposed cultural universality (Cultural universality refersto the belief that the origin, process, and manifestation of disorders are equallyapplicable across cultures (Bonnett 2000).) (global) of mathematical knowledge ortake on techniques, procedures, and practices of its cultural relativism. (Culturalrelativism 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 thanbe judged against the criteria of another (Todorov 1993).) In this context, the useof both emic and etic approaches deepens their understanding of important issuesin scientific research and investigations about ethnomathematics because they arecomplementary worldviews (Rosa and Orey 2013). Since these two approaches arecomplementary, it is possible to delineate forms of synergy between local and globalaspects 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 dimensionsof one culture to other cultures. This combined approach requires researchers andeducators to attain local knowledge developed by the members of distinct culturalgroups, which allows us to become familiar with the relevant cultural differences indiverse sociocultural contexts (Rosa and Orey 2015).

In the authors’ point of view, both local (emic) and global (etic) approaches areimportant to develop a clearer idea of what is needed for mathematics education in agiven context, mainly, to the conduction of ethnomodelling research. In this context,local knowledge and its interpretations (emic) are essential to the conduction ofthese studies as well as the promotion of debates related to the comparisons betweenmathematical knowledge developed in distinct cultural contexts (etic) which are alsonecessary 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 orientedto all comparable events (whether sounds, ceremonies, activities), of all peoples, of all partsof the earth; through the other lens, the emic one, he views the same events in that particularculture, 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 emicand etic approaches as a symbiotic process between two different mathematicalknowledge systems. Similarly, the resurgence of debates regarding cultural diversityin the mathematics curriculum has also renewed the classic emic-etic debatesince there is a need to comprehend how to build scientific generalizations whileunderstanding and making use of sociocultural diversity.

Yet, attending to unique mathematical interpretations developed by members ofeach cultural group often challenges fundamental goals of mathematics in which themain objective is to build a theoretical basis that can truly describe the developmentof 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 ofreality 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 componentsand the relationship between them, including sociocultural variables (Rosa and Orey2013).) taken from the sociocultural context of the members of distinct culturalgroups in order to enable its communication and transmission across generations.

The representation of this mathematical knowledge helps these members tounderstand, comprehend, and describe their world by using small units of infor-mation, named ethnomodels, which links their cultural heritage to diverse contextssuch as social, political, economic, environmental, and educational (Rosa and Orey2010). This approach helps them to develop techniques, processes, and methods tosolve problems they face daily.

This context allows ethnomodels to be defined as cultural artifacts that canbe considered as the pedagogical tools used to facilitate the understanding andcomprehension of systems taken from reality of the members of distinct culturalgroups (Rosa and Orey 2010). Hence, ethnomodels serve as external representationsof local phenomena that are both precise and consistent with the scientific andmathematical knowledge socially constructed and shared by the members of specificcultural groups. In the ethnomodelling process, ethnomodels can be emic, etic, anddialogic.

Emic ethnomodels are grounded in the mathematical features and characteristicsthat are important and valuable for members of distinct cultural groups since theirmodels 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 fromthe outsiders’ viewpoint. For example, etic ethnomodels represent how modellersthink the world works through systems taken from reality, while emic ethno-models represent how people who live in such world think these systems work intheir own.

The dialogic ethnomodels enable a translational process between emic and eticknowledge 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 toRosa and Orey (2017), this process involves a process of negotiating mathematicalmeanings expressed between local and global contexts through translation in theethnomodelling process.

Emic and Etic Ethnomodels of the Mangbetu Ivory Sculpture

It is useful to examine mathematical ideas found in an ivory hatpin from theMangbetu people, who occupy the Uele River area in the northeastern part ofthe Democratic Republic of Congo, and the geometric algorithm involved in itsproduction, which “gives explicit instructions for generating a particular set ofspatial patterns” (Eglash 1999, p. 61).

The creation of a Mangbetu design may reflect the artisans’ desire to “makeit 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 areused in many Mangbetu designs.

The combination of the 45-degree angle construction technique with the scalingproperties of the ivory carving may reveal its underlying structure, which has threeinteresting geometric features (Eglash 1999). However, this also suggests that ifthere 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 inwhich the left side shows a distinct angle about 20 degrees from the vertical. Thedecorative end of this ivory hatpin is composed of four scaled similar heads thatshows a scaling design (Fig. 7).

Figure 8 shows the geometric analysis of this sculpture in which the sequence ofshrinking squares can be constructed by an iterative process that bisects one squareto create the length of the side for the next square. However, Eglash (1999) statedthat it is not possible “to know if these iterative squares construction was the conceptunderlying the sculpture’s design, but it does match the features identified in thisprocess” (p. 68).

The mathematical idea implicit in this emic knowledge was passed to themembers of the Mangbetu people across generations, who were responsible for theconstruction and upkeep of this unique ivory cultural artifact.

Fig. 7 Mangbetu ivorysculpture. (Source Eglash1999)

18 M. Rosa and D. C. Orey

Fig. 8 Geometric analysis ofa Mangbetu ivory sculpture.(Source: Eglash 1999)

Fig. 9 Geometric relations inthe Mangbetu ivory sculpture(Source: Adapted fromEglash 1999)

On the other hand, Fig. 9 (This figure is not to scale.) shows the geometricrelations in the sculpture iterative square structure.

In this regard, it is possible to elaborate an etic ethnomodel to show that since α1and 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 =√

22

3√

22

=√

2

3√

2= 1

3and 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 thevertical (Eglash 1999). The construction algorithm of this etic ethnomodel can becontinued indefinitely, and the resulting structure can be applied to a wide variety ofmathematics teaching applications, from simple procedural construction to formaltrigonometry.

In this regard, D’Ambrosio (1993) affirmed that mathematical practices aresocially learned and transmitted to the members of cultural groups. In this example,an emic observation sought to understand this mathematical practice of makingthis sculpture from the perspective of the internal dynamics and relations withinthe Mangbetu culture by clarifying intrinsic cultural distinctions to the externalobservers 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 nightschool, ages ranging from 18 to 33 years old, in the second year of high school inthe 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 commontypes of Brazilian roller carts built with a wooden frame and steel bearings that arediscarded in automotive repair shops.

Etic ethnomodels enable students to analyze and interpret their data, to formulateand test their own hypotheses, and to verify the effectiveness of their elaboratedmathematical models taken from their own reality. In this approach, students intheir groups designed a model and constructed their roller carts by learning howmathematical concepts were used in the preparation, analysis, and resolution of theirmodels.

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 dimensionsof the roller carts that were suitable for them, regardless of their height and size sothat all of them could participate in the race. Figure 11 shows an example of an eticethnomodel of the roller carts developed by the students in each group by choosinga standardized model of the cart to be used in the race competition.

This approach helps students to move away from emotional arguments andto focus on and then apply data-based tools to build a model of a standardizedroller cart for a race competition. These students applied their etic ethnomodels todevelop a standardized roller cart for a competition by using mathematical contentto accomplish the proposed activities related to the design of the model and theconstruction 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 partsof the cart roller in order to standardize its dimensions. Then, they sent their notesto the woodworker for the validation of their results. For example, the majority ofthe students affirmed that the standardization of procedures enables the roller cartcompetitions 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 projectsrelated to the design and construction of roller carts in which they could participatein a race competition under equal conditions for all competitors. Figure 12 shows aroller 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 phenomenonrelated to the construction of the roller carts for understanding individuals fromdifferent cultural backgrounds so as to holistically comprehend and explain thismathematical practice from the viewpoint of the outsiders by seeking objectivityacross cultures.

Therefore, the focus of this ethnomodelling process was to apply data in aspecific sport competition related to roller carts that have been initially created bythe social, cultural, climatic, and economic influences in Brazil in which popularand 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 bythe students. (Source: Soares2018)

A Dialogic Ethnomodel of a Local Farmer-Vendor

A study that was conducted by Cortes (2017) in a public school in the metropolitanregion of Belo Horizonte and in a local farmers’ market, in the state of MinasGerais, Brazil, is an example of the application of a dialogic ethnomodel. The mainobjective of this study was to show how dialogic approaches of ethnomodelling cancontribute 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 yearsold, 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 groupof 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 toexamine local (emic) strategies applied by the farmer-vendor to his labor practicesin the farmer market, as well as the academic techniques (etic) employed by thestudents in their school context. These contexts constituted ambiences of effectiveexchange of local and academic mathematical knowledge reciprocally through theelaboration of emic, etic, and dialogic ethnomodels.

For example, the results of the study conducted by Cortes (2017) showed thatthe farmer-vendor developed through his observations and experiences an emicethnomodel 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 kilogramis sold at 4 reais, thus each 100 grams cost 40 cents, then you cannot sell it at that pricebecause 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 youdo not go to the market to buy the products and sell them at the same price. In this case, Iincreased 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. Thissystem is used to determine the price of any of the products I sell. For example, if I buy aproduct for 80 or 100 reais each box, then the price of the kilogram should be 16 or 12.80reais [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 hiscultural context. Thus, Rosa and Orey (2013) affirmed that the main objective of anemic approach is a descriptive idiographic orientation of mathematical phenomenabecause of the strength of the particularities of mathematical ideas, procedures, andpractices 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 newunderstandings related to how individuals from a different cultural background usetheir own mathematical thinking.

This etic approach is necessary to the holistic comprehension and explanation ofthis specific mathematical practice from the point of view of the students (outsiders).For example, Cortes (2017) affirmed that students developed an etic ethnomodel thatis 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$16.00. And a third product whose cost price is 100 reais, has a sale price between R$ 16.00and R$ 20.00. However, it is important to note that these sales prices may be increased byother 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, where5 ≤ v ≤ 6If CP(m) = 80, then SP(m) = v. m, where12 ≤ v ≤ 16If CP(m) = 100, then SP(m) = v. m, where16 ≤ v ≤ 20

CP = cost priceSP = Sale pricem = mass (kg) of the productv = variation of price including expenses andcharges

The interpretation of these results shows that the determination of these prices,besides being related to the quantity of products purchased, is also bounded tothe emic constructs developed by the daily labor experiences of the farmer-vendor.Table 1 shows the elaboration of a dialogic ethnomodel by the students, whichrepresents the sale process developed by the farmer-vendor.

This example shows that ethnomodelling allowed for the reconceptualizationand application of the function concept through the elaboration of mathematicalactivities originating in the sociocultural context of the farmer-vendor and schoolcommunity by applying the ethnomodelling process in the mathematics curriculumas 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 (emicapproach) and school mathematical contents (etic approach) with the use of problemsituations that emerged from the context of a farmer market (dialogic approach).

By using ethnomodels, students try to understand and comprehend their ownsurroundings through the development of explanations that are organized as proce-dures, techniques, methods, and theories in order to explain and deal with daily factsand phenomena (Rosa and Orey 2015). These strategies are historically organizedin every culture as knowledge systems, including mathematics.

Relevance of Ethnomodelling in a Mathematics Curriculum

In considering ethnomodelling as tool to study ethnomathematics, teaching is morethan the transference of knowledge because it becomes an activity that introducesthe creation of mathematical knowledge. For example, Freire (1970) argued thatthis approach is the antithesis of turning students into containers to be filled withinformation and that it is necessary for a mathematics curriculum to translate theinterpretations and contributions of ethnomathematical knowledge because studentsneed to be able to analyze the connection between both traditional and nontraditionallearning settings.

According to Rosa and Orey (2016), ethnomodelling applies mathematics as alanguage for understanding, simplification, and resolution of problems and activitieslinked to the students’ reality. Conversely, traditional mathematical modelling devel-oped in the academic mathematics curriculum aims at transmitting mathematicalcontent 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 suchways that they can only help memorization skills. These techniques and problemsare, for most students, boring, uninteresting, obsolete, and unrelated to their ownreality.

The characteristics of this kind of curriculum are responsible for the downgradingof school satisfaction and achievement in many countries. In this regard, Rosa andOrey (2016) affirm that ethnomathematics and modelling through ethnomodellingmay 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 mathematicalknowledge of members of diverse cultural groups found in many cultures (Rosa andOrey 2010).

In this regard, an ethnomodelling curriculum provides an ideological basis forlearning with and from the diverse cultural and linguistic backgrounds of themembers of distinct cultural groups. There are three reasons for the application ofethnomodelling 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 schoolor 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 tovalue the contributions of other mathematical knowledge traditions. Thus, in orderto achieve this goal, it is recommended that teachers interpret alternative mathemat-ical ideas, procedures, and practices by starting with the outside sociocultural realityof students.

It is important to emphasize here how ethnomodelling is not considered as away to reach academic mathematical concepts because an emic approach does notwork as a kind of scaffolding to school mathematics. Admittedly, this is a possibleperspective, but in our investigations, this is different from perceiving emic andetic descriptions as parallel in such a way that ethnomodelling is much more as anencounter of distinct cultures than a scaffolding educational process.

Consequently, it is beneficial to apply an ethnomathematical ethnographicperspective in order to come to an understanding of, and respect for, mathematicalknowledge of the members of a given cultural group and having a clear purpose ofthis educational activity. Thus, the implementation of an ethnomodelling perspectivemust be preceded by an inventory of students’ tacit knowledge (Rosa and Orey2013).

In this regard, coming to comprehend students’ contexts, ethnomodelling pro-vides a deeper appreciation of mathematical beauty and utility. Thus, it is usefulto 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 andhistorical contexts.

Conclusion

This chapter sought to outline ongoing research related to cultural perspectives inmathematical modelling. Contemporary academic mathematics is predominantlyEurocentric. This Eurocentrism, which is not necessarily bad, is insufficient toconnect this kind of mathematics to the local realities of learners and educatorsas it facilitates an ongoing divide that has hindered the mathematics coming fromnon-Western traditions. The motivation towards a cultural approach presents uswith an accompanied assumption that makes use of cultural perspectives throughethnomathematics and uses mathematical modelling to bring local issues into globaldiscussion.

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 betweenmathematical knowledge and society. Moreover, they have presented modern orwesternized mathematics as primarily dominated by the preferences of science andcapitalism of the West (European-North American) and that this Eurocentrism posesmany 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 asthe study of mathematical phenomena within a culture. In this context, ethnomod-elling differs from traditional definitions of modelling that considers the foundationsof mathematics education as constant, universal, and applicable everywhere. Hence,it is necessary to point out that the study of ethnomodelling takes the position thatmathematics curriculum, and the many unique problems that it may model, is asocial construct and thus culturally bound.

In order to keep up with modern Western developmental models, other cultureshave been forced to adapt or perish. Relying primarily on constructivist theories, theauthors argue that universal theories of mathematics take different forms in differentcultures and that Western views on abstract ideas of modelling are culturally bound.

The study of ethnomodelling is considered a powerful tool used in the translationof a problem-situation of mathematical ideas and practices within a culture. Thesenew-found ethnomathematical lenses lead to new findings in the development of aninclusive model of the connection between mathematics and culture.

Ethnomodelling is also a pedagogical action that enables students to link schoolmathematics and the mathematics as used by the members of other culturalgroups. It has to do with developing an understanding or creating a sense ofmathematical knowing of and between our own cultures and translating them intoschool perspectives and vice versa.

In order to do so, it is important to develop the concepts of emic, etic, anddialogic 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 ithappens in mathematics education practices in the schools. Thus, three examplesillustrate the use of emic, etic, and dialogic approaches as part of the pedagogicalaction 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 thisknowledge through the proposition of contextualized mathematical activities. In thiscultural dynamism, local knowledge has interacted dialogically with the knowledgeconsolidated by the academy, developing a reciprocal relationship between thesetwo approaches.

In an increasingly glocalized world, educators must consider the cultural andphilosophical 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 thesedifferences enriches the curriculum and increases understanding between peoples,which can only be a good thing!

The adoption of an ethnomodelling perspective in a mathematics curriculumrecognizes the importance of local cultures to the development of mathematics. Thispedagogical aspect produces student-researchers who are active participants in theirown mathematics education as they learn that they themselves can contribute to thedevelopment 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 doteachers 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 learningwith mathematical modelling]. Editora Contexto, São Paulo

Berry J (1989) Imposed etics-emics-derived etics: the operationalization of a compelling idea. IntJ 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 Brasileirode Etnomatemática – CBEm-1. FE-USP, São Paulo, pp 132–141

Biembengut MS, Hein N (2000) Modelagem matemática no ensino [Teaching mathematicalmodelling]. 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 YorkCortes DPO (2017) Re-significando os conceitos de função: um estudo misto para entender as

contribuições da abordagem dialógica da Etnomodelagem [Re-signifying function concepts: amixed method study to understand the contributions of the ethnomodelling dialogic approach].

Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 27

Dissertação (Mestrado Profissional em Educação Matemática). Instituto de Ciências Exatas eBiológicas – ICEB. Departamento de Educação Matemática – DEEMA. UFOP, Ouro Preto,Brazil

D’Ambrosio U (1985) Ethnomathematics and its place in the history and pedagogy of mathematics.Learn Math 5(1):44–48

D’Ambrosio U (1990) Etnomatemática [Ethnomathematics]. Editora Ática, São PauloD’Ambrosio U (1993) Etnomatemática: um programa [Ethomathematics: a program]. A Educação

Matemática em Revista 1(1):5–11D’Ambrosio U (1995) Multiculturalism and mathematics education. Int J Math Educ Sci Technol

26(3):337–346D’Ambrosio U (1997) Foreword. In: Powell AB, Frankenstein M (eds) Ethnomathematics:

challenging Eurocentrism in mathematics education. State University of New York Press,New York, pp 15–21

D’Ambrosio U (1999) Educação para uma sociedade em transição [Education for a society intransition]. Papirus Editora, Campinas

D’Ambrosio U (2000) Etnomatemática e modelagem [Ethnomathematics and modelling]. In:Domite MC (ed) Anais do Primeiro Congresso Brasileiro de Etnomatemática – CBEm-1.FEUSP, São Paulo, p 142

D’Ambrosio U (2002) Prefácio [Forward]. In: Bassanezi RC (ed) Ensino-aprendizagem com mod-elagem matemática [Teaching and learning with mathematical modelling]. Editora Contexto,São Paulo

D’Ambrosio U (2006) The program ethnomathematics: a theoretical basis of the dynamics of intra-cultural encounters. J Math Cult 1(1):1–7

Eglash R (1997) When math worlds collide: intention and invention in ethnomathematics. SciTechnol Hum Values 22(1):79–97. Rutgers University Press

Eglash R (1999) African fractals: modern computing and indigenous design. Rutgers UniversityPress, New Brunswick

Eglash R, Bennett A, O’Donnell C, Jennings S, Cintorino M (2006) Culturally situated designedtools: ethnocomputing from field site to classroom. Am Anthropol 108(2):347–362

Eglash R, Odumosu TB (2005) What mathematics from Africa? In: Sica G (ed) Fractals,complexity, and connectivity in Africa. Polimetrica International Scientific Publisher, Monza,pp 101–109

Ferreira ES (2004) Os índios Waimiri-Atroari e a etnomatemática [The indigenous people Waimiri-Atroari and ethnomathematics]. In: Knijnik G, Wanderer F, Oliveira CJ (eds) Etnomatemática:currículo e formação de professores [Ethnomathematics: curriculum and teacher’s education].EDUNISC, Santa Cruz do Sul

Freire P (1970) Pedagogy of the oppressed. Herder and Herder, New YorkGerdes P (1991) Lusona: geometrical recreations of Africa. E. M. University Press,

MaputoGraham D, Bachmann T (2004) Ideation: the birth and death of ideas. Wiley, HobokenHelfrich H (1999) Beyond the dilemma of cross-cultural psychology: resolving the tension between

etic and emic approaches. Cult Psychol 5:131–153Hodgson T, Harpster D (1997) Looking back in mathematical modelling: classroom observations

and instructional strategies. Sch Sci Math 97(5):260–267Huntington SP (1993) The clash of civilizations? Foreign Aff 72(3):22–49Jonson B (2005) Design ideation: The conceptual sketch in the digital age. Design Studies, 26(6):

613–624.Khondker HH (2004) Glocalization as globalization: evolution of a sociological concept.

Bangladesh J Sociol 1(2):1–9Kloos P (2000) The dialectics of globalization and localization. In: Kalb D, van der Land M,

Staring R, van Steenbergen B, Wilterdink N (eds) The ends of globalization: bringing societyback in. Rowman & Littlefield, Lanhamp, pp 281–298

28 M. Rosa and D. C. Orey

Lett J (1990) Emics and etics: notes on the epistemology of anthropology. In: Headland TN,Pike KH, Harris M (eds) Emics and etics: the insider/outsider debate. Sage, Newbury Park,pp 127–142

Lumsden CJ, Wilson EG (1981) Genes, mind and culture. Harvard University Press, CambridgeOrey DC (2000) The ethnomathematics of the Sioux tipi and cone. In: Selin H (ed) Mathematics

across culture: the history of non-western mathematics. Kluwer Academic Publishers, Dor-drecht, pp 239–252

Orey DC, Rosa M (2014) How we came to use a combination of emic, etic, and dialogicalapproaches in the field research ethnomodeling. In: Tsijli MM (ed) Memória IX FestivalInternacional de Matemática. CIENTEC, Quepos, pp 167–179

Pike K (1967) Language in relation to a unified theory of the structure of human behavior. Mouton,The Hague

Powell AB, Frankenstein M (1997) Introduction. In: Powell AB, Frankenstein M (eds) Ethnomath-ematics: challenging Eurocentrism in mathematics education. State University of New YorkPress, New York, pp 1–4

Rosa M (2000) From reality to mathematical modelling: a proposal for using ethnomathematicalknowledge. Master thesis, College of Education, California State University, Sacramento,Publication No. R7880 2000

Rosa M (2010) A mixed method study to understand the perceptions of high school leaders aboutEnglish language learners (ELL): the case of mathematics. Doctorate Dissertation, College ofEducation. California State University, Sacramento

Rosa M, Orey DC (2003) Vinho e queijo: etnomatemática e modelagem! [Wine and cheese:Ethnomathematics and modelling!]. BOLEMA 16(20):1–16

Rosa M, Orey DC (2006) Abordagens atuais do programa etnomatemática: delinenando-se umcaminho para a ação pedagógica [Current approaches in the ethnomathematics as a program:delineating a path toward pedagogical action]. BOLEMA 19(26):19–48

Rosa M, Orey DC (2007) Cultural assertions and challenges towards pedagogical action of anethnomathematics program. Learn Math 27(1):10–16

Rosa M, Orey DC (2010) Ethnomodeling: an ethnomathematical holistic tool. Acad Exch Q14(3):191–195

Rosa M, Orey DC (2013) Ethnomodelling as a methodology for ethnomathematics. In:Stillman GA, Gabriele K, Blum W, Brown JP (eds) Teaching mathematical modelling: con-necting to research and practice. Springer, Dordrecht, pp 77–88

Rosa M, Orey DC (2014) Etnomodelagem: a abordagem dialógica na investigação de saberes etécnica êmicas e éticas [Ethnomodelling: the dialogic approach in the investigation of emic andetic knowledge]. Contexto e Educação 29(94):132–152

Rosa M, Orey DC (2015) Evidence of creative insubordination in the research of pedagogicalaction of ethnomathematics program. In: D’Ambrosio BS, Lopes CE (eds) Creative insubordi-nation in Brazilian mathematics education research. Lulu Press, Raleigh, pp 131–146

Rosa M, Orey DC (2016) Ethnomodelling: exploring glocalization in the contexts of local (emic)and global (etic) knowledge. Int J Res Math Educ 6(1):196–218

Rosa M, Orey DC (2017) Concepts of positive deviance in pedagogical action in an ethnomathe-matics program. REnCiMa 9(4):62–79

Schildkrout E, Keim CA (1990) African eflections: art from northeastern Zaire. The University ofWashington Press, New York

Skovsmose O (2000) Cenários para investigação [Scenarios for investigation]. BOLEMA13(14):66–91

Soares RB. (2018) Modelagem matemática como um ambiente de aprendizagem para o desenvolvi-mento de competências de modelagem matemática de um grupo de estudantes ao transformaruma brincadeira em uma prática esportiva [Mathematical modelling as a learning environmentto the development of mathematical modelling competencies of a group of students by turninga street play into a sports practice]. Dissertação de Mestrado. Departamento de EducaçãoMatemática. UFOP, Ouro Preto, Minas Gerais

Ethnomodelling as the Translation of Diverse Cultural Mathematical Practices 29

Sue DW, Sue D (2003) Counseling the culturally diverse: theory and practice. John Wiley & Sons,New York

Todorov T (1993) On human diversity: nationalism, racism, and exoticism in French thought.Trans. C. Porter, Cambridge, Boston