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