Academic
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Ethnomodeling: An Ethnomathematical
Holistic Tool

**Milton Rosa, Encina
Preparatory HS, San Juan USD, Sacramento, CA**

**Daniel Clark Orey, ****California**** ****State**** ****University****, ****Sacramento****, ****CA**

*Rosa,
Ed.D., is a mathematics teacher, in the Mathematics Department, and Orey, PhD.,
is a professor of mathematics and multicultural education, in the College of
Education*

**Abstract**

Mathematics used
outside of the school may be considered as a process of modeling rather than a
mere process of manipulation of numbers. The application of ethnomathematical
techniques and the tools of modeling allow us to see a different reality and
give us insight into mathematics done in a holistic way. The pedagogical
approach that connects the cultural aspects of mathematics with its academic
aspects is denominated ethnomodeling, which is a process of translation and
elaboration of problems and questions taken from systems that are part of the
students’ reality.

Introduction

There are hundreds of reasons for teaching mathematics. In our work, one
of the most relevant, reasons involves the consideration of mathematics as an
expression of human development, culture and thought and that it is an integral
part of the cultural heritage of humankind. Contemporary society places great
value on a capitalistic scientific western oriented science and mathematics. On
the other hand, ethnomathematics has demostrated that mathematics is composed
of many diverse and distinct cultural traditions, not just those emerging from
the Mediterranian basin mathematics tradition (D’Ambrosio, 1993).

Mathematical thinking has been influenced by the vast diversity of human
characterisitcs such as languages, religions, morals, and
economical-social-political activities. In concert with these, humans have
developed logical processes related to universal needs to quantify, measure,
model and explain, allshaped and operating with in different socio-historical
contexts. Because each cultural group has its own way of doing mathematics,
these connections often came to represent a given cultural system, especially in
the way that they quantified and used numbers, geometric forms and
relationships, measured or classified objects in their own environment.

For all these reasons, each cultural group has developed its own way to *mathematize*[1] their own realities. Western
scientific arrogance, that is a disrespect of and outright refusal to
acknowledge a cultural identity by some scientists and mathematicians puts all
processes of understanding and comprehension of many non-Western cultural systems
at risk (D’Ambrosio, 1985, 1990; Zaslavsky, 1996). According to Bassanezi
(2002), these particularities should not be ignored and they should be
respected when individuals attend school because this aspect gives confidence and
dignity to students when their previous knowledge is acknowledged.

The purpose of this article is to show to the readers that it is
possible to apply ethnomathematical techniques and the tools of modeling in
real-world situations. This approach allows us to see reality in different
ways, which give us insight into mathematics that is done in a holistic and
distinct manner. In this article we provide details on how pedagogical
approaches can connect the cultural influences of mathematics with its academic
aspects. We refer to this process as ethnomodeling, which can be described as the
translation and elaboration of problems and questions taken from systems that
are part of the students’ reality.

In so doing, a
search for new methodological approaches is necessary to record historical
forms of mathematical ideas that occur in different cultural contexts.

**Ethnomathematics as a Holistic Approach to
Mathematics Education**

One of the most important concepts of ethnomathematics is the
association of the mathematics found in diverse cultural contexts.
Ethnomathematics as a research paradigm is much wider than traditional concepts
of mathematics* *and ethnicity or any current sense of
multiculturalism. D’Ambrosio (1990) referred to *ethno* as that related to distinct cultural groups identified by
cultural traditions, codes, symbols, myths, and specific ways of reasoning and
inferring. In so doing, ethnomathematics may be considered as the way that
various cultural groups mathematize because it examines how both mathematical
ideas and mathematical practices are processed and used in the daily
activities. It can be also described as the arts or techniques developed
by diverse students to explain, to understand, and to cope with their own
environment (D'Ambrosio, 1992).

In accordance to Barton (1996) ethnomathematics embraces the
mathematical ideas thoughts and practices as developed by all cultures. From
his perspective, a body of anthropological research has come to focus on both
the intuitive mathematical thinking and the cognitive process that are largely
developed in minority cultural groups. Ethnomathematics may also be
considered as a program that seeks to study how students have come to
understand, comprehend, articulate, process, and ultimately use mathematical
ideas, concepts, and practices that may solve problems related to their daily
activity.

In this context, Barton (1996) stated that ethnomathematics is not only
the study of mathematical ideas because it is also the study of anthropology and
history. This means that the study of the history of mathematics and
mathematics attempts to identify the cultural and mathematical contributions of
different cultures across the world. Seen in this context, the focus of
ethnomathematics consists essentially of a serious and critical analysis of the
generation and production of the mathematical knowledge and intellectual
processes, the social mechanisms in the institutionalization of knowledge; and
the diffusion of this knowledge (Rosa & Orey, 2006). In this much more
holistic[2] context of mathematics that uses an anthropological perspective to
include diverse perspectives, patterns of thought, and histories, the study of
the systems[3] taken from reality help students to come to reflect, understand,
and comprehend extant relations among all of the components of the system.

All
individuals and students as well possess and develop both anthropological and
mathematical concepts. These concepts are rooted in the universal human
endowments of curiosity, ability, transcendence, life, and death. They all
characterize our very humanness. Awareness and appreciation of cultural
diversity that can be seen in our clothing, methods of discourse, our religious
views, our morals, and our own unique world view allow us to understand each
aspect of the daily life of humans (Rosa & Orey, 2006).

The
unique cultural background of each student represents a set of values and the
unique way of seeing the world as it is transmitted from one generation to
another. The principals of anthropology that are relevant to the work of
ethnomathematics includes the essential elements of culture such as language,
economy, politics, religion, art, and the daily mathematical practices of
diverse groups of students. Since, cultural anthropology gives us tools
that increase our understanding of the internal logic of a given society;
detailed anthropological studies of the mathematics of distinct cultural groups
most certainly allows us to further our understanding of the internal logic and
beliefs of diverse group of students.

**Ethnomathematics and Modeling**

Historically, models
that arise from reality have been the first paths towards providing abstractions
of mathematical concepts.
Ethnomathematics that uses the manipulations of models of reality and
modeling as a strategy of mathematical education uses the codifications
provided by others in place of formal language of academic mathematics. Within this
context, D’Ambrosio (1993), Bassanezi (2002); Monteiro (2004); Rosa & Orey
(2006) stated that mathematical modeling is a methodology that is closer to an
ethnomathematics program. ethnomathematics may be defined as the
intersection between cultural anthropology and institutional mathematics, that
utilizes mathematical modeling to interpret, analyze, explain and solve real
world problems or mathematize existing phenomena (D’Ambrosio, 1993; Rosa, 2000,
Orey & Rosa, 2003).

Investigations in modeling
have been found to be useful in the translation of ethnomathematical contexts
by numerous scholars in

Outside of the ethnomathematics related research paradigm, it is known
that many scientists search for mathematical models that can translate their
deepening understanding of both real world situations and diverse cultural
contexts. This enables them to seek and take possible (i.e. political)
positions in relationship to the objects of the study (Bassanezi, 2002;
D’Ambrosio, 1993; Rosa & Orey, 2006). Using modeling as a tool towards
pedagogical action, students have been shown to learn how to find and work with
authentic situations and real-life problems.

**Ethnomodeling**

Ethnomodeling is a process of elaboration
of problems and questions growing from real situationsthat form an image or
sense of an idealized version of the *mathema*.
The focus of this perspective essentially forms a critical analysis of the
generation and production of knowledge (creativity), and forms an intellectual
process for its production, the social mechanisms of institutionalization of
knowledge (academics), and its transmission (education). According to D’Ambrosio (2000), “this process
is modeling” (p. 142). In this
perspective, by analyzing their role in reality as a whole, this holistic context
allows those engaged in the modeling process to study systems of reality in
which there is an equal effort made by them to create an understanding of all
components of the system as well as the interrelationships among them
(D’Ambrosio, 1993; Bassanezi, 2002).

The use of modeling as pedagogical action for an etnomathematics program
values the previous knowledge of the community by developing student capacity
to assess the process of elaborating a mathematical model in its different
applications and contexts by having started with the social context, reality
and interests of the students and not by inforcing a set of external values and
curriculum without context or meaning for the learner. Bassanezi (2002)
characterizes this process as “ethno-modeling” (p. 208), and defines
ethnomathematics as “the mathematics practiced and elaborated 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).

In considering ethnmodeling, teaching is much more than the transferance
of knowledge because teaching becomes an activity that introduces the creation
of knowledge (Freire, 1998). This
approach in mathematics education is the antithesis of turning students into containers
to be filled with information (Freire, 1970).

It is necessary for school curriculum, to translate the interpretations
and contributions of ethnomathematical knowledge into systemized mathematics
because students will be able to analyze the connection between both
traditional and non-traditional learning settings.

**Final
Considerations**

Any study of ethnomathematics and mathematical
modeling represents a powerful means for validating a student’s real life
experiences, and gives them the tools to become critical participants in
society. In so doing, educators should be empowered to analyze the role of what
Borba (1990) refers to as a student’s ethnoknowledge[4] in the mathematics
classroom. There exists a need to create a new role to mathematics instruction
that empowers students to understand power and oppression more critically by considering
the effect of culture on mathematical knowledge by working with their students
to uncover the distorted and hidden history of mathematical knowledge.

This perspective forms the basis for significant
contributions of a Freirean-based ethnomathematical perspective in
re-conceiving the discipline of mathematics and in a pedagogical practice. The use of Freire’s (1970) dialogical
methodology is seen as essential in developing the curricular praxis of
ethnomodeling by investigating the ethnomathematics of a culture in
constructing a curriculum with people from other cultures to create curricula
that enable the enrichment for all people’s knowledge of mathematics.

Seen in this context, we would like to broaden the
discussion of** **possibilities for the
inclusion of ethnomathematics and** **mathematical
modeling perspectives that respect the social and cultural diversity of all
people with guarantees for the development of understanding our differences
through dialogue and respect. This is how ethnomodeling can empower students in
this century against all kinds of domination and oppression.

**Endnotes**

[1] Mathematization is a process in which individuals from
different cultural groups come up with different mathematical tools that can
help them to organize, analyze, comprehend, understand, and solve specific
problems located in the context of their real-life situation. These tools allow them to identify and
describe a specific mathematical idea or practice in a general context by
schematizing, formulating, and visualizing a problem in different ways,
discovering relations, discovering regularities, and transferring a real world
problem to a mathematical idea through mathematization.

[2] A holistic context consists essentially of
a critical analysis of the generation (creativity) of knowledge, and the
intellectual process of its production. The focus on history analyzes the
social mechanism and institutionalization of knowledge (academics), and its
transmission through the educational process (D’Ambrosio, 1990).

[3] A system is a part of reality considered
integrally. It is a set of components taken from the reality, which analyses
components interrelationships between these components (D’Ambrosio, 1990).

[4]
Ethnoknowledge is acquired by students in the pedagogical action process of
learning mathematics in a culturally relevant educational system. In this
process, the discussion between teachers and students about the efficiency and
relevance of mathematics in different contexts should permeate instructional
activities. The ethnoknowledge that students develop must be compared to their
academic mathematical knowledge. In this process, the role of teachers is to
help students to develop a critical view of the world by using mathematics.

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