La educación intercultural y la
etnomatemática
en la formación del
docentede Matemática y Física
Intercultural
education and ethnomathematics in the teacher
training in Mathematics and Physics
Iván Dávila-Garzón
Universidad Central del Ecuador, Quito, Ecuador
lidavilag@uce.edu.ec
https://orcid.org/0000-0002-1541-7875
Ximena Pinos-Benavides
Universidad Central del
Ecuador, Quito,
Ecuador
cxpinos@uce.edu.ec
https://orcid.org/0000-0001-6441-3590
(Received
on: 14/10/2018; Acepted on: 24/10/2018; Final version received on: 29/11/2018)
Suggested citation: Dávila-Garzón, I. and
Pinos-Benavides, X. (2019).
Intercultural education and ethnomathematics in teacher training
in Mathematics and Physics. Revista Cátedra, 2(1),15-25.
Resumen
En el rediseño curricular de la Carrera de Pedagogía de la Ciencias Experimentales, Matemática y Física, de la
Facultad de Filosofía, Letras y Ciencias de
la Educación, de la Universidad Central del
Ecuador se incluyen dos disciplinas:
Educación Intercultural e
Inclusiva y la Etnomatemática. La
educación intercultural e inclusiva
se presenta con lineamientos para preservar
los saberes y cosmovisión de
los pueblos plurinacionales. La etnomtemática
usa sus aplicaciones en el campo de la matemática. El presente artículo
tiene como objetivo estudiar el aporte de las disciplinas
mencionadas en la formación académica
de los estudiantes y docentes. El estudio se hará a partir de los principales postulados de Osuna
y D´Ambrosio quienes realizan sus aportes en torno a la
educación intercultural y al inicio del estudio
de la etnomatemática, de aquí se
determinan los principales insumos teóricos para comprender
procesos de construcción y reconstrucción en el proceso de enseñanza-aprendizaje.
En el proceso de formación de
docentes se analizará los aspectos principales para que su enseñanza sea intencional y orientada a las condiciones de la sociedad
actual. En el proceso de
formación de los estudiantes que serán los futuros
docentes de Matemática y Física
se desarrollarán investigaciones prácticas para impulsar
la construcción del conocimiento
y fortalecer de la identidad nacional, pues, está
desapareciendo. Los resultados del aprendizaje
de estas asignaturas fortalecerán la formación integral y las prácticas interculturales
educativas con compromiso social
y ético.
Palabras clave
Campo de formación, diseño curricular, Educación intercultural, Etnomatemática, formación docente.
Abstract
The research on intercultural education and ethnomathematics
in teacher training in Mathematics and Physics
is important because of the new curricular design of the Pedagogy Career of Experimental Sciences,
Mathematics and Physics at the Faculty
of Philosophy, Letters and Education Sciences, Universidad Central del
Ecuador. Two disciplines are included in
the field of Integration of knowledge
and culture contexts, these disciplines are
Intercultural and Inclusive Education
and Ethnomathematics, and present guidelines for preserving the knowledge and worldview of the plurinational people; ethnomathematics
use their applications in the field of mathematics. The objective of this article is to study the contribution of the disciplines mentioned in the academic training
of students and teachers,
and will be based on the
main postulates of Osuna and D'Ambrosio who made their
contributions in the intercultural
education. The main theoretical inputs to understand construction and reconstruction
processes are determined. In the process of
teacher training, the main aspects will be analyzed so that their teaching is
intentional and oriented to the conditions of today's society. In the process of training students practical research will be developed to promote the construction of knowledge and strengthen
national identity, a topic that is
being lost.
Keywords
Training field, curricular design, intercultural
education, ethnomathematics,
teacher training.
1.
Introduction
Bibliographic and documentary research on intercultural education and etnomathematics in the teaching of mathematics and physics is very important,
reason for which these subjects are included in the new curricular design and their
contents should be proposed and developed in the learning process of the new professor. The
study of intercultural education will preserve
the knowledge and worldview of
the indigenous population.
Etnomathematics proposes the use of ethnoscience and the application of the resources applied in the field of mathematics.
The main problem of the study is to rescue and
recover the ancestral knowledge and the worldview of people. The identification of basic definitions to initiate the
knowledge of these subjects with a
formative intentionality will be the main target for
the
development of
construction and reconstruction processes of meanings and senses that
could work as a foundation in
the dialectic relation between human nature
and its
transformative capacity of the student.
The training process
of professors is intentional and oriented to create
citizens who meet the conditions that the
current society demands. The formation of new professors carries
with
it activities to promote the development
of people committed to the history and traditions of
their environment, deeply reflective
with full analysis capacity, argumentation, and someone prepared to
assume the work and professional
performance required, i.e., committed,
flexible and transcendent professionals.
The aim of this
research is to collaborate
with the education career of experimental
sciences, mathematics and physics, in the knowledge
organization of the conceptual and self- reference system favoring the interaction
between the disciplines. In order to achieve this purpose, methods, processes and procedures will be used, and
the professors and students will receive the preliminary
guidelines of the intercultural education and the etnomathematics
in the new curriculum. In addition and as proposed by
Granados (2012), the new
approach of higher education is established on the basis of the
principle of complementary and interdependent
adaptation to the
transformations that
have occurred in recent years
concerning the organization of knowledge and learning
of the new epistemological horizons
of the complexity and the ecology of
knowledge that generate creativity with collaborative and intercultural environments (curricular design
of the mathematical and physical career, 2016, p. 22).
The basic nuclei
that make up the
theoretical-methodological constructs of the disciplines that underpin
the profession in relation to
intercultural education and etnomathematics are related
to contemporary society and educational policy. The strategies and teaching
methods will be intended to propose a holistic education with the integration of
knowledge to create an active subject committed to the national identity.
This
study is important, interesting and relevant since
the student of mathematics
and physics once graduated will provide his/her
professional services in Elementary and High School; thus, with this new knowledge the student will receive better education. The proposal of the redesign is an essential alternative
to meet the demands of the institutions
of the Ecuadorian Education System, which require the formal inclusion of
professionals of these areas in the educational levels indicated.
Mathematics and Physics are disciplines that must be adequately taught, since these contribute significantly to local and regional development.
Only the teachers specialized in these subjects are highly qualified and contribute to the improvement of the academic
quality of the students
and to the formation of good citizens. Thus, they will promote the production
of knowledge and the learning, the recognition of
reality, the project of life, the learning environments and the integrative
nature of the curriculum,
aspects that are essential and topical.
The present study
presents data related to
intercultural education,
theories and approaches, inter-cultural teaching practices, intercultural in the Constitution, importance
and strategies. Additionally, the importance and object
of Etnomathematics are indicated. Finally, conclusions is presented, which will help teachers
and students to face this new challenge.
2.
Education
2.1
Interculturality:
theory and approaches
Interculturality
is a way of life and coexistence among individuals that determines the
interrelation and interaction of people in a social group. In this regard,
Nomberto (2010) states that:
Interculturality
refers to the communicative interaction that occurs between two or more human
groups of different cultures. If one or more of the groups in mutual
interaction are going to be called ethnicities, societies, cultures or
communities it is rather a subject of preferences of schools of social sciences
and in no case it refers to epistemological differences (Millán, 2010, p. 18).
In
the virtual library of indigenous people, it is indicated that interculturality
is the contact and exchange between cultures in equal terms. This contact and exchange
should not be thought in ethnical terms but from the
relationship, communication and lifelong learning between people, groups, knowledge,
values and traditions. All of the above should be oriented to
generate, build and promote mutual respect,
and a development of the
capacities of individuals and community above their cultural and social
differences. Therefore, interculturality tries to break with hegemonic history and reinforces
traditional identities.
In
the intersectional analysis and
intercultural approach, Walsh, (2010), states that
there are three approaches to interculturalism. The first approach refers
to the contact and exchange
between cultures, whose problem hides in the domination in which the relationship
is carried out. The second approach is called functional, which demands the
recognition of diversity and cultural
difference as goals of inclusion
in the established social structure. The third and final approach
is the critical interculturality which recognizes that the difference relies in a colonial structure and matrix of power (Walsh, 2010, pp.
31-52).
Therefore,
interculturality means the
construction of equitable relationships
between people, ethnicities and cultures. Some elements that
contribute to intercultural
education, as indicated by the United Nations, relate
to educational, political,
economic, cultural, environmental and legal aspects. In
the educational field, interculturalization of education refers to fundamental issues such
as education
laws, educational projects, objectives, policies, plans and programs,
curriculum, teacher training, school textbooks, school culture and the
exchange with the community
and the context.
In this regard,
Walsh (2010) says:
Interculturality does not separate from
identity. The fact of relating symmetrically
with people, knowledge, senses and different cultural practices requires a self-awareness of
the person,
the identities that are formed and emphasize the own and
the differences. Self-identity is
not something we can choose from, but something
that has to be negotiated socially
with all the other meanings and
images built as knowledge activated by our own use of identity
(Hall, 1997), i.e., identifying
ourselves with the family and cultural environment requires at the same time
to differentiate us from different identification processes that are often unconscious (pág. 54).
From the above, the
concepts of location, town or city are mentioned, which are characterized by
different cultural traits of each other that allow their distinction. In
addition, there can be social subdivisions in each of these groups that
maintain their own characteristics that characterize them.
2.2.
Intercutural teaching practices
In
accordance with the new curricular design (2016), the mathematics and physics career creates inter-cultural practices by introducing the subject of cross-cultural education that benefits
the formation of the future professor, one of these practices are: the
knowledge of the cultural diversity of the nationalities
and people of the country, the valorization of the cultural
manifestations of the various members and the application
of strategies and creation of didactic resources that allow
an intercultural education.
To reach the goal it is possible to develop the creation
of projects that promote intercultural
education; research that proposes the development of the student in
intercultural aspects; to foster interdisciplinarity with etnomathematics
to achieve the recovery of ancestral knowledge and worldview, our language and national
identity.
2.3 Interculturality in the Constitution of Ecuador
In
the Political Constitution of Ecuador
of 2008, there are some articles referring to the intercultural education, particularly Article
27 states that education will focus on human beings and ensure their holistic
development in the context of
respect for human rights, the sustainable
environment and democracy it
will be participatory, compulsory, intercultural,
democratic, inclusive and diverse, of quality and warmth, i.e., it highlights the
anthropological character of education, which includes all the dimensions inherent in
human beings.; and the intercultural
nature of education is also highlighted.
Article
29 states
that it is the duty of the State to guarantee
the right of people and to learn in their
own language and cultural sphere.
This requires that all people
living in the Ecuadorian territory be taken into account, including
those who come from diverse cultural
and linguistic realities.
Finally,
in article 57, numeral 14, reference is made to develop, strengthen and enhance
the intercultural bilingual education system, with quality criteria from early
stimulation to the higher educative level. The identities should also be cared
for and preserved in accordance with teaching and learning methodologies.
What
is indicated in the Magna Carta makes it possible that these aspects are taken
into account in the career of mathematics and physics. The student must be led
to the preservation of the cultural identities of our country.
2.4 Importance of intercultural education
In
accordance with Akros (2018), intercultural education "has gained
importance since it proposes that diversity be valued more so that there is
more respect for people and their freedom of thought” (p. 2). In this sense it
is confirmed that interculturality is a form of coexistence and interaction
between people and therefore, in education, communication. With intercultural
education, flexibility and reflection are put into practice as part of their
identity. It can also be implemented through active methodologies, group work,
to develop equality, equity and respect.
In order to achieve the objectives of the teacher training,
workshops, trainings and courses
must be included, in whichthe future teacher has a preponderant role and adapts
according to the reality of the communities, and where permanent
socio-cultural changes are experienced with people of different backgrounds
and expressions that allow to have links
between
cultural groups. In addition, intercultural education is important because of
the positive assessment of diversity and respect for people.
Regarding the cultural aspect, every person expresses his/her interactions with others. Therefore, intercultural
education develops a reflection and an
educational practice that considers diversity to be normal in any group. In this sense,
intercultural education can put into
practice the flexibility and reflection of each person in its essence and culture because it
interacts with others, and
is part of their identity.
2.5
Strategies
for the intercultural education
In
the career of mathematics and physics career values
such as equality, respect, pluralism,
tolerance, cooperation,
responsibility, and the recognition of each student's personal right by
fostering their personal identity should be promoted to facilitate learning
and achieve objectives. The positives
of cultural diversity and languages must be recognized, giving attention
to diversity and respect for differences. In addition, demonstrations of racism and discrimination should be
avoided, and the promotion of ethnic groups.
To
reach the stated objectives, some
intercultural strategies should be used, such as avoiding
ethnic discrimination in and out of the classroom. Interculturality is treated
as a transversal axis of education
and is based on intercultural practices, for example, the
etnomathematic project. The
knowledge and cultural manifestations of the
students must
be taken into account in the classroom
so that through lectures
they discover or rediscover the ancestral knowledge of people
and nationalities of our country.
3. Etnomathematics
The
term "Etnomathematics" was coined by the Brazilian educator and
mathematician Ubiratàn D'Ambrosio in 1977, during a presentation for the
American Association for the Advancement of Science. From this date, numerous
proposals have been made to obtain a more precise definition, included by the
same D'Ambrosio in 1999, who indicates that the etymological abuse leads him to
use the words ethno and mathema for their categories of analyses and Ticas
representing technique (Martínez, 2013, p. 429).
Rohrer and Schubring (2011) indicate that Falsirol used the term
Etnomathematics as a combination
of ethnology and mathematics, and he used it before
D´Ambrosio and referred to
the work of Ewald Fettweis work (German
mathematical educator: 1881-1967). The conceptualization managed by Fettweis considered mathematics as a cultural
element that connected with other disciplines, such as: ethnology, history of mathematics,
history of culture and mathematical education
(Martínez, 2013, p. 429).
Etnomathematics of D´Ambrosio refers to
the mathematics practiced by groups such as urban communities, rural, workers, professional classes, specific ethnicities and
community of professional mathematicians. He points out that this discipline was created by specific
ethnicities and by other cultural groups according to their own mathematical processes,
symbols, jargon,
mythologies and reasoning models (Martínez, 2013, pp. 429 – 439).
Etnomathematics in its etymology is based on three
roots: ethno, matema
and tica. Ethno, which
comes from the Greek root ethnos that means peoples/races refers to the various natural, social,
cultural and imaginary environments. Mathema
from the Greek to
explain, to understand, to teach, and Thica
linked to the Greek
root tecni, related to the arts,
techniques and ways (Martínez, 2013, p. 430).
From the previous consideration, Gilmer (1995, p.
188) assumes etnomathematics as the "study of the mathematical
techniques used by cultural groups
identified to understand, explain and manage problems
and activities that arisen in their own environment". Additionally, Gerdes (2007), indicates that the etnomathematics
is derived from the overlap that occurs between cultural
anthropology, mathematics and mathematical
education, mentioning the need to be aware on the existence of various mathematics, according to different cultures.
Thus, it can be indicated that
the etnomathematics is practiced by cultural groups, identified social groups and guilds, in which
its reality is better understood.
In addition, this mathematics
is usually practiced since they
are children.
It
can also be said that etnomathematics is a variation of mathematic didactics, and it is interested
in studying the knowledge of a
cultural group and their behaviors for
interpreting the world and their
Cosmo. Therefore, the etnomathematics thought
is a way of understanding the learning process of mathematics according to ancestral knowledge.
Gómez and Ortiz refer to Gavarrete (2012) by pointing out
that at the end of the last century, several
discussions were developed on the linguistic and sociocultural aspect of mathematics,
such as the language used,
the semantics, the influence of culture and society. In the present century, it reflects on the most appropriate way
of legitimizing the knowledge of the etnomathematics. Epistemological and political aspects are taken into
consideration in order to generate
more democratic access to mathematics
(Gómez and Ortiz, 2016, p. 23). In addition, it should be
noted that in the learning
process, the etnomathematics should consider the ancestral inheritance and the particularities
of the group's worldview, and this must be
conceptualized and worked for indigenous communities, people or cultures. Mathematical concepts
must be linked to the social and economic structures of them, and learning methods and arithmetic and geometric
techniques should be used, and
should be contextualized to their surroundings in order to obtain
the knowledge of the students.
4. Target of Etnomathematics
According
to Oliveras (2000), the object of study of the etnomathematics refers to three
thematic areas that are well defined:
The first corresponds to the cultural-mathematical anthropology
with respect to
the elements that are needed to theoretically define the terminology and the
anthropological and epistemological approach of etnomathematics,
and to be able to understand
them. The second thematic area
refers to the mathematical
cognition contextualized towards
the groups, and
corresponds to the elements
of the cognitive psychology related to the mathematics in the daily life
and the
mathematical learning in the
school and out of it. The third are the curricular aspects and processes that correspond to the socio-cultural and political
conditions related to the problem
of curriculum and learning (Gavarrete,
2012, p. 49).
Etnomathematic knowledge has been present
in most cultures throughout history, since cultural diversity is one of the characteristics
of every civilization in the world.
The origin
of etnomathematics relies to the emergence of the ethnosciences
and anthropological research that has developed since the nineteenth century (Trujillo, 2016).
To understand the intrinsic relationship between the ancestral knowledge of people and fields such as mathematics
corresponds to traditional activities, for example, working with plants, animals,
architectural conceptions, graphic representations. Finally,
etnomathematic is centered on thenature.
4.1
The training
of the math professor
In the study carried out by Juan Ramón Cadena
entitled "Insertion of the etnomathematics in the teacher training in
the Ecuadorian higher education",
indicates that in Ecuador the careers of
teacher training in Mathematics have
not considered this discipline in any case. The results
of a survey of fifth-semester students of the teaching career in mathematics, Universidad Central del
Ecuador, determines that the level of basic knowledge of mathematical concepts, history, etnomathematics is deficient. A teaching practice must be developed involving
two independent but non-dichotomous processes, teacher training and classroom practice.
In the teacher
training, the insertion of etnomathematics should be
taken into account as a process
that allows the extrapolation of practices towards the construction of a theoretical construct. Thus, there must be a dialectical bidirectional
relationship of praxis and theory. As indicated, one of the most pressing problems in mathematics teachers is the search for problems
and didactic situations. All the methodologies used should propose a meaningful
learning in the students.
Didactic intervention
is a very delicate process. According
to the analysis of Cantoral (2016),
three planes are articulated in
the creation of conceptual systems:
· The first is the problem as a consequence of the nature of mathematical knowledge.
· The second, for the knowledge is the "local", which implies
a social praxis adjacent to the characteristics of the environment,
the context and theidea.
· The third is the search
for an own epistemology of mathematical knowledge, the set of categories
from didactic activity and systematized experiences that produce a
theory of authentic and autonomous learning.
The teaching-learning process of mathematics in
Ecuador has mainly been based on the uncritical acceptance and reproduction of the western rationality model. The Andean rationality recognizes
the otherness or difference as something essential that admits
and is enriched with other
forms of sensibility in the comprehension of the world. There is symmetry,
which is reciprocity, reflected in a dualistic
view of reality and non-arbitrariness.
Therefore,
the objective is to know the conceptions of contemporary culture and
education in a dialogue with anthropology, history, sociology and educational psychology. These constructs help the learning process and develop the pedagogy of mathematics. In addition, the aspects relating to the characteristics of each cultural sector should be taken into account, considering the specific conditions of cognition and conceptualization of mathematical
elements.
5. Conclusions
The career of
Education in the experimental sciences, mathematics
and physics practices the intercultural education inside its classrooms, and proposes to
develop the concepts in
the
practice and in the linkage with the society through its practical application
in the individual and group research.
Using active methodologies makes necessary to realize the discovery and rediscovery of the contents of the etnomathematics;
thus, bibliographical and documentary research will be used with individual and group work,
exhibitions and elaboration of graphicorganizers.
The
use of interculturality will be evident in the knowledge of its principles, and
it will be carried out in educative institutions through intercultural
educational practices. Etnomathematics will contribute significantly to the
rescue of ancestral knowledge.
Interculturality and etnomathematics can be carried out with field work that contributes to meaningful learning to
assess the cultural identity of
the country in an efficient way. The basic
definitions will be identified to
initiate the knowledge of these
subjects through the development
of construction and reconstruction processes.
The teacher training
process will be intentional and oriented to
the conditions demanded by
today's society, and will be related to the educational policy, the training of the person and
the professional development of the teacher as a reflective, analytical, critical,
committed, flexible and transcendent entity.
It
will be necessary to improve the academic quality
of the students, future teachers, in the formation of good citizens. One of the strategies will be the implementation of intercultural education in the
classrooms. The application of etnomathematics in
the Elementary and High Schools through educational
intercultural practices and the
rescue of the ancestral knowledge. Bibliographical
and field research will contribute to meaningful learning to assess
the cultural identity of thecountry.
Bibliography
Asamblea Nacional del Ecuador. (2008). Constitución de la
República del Ecuador. Quito - Ecuador: Lexis.
Blanco, H. (2008).
Entrevista al profesor Ubiratán D’Ambrosio.
[Documento el Línea]. Revista
Latinoamericana de Etnomatemática, 1(1). 21-25. Disponible en http://www.etnomate-matica. org/v1-n1-febrero2008/blanco.pdf.
Blanco-Álvarez, H., Higuita Ramírez, C., & Oliveras, M. L.
(2014). Una mirada a la Etnomatemática y la Educación Matemática en Colombia: caminos recorridos. Revista Latinoamericana
de Etnomatemática, 7 (2), 245 – 269.
Cadena Juan Ramón. (s/f). Inserción de la Etnomatemática en
la formación docente en la Educación Superior Ecuatoriana. Quito. Ecuador.
Carrera de Matemática y Física. (2016). Diseño Curricular de
la Carrera de Matemática y Física, Quito, Ecuador.
Durán R. Rolando. Teoría de
la interculturalidad. uto=download
Falconi G. José. Principios de
interculturalidad. Recuperado el 21 de octubre del 2018, de
https://derechoecuador.com/principios-de-interculturalidad.
Gavarrete María Elena. (2012). Modelo de
aplicación de Etnomatemáticas en la
formación de profesores en el contexto indígena de Costa Rica.
Gerdes, P.
(2007). Etnomatemática. Reflexões sobre Matemática e diversidade cultural.
Famalicão:
Edições Húmus.
Gilmer, G. (1995). Una definición de Etnomatemática.
Boletín ISGEm, 11 (1), p. 188. En
H. Blanco (Comp.). Boletines del grupo de
estudio internacional de Etnomatemática: ISGEm, 1985-2003, disponible en http://www.etnomatematica.org/home/?page_id=112.
Gómez Guerra Enrique Octavio, Ortiz Lucero María Fernanda,
(2016), Incorporación participativa de formas de pensamiento
etnomatemático en programas curriculares de 5to, 6to y 7mo año de educación básica de dos colegios particulares de Quito para el año lectivo 2016 – 2017, PUCE,
Quito, Ecuador.
La importancia de la
educación intercultural. Recuperado el 21 de octubre del 2018, de https://akroseducational.es/blog/la-importancia-de-la-educacion-intercultural/
Marga Carol. (2017). La importancia de la educación intercultural
en un mundo globalizado. Recuperado el 21 de octubre del 2018, de http://colegioedwardconcepcion.cl/la- importancia-de-la-educacion-intercultural-en-un-mundo-globalizado/.
Martínez Padrón Oswaldo Jesús, (2013), Etnomatemática: una
reseña crítica de sus acepciones, Revista científica, Educación científica y
tecnológica / ISSN 0124 2253/ Octubre de 2013 / Edición especial / Bogotá, D.C.
Nomberto, Víctor. (2010). Teoría de la interculturalidad. Recuperado de
http://blog.pucp.edu.pe/blog/victornomberto/2010/02/18/teor-a-de-la- interculturalidad/
Oliveras María Luisa.
(2000). Etnomatemáticas. En J. Fuentes y M. L. Olivares. Matemáticas en la
Sociedad. Granada. España. Repro-digital.
Prieto Martín.(2017).
Enfoques de la interculturalidad. Recuperado el 2018-10-21 en https://prezi.com/qjqy60lojrsn/enfoques-de-la-interculturalidad/
Trujillo Pablo. (2015). Cultura,
Identidad e Interculturalidad. Recuperado el 21 de octubre del 2018, de https://es.slideshare.net/pablotrujillo3956/cultura-identidad-e- interculturalidad
Trujillo, J. (2016). La Etnomatemática Runayupay.
Recuperado el
08 de agosto de 2016,
de Proyecto Etnomatemática Runayupay: http://www.etnomatematica- ecuador.runayupay.org/assets/doc-141128-la_etnomatematica_(jt).pdf
Walsh, Catherine.
(2010). Análisis interseccional y enfoque intercultural.
Authors
LUIS IVAN DAVILA-GARZON holds a Master's degree
in Education with a specialization in
Educational Management, Universidad Politécnica Salesiana (Ecuador) in 2012. He
obtained a Master in University Teaching
and Educational Administration from
Universidad Tecnológica Indoamérica (Ecuador) in 2003. He obtained
his PhD in Education sciences, Universidad Central del Ecuador (Ecuador) in 2001. He got
a bachelor's degree in Education Science in the specialization
of Physics and Mathematics, Universidad Central del (Ecuador) in
1981.
He
is currently a professor of mathematics and physics
at the Faculty of Philosophy, Letters and Education Sciences of Universidad Central del Ecuador. He is the
author of
several books and articles published.
XIMENA PINOS-BENAVIDES obtained her master's degree in criminal law, Universidad Tecnológica Indoamérica
(Ecuador) in 2012. She obtained
a title of procedural law,
Universidad Tecnológica Indoamérica (Ecuador)
(Ecuador) in 2009. She has
a Diploma in Procedural Law, Universidad Tecnológica Indoamérica (Ecuador) (Ecuador) in 2008. She
holds a degree of specialist in Applied Geography, Universidad Andina Simón Bolívar (Ecuador) in 2007. She obtained
the title of Docto in law of the courts and judgements of the Republic,
Universidad Central del Ecuador (Ecuador) in 2005. She
obtained a Bachelor of Science Education in the specialization
of History and Geography,
Universidad Central del
Ecuador (Ecuador) in 2004.
She is currently a professor of Mathematics and Physics
at the Faculty of Philosophy, Letters and Education Sciences of Universidad Central del Ecuador. She is the
author of several books and published articles.