Como citar APA:
Puyol, J., Benítez, F. (2023) Estudio comparativo de investigación basada
en el uso de la técnica álgebra geométrica a dos colores para desarrollar el pensamiento
espacial. Repique, 5(2), 112-125
Vol. 5 Núm. 2
July - December 2023
e-ISSN: 2550-6676
pp 112-125
Comparative research study based on the use of the two-color
geometric algebra technique to develop spatial thinking
Jorge Luis Puyol Cortez
*
Félix Benítez Mero*
Abstract
The comparative analysis carried out in this work allows us to
determine the differences and similarities of two investigations, in
which didactic proposals are presented with the teaching strategy:
Geometric Algebra in Two Colors. This analysis, together with the
results of the interviews with the teachers of Mathematics I, and the
results obtained in the evaluation of the first semester students of
Business Administration and Accounting and Auditing, allow us to
prove that the application of this didactic proposal helps to develop
spatial thinking and to prove the hypothesis, which assumes that
two-color geometric algebra helps to improve academic
performance in mathematics.
Keyword:
methodology, didactic proposal, mathematics education
* Universidad Técnica “Luis Vargas Torres” De Esmeraldas, jpuyol@ulvt.edu.ec,
https://orcid.org/0000-0002-0734-694X
*
Universidad Técnica “Luis Vargas Torres” De Esmeraldas, fmerol@ulvt.edu.ec,
https://orcid.org/0000-0003-1877-7785
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113
Técnica álgebra geométrica a dos colores para desarrollar el
pensamiento espacial
Resumen
El análisis comparativo realizado en este trabajo nos permite
determinar las diferencias y similitudes de dos investigaciones, en
las que se presentan propuestas didácticas con la estrategia de
enseñanza: Álgebra Geométrica a Dos Colores. Este análisis junto
a los resultados de las entrevistas dirigidas a los docentes que
imparten la asignatura de Matemáticas I, y a los resultados
obtenidos en la evaluación tomada a los estudiantes de primer
semestre de las carreras de Administración de Empresas y
Contabilidad y Auditoría, nos permiten comprobar que la
aplicación de esta propuesta didáctica ayuda a desarrollar el
pensamiento espacial y a comprobar la hipótesis planteada, la
misma que tiene como supuesto que el álgebra geométrica a dos
colores ayuda a mejorar el rendimiento académicos en las
matemáticas.
Palabra clave:
metodología, propuesta didáctica, enseñanza
matemática
Received :
06-01-2023
Approved:
08-03-2023
INTRODUCTION
One of the main obstacles in the learning of mathematics is the
poor mastery of basic concepts, which leads to insecure and
ineffective processes for solving exercises and makes it difficult to
anchor with the topics foreseen in higher levels of the curricula.
Jorge Luis Puyol Cortez
Félix Benítez Mero
114
The effective learning of students and the formation of
professionals capable of solving real problems of society are
compromised when the main actors of education, teachers, do not
know or do not use effective methodologies and tools to match the
mathematical knowledge of young people entering university.
The use of the technique called two-color Geometric Algebra is
proposed as an alternative to equalize the knowledge of first
semester students, knowing its effectiveness to reach the solution
and understanding of exercises with basic Algebra polynomials,
which are the basis of the knowledge of higher Algebra, thus
relating it to what is established in the research line "Education and
Pedagogy" to which the research project belongs FAPED, the
faculty to which this master's degree is attached, promoting
educational development, culture, The FAPED faculty, to which this
master's degree is attached, promotes educational development,
culture, ancestral knowledge and recreation in the Province of
Esmeraldas, thus aiming to contribute to the fulfillment of the fourth
sustainable development goal "Ensure inclusive, equitable and
quality education and promote lifelong learning opportunities for
all, intertwined with goal 1, 6 of the National Development Plan
creating opportunities which "guarantees the right to health,
education and integral care during the life cycle under quality
criteria", landing this work in the fulfillment of the institutional
objectives of UTELVT such as "e, h, i, and j". In this way, we intend
to contribute to the solution of the low appropriation of basic
concepts and poor understanding of the solution processes in
algebraic subjects, which produces an inefficient mastery and low
academic performance of students in the first levels of the careers
in the administrative field of the Universidad Técnica Luis Vargas
Torres.
An educational project presented by the government of Ecuador in
2007 indicated the following:
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In recent years, and as a consequence of the great changes
produced by the processes of globalization, the irruption of
information and communication technologies and other
phenomena and processes associated with the information society,
there is an insistent questioning of what are the learning, the
cultural knowledge, that have to form the frame of reference for the
full exercise of citizenship. (Ministry of Popular Power for Education,
2007).
But the importance of mathematics was never doubted, since its
use is fundamental in every field; in any type of business it is
required to elaborate budgets, projections and analyze results; to
model and build houses it is required to know the dimensions of a
land; an anesthesiologist must know what dose to assign to the
patient so that he can operate him; and this would not be possible
without the application of the laws and procedures dictated by
mathematics.
Understanding mathematics and its rules to complete the resolution
processes that a problem requires is not easy for students, because
most of them generally present an aversion or resistance to
traditional teaching methods, considering the subject as a difficult
and not very understandable subject. On the other hand, students
who manage to understand the subject with ease lose interest in it,
since they will have to wait until most of their classmates have
understood the last topic for the teacher to move on to a new one.
The diagnostic test conducted in the academic period 2s-2021,
applied to students of the Universidad Técnica Luis Vargas Torres
de Esmeraldas who entered the first level of the career of Business
Administration and Accounting and Auditing, yielded clear
information, revealing a poor command of basic knowledge of
mathematics. In order to advance in their higher studies, students
must have a logical mathematical thinking and be able to solve
problems with complex variables, therefore, it is essential to design
Jorge Luis Puyol Cortez
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a methodological didactic proposal that ensures a rapid
assimilation of the concepts and basic rules of mathematics for the
resolution of algebraic problems. The non-use of creative
techniques such as two-color geometric algebra causes a poor
understanding of the basic algebraic fundamentals that serve as an
anchor to consolidate higher mathematical knowledge and limit the
academic progress of students who study majors related to
mathematics.
Achieving the leveling of mathematical knowledge in a course is a
complex task, since its fulfillment can be affected by different
variables, among them we can indicate the individual desire to
learn, the teacher's capacity to transmit his knowledge, the time
available, the methodologies used, the maturity in the
development of the student's autonomous work, the examples and
exercises presented and the tasks assigned.
The mastery of the mathematical foundations and the resolution of
problems with different unknowns, such as the majority of
mathematical exercises proposed in the careers of business
administration and accounting and auditing, is fundamental for the
appropriation of concepts that serve as support in the mastery of
higher knowledge, In the curricular mesh of the career many
professionalizing subjects have as a prerequisite the approval of
"Mathematics I" and "Mathematics II", sometimes the grades that
students get in the exam to pass the subject do not reflect the real
knowledge acquired, and this can be detrimental to the student,
especially when he/she fails to understand all the topics.
Based on the search performed it can be indicated that in Ecuador
at the moment there is only one research entitled <<Factorization
with two-color Geometry>>, to deal with the same subject they use
other names such as, for example; geometric algebra for
polynomials or geometric algebra of Euclid, in addition there are
many researches that aim to design a didactic proposal for the
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teaching of algebra. This may be motivated by what Gyves, (2006)
indicates in his research:
...in the last two decades the use of geometric methods has
gradually increased (with the rise of computers and educational
research). One of the reasons for this boom is the existence of
studies in favor of the multiple use of representation registers. (p.
4).
He also mentions that "These studies explain that the plurality of
semiotic systems -which consist of diverse representations of the
same object- makes it possible to increase the cognitive capacity of
the subjects" (Gyves, 2006).
The review of approximately forty publications has made it possible
to delimit the problem and to evaluate research of greater
relevance for the present work. Reviewed publications such as
"Euclid's Geometric Algebra: an experience in the teaching of
algebra", "Validation of the Delaunay triangulation using conformal
geometric algebra", "Design of the didactic unit: Factorization with
two-color geometry" and many more, demonstrate the great
interest of the scientific community in the topic chosen for this work.
All these researches are important because they allow us to know
the results obtained with the application of algebra teaching
methodologies with the representation of geometric figures and
mark a starting point to achieve the objectives set for this research.
The importance of the implementation of geometric figures for the
explanation of algebra lies in the possibility of developing algebraic
and geometric thinking, as indicated by Gómez (2021) in his
research in which he states that geometric algebra "helps to
develop two important thoughts for mathematics, the algebraic and
the geometric" (p. 2). Villella (2001) mentions that the teaching of
geometry involves offering elementary school students the
possibility of describing, understanding and interpreting the world
and its phenomena.
Jorge Luis Puyol Cortez
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118
In the research conducted by Delgado and Butto (2015) one of the
objectives is to investigate the feasibility of establishing a didactic
sequence based on Euclid's geometric algebra and to investigate
the algebraic representations of the relationship between different
areas, for this, a qualitative methodology is used with the intention
of understanding the meaning of the students' actions and to
describe the phenomena that occur in the classroom, such as the
difficulties presented by each student in the development of the
proposed exercises. With this study, it is concluded that the didactic
sequence for problem solving should start with the interconnection
of the algebraic-geometric part from the algebraic knowledge and
establish through geometry, the areas and their equation for
rectangles and squares, with the purpose of interacting easily in the
algebraic processes.
Guerra (2014) also presents a didactic proposal for factoring with
two-color Geometry, in his final research work for obtaining the
degree of Master in Education, In his work he uses the experimental
method that allows him to observe, manipulate and record all the
variables of study. For her proposal she considers necessary to
develop 7 sessions, each one with a time of forty minutes, to deal
with the topics presented:
Review and revision of contents. Remarkable products and
remarkable quotients.
Common Factor and Common Factor by Grouping Terms.
Trinomials.
Difference of perfect squares and Sum and difference of perfect
cubes. Presentations and defense of the procedures of an exercise.
Recovery with two-color Geometric Algebra (Guerra, 2014, p.18).
METHODOLOGY
In order to achieve each of the specific objectives of this study, it
has been necessary to implement various research methods and an
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instrument for data collection, which guarantees the reliability of
the information.
A mixed methodology is used, which will allow the combination of
qualitative and quantitative methods to achieve the objectives set
out in this research.
For the development of this research, the problem was posed, a
review of the state of the art was carried out, a hypothesis was
formulated, data were collected from students, teachers and
comparative analysis, applying an evaluation, an interview and a
comparative table, finally, the data obtained were analyzed with
which conclusions were drawn. With the comparative analysis of
the two didactic proposals, the similarities and differences were
determined, these were used to design and implement a new
didactic proposal that was applied to the students of the university
headquarters to verify the hypothesis raised. Therefore, we can
indicate that the deductive method and the inductive method were
applied.
Due to the nature of the present research work, the main method
implemented is the Comparative Method, about this method Rus
(2020) mentions that it is a way of elaborating or rejecting theory
and hypotheses using comparisons based on procedures similar to
scientific methods. So it is oriented to test the validity of an
argument using science and the study of differences and
similarities.
From the above we can affirm that the comparative method gives
us the facility to determine, based on the study of the similarities
and differences of a real argument, if a hypothesis or theory is
approved or rejected by omitting a statistical procedure. In this
case, thanks to the comparative method it will be possible to
determine the differences between the didactic proposals
reviewed and finally if the two-color geometric algebra improves
the academic performance in mathematics of first semester
Jorge Luis Puyol Cortez
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students of Business Administration and Accounting and Auditing
of the Luis Vargas Torres Technical University of Esmeraldas.
Another indispensable methodology for the development of this
research is the descriptive methodology, according to Cerda
(1998, as cited in Bernal, 2010), "traditionally the word describe is
defined as the act of representing, reproducing or depicting
people, animals or things..."; and adds: "One must describe those
most characteristic, distinctive and particular aspects of these
people, situations or things, that is, those properties that make
them recognizable to the eyes of others" ( p.
71). Bernal (2010) indicates that "one of the main functions of
descriptive research is the ability to select the fundamental
characteristics of the object of study and its detailed description of
the parts, categories or classes of that object" (p. 129).
The ethnographic technique and documentary analysis were
essential for the process of obtaining and classifying the
information, with the first one it was possible to obtain the
necessary information to analyze the impact of two-color geometric
algebra in the first semester courses of Business Administration and
Accounting and Auditing of the Universidad Tecnica Luis Vargas
Torres of Esmeraldas and with the bibliographic review it was
possible to compare the revised methodological proposals.
RESULTS
One of the objectives of this research aims to identify the minimum
knowledge of the population about two-color geometric algebra,
to meet this objective it was essential to conduct an interview with
4 teachers of the Technical University Luis Vargas Torres of
Esmeraldas who teach or have taught the subject Mathematics 1.
The first question addressed to the teachers was whether they had
implemented the two-color geometric algebra didactic strategy for
solving algebraic problems, the answer of all teachers was that they
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had not used this strategy before, three teachers indicated that
they had heard about this strategy, but did not know it in detail,
one teacher said that he did not use it because there were no
resources to apply this strategy.
The second question addressed to the teachers dealt with the
methodologies used for teaching the operations of algebraic
expressions. One teacher mentioned that he used the problem-
based methodology, indicating that he posed problems with one
or two unknowns, so that the student would first write down the
linear equation that represents the problem, and then proceed to
solve it. The other teachers stated that they use the classical
methodology, which consists of explaining the concepts,
presenting examples of solving the exercises, and proposing
similar exercises.
In the third question, we asked about the difficulties that students
have in the subject. All teachers agreed and stated that students
have difficulties in solving exercises with rational whole and
fractional algebraic expressions with a basic degree of complexity.
In some cases, a minority of students also present difficulties in
solving exercises due to the lack of knowledge of the rules of
precedence of operations.
To the question "What are the most common errors in solving
exercises? They responded that based on their experience,
students do not manage to reach the solution of an exercise or
problem because they do not apply a reflective strategy to reach
the solution, so they make calculation errors, do not identify the
available data, fail to identify the unknown, or do not complete the
exercise.
In the penultimate question, "What are the topics planned in the
syllabus and the expected achievements? The interviewees were
able to state that in the first unit there is a review of the concepts,
properties and relationships of natural numbers and integers
Jorge Luis Puyol Cortez
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proper to arithmetic, and then they go on to deal with topics such
as linear equations, fractional equations, equations with radicals,
quadratic equations.
The last question of the survey was asked only to the teachers who
currently teach Mathematics I. This was done after applying the
didactic proposal, the question asked the teacher to indicate the
benefits and results he/she could observe. The textual responses
of the teachers are reported:
● Teacher 1
Students show greater interest in the classes, a greater
predisposition to solve the proposed exercises is appreciated, and
they manage to complete the exercises in less time.
● Teacher 2
With the implementation of geometric figures it is easier for
students to solve the proposed exercises in a faster and safer way,
greater enthusiasm is appreciated, the planned topics can be dealt
with in less time, but it is necessary to have software that allows
solving and saving the exercises.
CONCLUSIONS
The conclusions derived from the bibliographic analysis, the
observation and verification of the research selected for analysis,
and the results obtained from the surveys conducted with teachers
and the evaluations taken from the students of Business
Administration and Accounting and Auditing.
Based on the bibliographic review carried out to determine the
minimum knowledge of the population on the topic: two-color
geometric algebra, we can indicate that in Ecuador to date there is
only one research that deals with the topic under this name, even
so, we were able to determine that the topic of analysis is
investigated only under the name of Geometric Algebra. There are
numerous investigations that talk about the importance of
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geometric algebra for and its applications in engineering, however,
the investigations are limited in the analysis of the benefits of
geometric algebra for the teaching of mathematics, it is highlighted
that there is a strong separation of algebra and geometry in
compulsory education. It has been observed that other Spanish-
speaking countries have dealt with the subject and have elaborated
didactic proposals that implement the geometric algebra strategy
for teaching mathematics. In this study, the results and conclusions
show that the implementation of a didactic proposal with the two-
color geometric algebra strategy favors the teaching and learning
process of algebraic problems and exercises.
REFERENCES
Alvarez de Zayas. (1984). Fundamentos teóricos de la dirección del
proceso de formación del profesional de perfil ancho. Pueblo
y Educación.
Asamblea Nacional de la Republica del Ecuador. (2008).
Constitución del ecuador. Registro Oficial, 449, 1–132.
http://www.asambleanacional.gov.ec/documentos/constitucio
n_de_bolsillo.pdf
Ley Orgánica de Educación Superior, (2010) (testimony of
Asamblea Nacional del Ecuador).
Báez, R., & Iglesia, M. (2007). Principios didácticos a seguir en el
proceso de enseñanza y aprendizaje de la geometría en la
UPEL “EL MÁCARO” . Enseñanza de La Matemática, 12, 76–
87.
http://funes.uniandes.edu.co/14702/1/Baez2007Principios.pd
f
Baldor, A. (1983). Álgebra de Baldor (1st ed.). Compañía Cultural
Editora y Distribuidora de Textos.
Bernal, C. (2010). Metodología de la investigacion: administración,
economía, humanidades y ciencias sociales (PEARSON
Jorge Luis Puyol Cortez
Félix Benítez Mero
124
EDUCACIÓN, Ed.; 3ra ed.).
http://library1.nida.ac.th/termpaper6/sd/2554/19755.pdf
Cáceres, L., Prisco, C. di, Peña, J. A. D. la, Pineda, Á., & Solotar, A.
(2014).
MATEMÁTICAS EN AMÉRICA LATINA Y EL CARIBE. Interciencia,
39(11), 761-undefined.
https://www.redalyc.org/articulo.oa?id=33932572001
Carretero, M. (2005). Constructivismo Y Educacion (Vol. 2).
Cepeda, J. (2013). Estrategia de enseñanza para el aprendizaje
por competencias (Primea).
Colom, Sureda, & Salinas. (1988). Tecnología y Medios Educativos.
Cincel. https://www.iberlibro.com/TECNOLOG%C3%8DA-
MEDIOS-EDUCATIVOS- Colom-Canellas-
Sureda/798849110/bd
Estatuto Universitario , Universidad Técnica Luis Vargas Torres de
Esmeraldas (2019).
https://drive.google.com/file/d/1emW1FbnsBD8DCyY9OhXF
5wuGgeDMgMtm/view
Díaz Barriga, Á. (2006). El enfoque de competencias en la
educación. ¿Una alternativa o un disfraz de cambio? Perfiles
Educativos, XXVIII, 7–36.
https://www.redalyc.org/pdf/132/13211102.pdf
Dirección de Desarrollo Curricular y Docente. (2018). Manual de
orientaciones: Estrategias Metodológicas de Enseñanza y
Evaluación de Resultados de Aprendizaje.
https://pregrado.ufro.cl/images/files/2018/documentos-
desarrollo- curricular/orientaciones-metodologicas.pdf
Dominguez, M., & Stipcich, M. (2010). Una propuesta didáctica
para negociar significados acerca del concepto de Energía.
Revista Eureka Sobre Enseñanza y Divulgación de
Repique. Revista de Ciencias Sociales.
Vol. 5 No. 2 – 2023
July - December
125
Las Ciencias, 7, 75–92.
https://www.redalyc.org/articulo.oa?id=92013011006
García, J. (2019). Estrategias en la resolución de problemas
algebraicos en un contexto intercultural en el nivel superior.
Bolema: Boletim de Educação Matemática, 33(63),
205–225. https://doi.org/10.1590/1980-4415V33N63A10
Gavilán, P. (2011). Dificultades en el paso de la aritmética al
álgebra escolar: ¿puede ayudar el Aprendizaje
Cooperativo? Investigación En La Escuela, 73, 95–108.
Godino, J., & Ruíz, F. (2002). Geometría y su didáctica para
Maestros. In Matemáticas y su Didáctica para Maestros.
Departamento de Didáctica de la Matemática.
http://www.ugr.es/local/jgodino/edumat-maestros/
Gómez Zapata, J. (2021). Uso del Álgebra Geométrica como
estrategia metodológica para el de desarrollo del
pensamiento numérico y geométrico de los estudiantes del
grado octavo de la Institución educativa Celestin Freinet
[Universidad Cooperativa de Colombia].
https://repository.ucc.edu.co/bitstream/20.500.12494/3370
0/4/2021_uso_algebra_geo metrica.pdf
Guerra, M. (2018). Diseño de la unidad didáctica:
“Factorización con Geometría a dos colores.”
Gyves, N. (2006). Discursos en los registros algebraico y
geométrico de las ecuaciones diferenciales ordinarias.
Educación Matemática, 18(2), 123–147.
https://www.redalyc.org/articulo.oa?id=40558507006
Kieran, C., & Filloy, E. (1989). El aprendizaje del Álgebra escolar
desde una perspectiva psicológica. INVESTIGACIÓN Y
EXPERIENCIAS DIDÁCTICAS , 7(3), 229–240.
