=Paper=
{{Paper
|id=Vol-3691/paper32
|storemode=property
|title=Systematic Review: State of Knowledge on Learning Difficulties and Teaching Strategies in Linear Algebra
|pdfUrl=https://ceur-ws.org/Vol-3691/paper32.pdf
|volume=Vol-3691
|authors=Alethia Piñón Jiménez,Diana Margarita Córdova Esparza
|dblpUrl=https://dblp.org/rec/conf/cisetc/JimenezE23
}}
==Systematic Review: State of Knowledge on Learning Difficulties and Teaching Strategies in Linear Algebra==
https://ceur-ws.org/Vol-3691/paper32.pdf
Systematic Review: State of Knowledge on Learning
Difficulties and Teaching Strategies in Linear Algebra
Alethia Piñón Jiménez1 and Diana Margarita Córdova Esparza2
1 Facultad de Informática, Universidad Autónoma de Querétaro, Av. de las Ciencias S/N, Juriquilla, Querétaro 76230,
México
2 Facultad de Informática, Universidad Autónoma de Querétaro, Av. de las Ciencias S/N, Juriquilla, Querétaro 76230,
México
Abstract
This paper presents a systematic review focusing on diagnosing learning difficulties and implementing
didactic strategies in linear algebra. We aim to deepen the understanding of this topic over the past
decade. Our research, guided by four questions, analyzed 78 articles, ultimately including 37 in this
review. We based our search strategy on the PRISM protocol and used specific indicators. Our findings
indicate that most authors in this review primarily use the APOE theory and genetic decomposition for
formal diagnosis of learning problems. This approach helps build knowledge frameworks, especially in
vector spaces and linear transformations. A key finding is the prevalent use of digital technology in both
the models and strategies proposed in these studies. This review highlights opportunities for future
research in diagnosing learning problems and developing innovative, technology-integrated strategies
in education.
Keywords
Education, didactic strategy, linear algebra 1
1. Introduction
In the field of education, teaching and learning mathematics often presents significant challenges
for teachers. These challenges include covering the subject's content within the allotted time and
addressing the diverse learning difficulties that students face. Additionally, teachers must
develop effective teaching strategies to enhance learning outcomes in mathematics.
Each researcher in this field brings their unique perspective, knowledge, and experience to
analyze the state of knowledge on teaching and learning mathematics. Despite these efforts,
learning problems in linear algebra, especially in abstract topics like vector spaces and linear
transformations, persist (31).
This paper aims to conduct a systematic review to better understand how learning difficulties
in linear algebra are formally diagnosed and what teaching strategies are being implemented. The
importance of this review becomes evident when considering that linear algebra is a fundamental
subject in science and engineering courses. It contributes significantly to developing students'
logical, heuristic, and algorithmic thinking skills by using linear models to predict and control
system behaviors.
Therefore, this review will analyze current knowledge on diagnosing student learning
problems in linear algebra and the recent implementation of didactic strategies to improve
teaching and learning in this field.
CISETC 2023: International Congress on Education and Technology in Sciences, December 04–06, 2023, Zacatecas,
Mexico
alethia463@gmail.com (A. Piñón-Jiménez); diana.cordova@uaq.mx (D. M. Córdova-Esparza)
0000-0003-0326-4741 (A. Piñón-Jiménez); 0000-0002-5657-7752 (D. M. Córdova-Esparza)
© 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ht
tp:
//
ceur
-ws
.or
g
Works
hop I
SSN1613-
0073
Pr
oceedi
ngs
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
2. Method
Our search strategy used the PRISM (Preferred Reporting Items for Systematic Reviews and
Meta-Analyses) protocol as a reference and followed specific indicators. We guided our research
with four key questions:
1. What are the main factors influencing learning problems in linear algebra?
2. Which learning theories have been applied to formally diagnose these learning problems
and design teaching strategies for linear algebra?
3. What are the developed thematic strategies for linear algebra, and do they share any
common characteristics?
4. What were the sizes of the groups used to validate the formal diagnoses or as pilot groups
for implementing teaching strategies?
To address our research questions and achieve the study's objective, we conducted a
systematic literature review. This method is known for systematically integrating empirical
results related to a specific research problem (34). We developed our research methodology in
four distinct stages, which we detail in the subsequent paragraphs.
Stage 1: Setting Inclusion and Exclusion Criteria for Research Studies
In this first stage, we established specific criteria for including and excluding studies in our
research. For inclusion, we focused on research articles, excluding other document types like
theses and book chapters. We considered articles published from 2013 to 2022, ensuring the
research was no more than 10 years old. Additionally, we included studies written in Spanish,
English, or Portuguese. The final inclusion criterion was that the articles must be related to
teaching or learning linear algebra; we excluded articles on topics outside this specific
educational area.
For exclusion, we omitted any articles that did not meet all our inclusion criteria. This also
included articles that were duplicates in our study.
Stage 2: Developing the Search Strategy
In this stage, we executed our search strategy across various databases, yielding 71 articles for
analysis. Our search criteria varied depending on the database to maximize results (see Figure 1).
We selected databases that showed the highest number of relevant results for our topic. The
databases and their respective search formulas were:
1. ERIC: Using the formula (“Education”) AND (“Linear Algebra”), we obtained 9 articles.
2. Scielo and DOAJ: We used (“Education”) AND (“Linear Algebra”) and (“Education”) AND
(“Linear Algebra”), obtaining 8 and 30 articles, respectively.
3. Redalyc: With the formula (“Education”) AND (“Linear Algebra”), we found 8 articles.
4. Science Direct: We used (Teaching OR Learning) AND (“Linear Algebra”), leading to 9
articles.
5. Dialnet: The formulas (Teaching OR Learning) AND (“Linear Algebra”) and (Didactics) AND
(“Linear Algebra”) resulted in 7 articles.
Additionally, we identified 49 articles through references. After applying our exclusion
criteria, 7 of these were ultimately included in our study.
Figure 1: Overview of the Information Search and Data Collection Process. This flowchart details
the search terms used across various databases, the number of articles retrieved from each, and
the filtering process leading to the final selection of articles included in the review. It also outlines
the exclusion criteria applied and the total number of articles analyzed.
Stage 3: Information Purification
In this stage, we conducted an initial review of the 78 articles gathered from the databases and
references mentioned earlier. The purpose of this review was to assess each article's relevance
to our research objectives. We rejected 34 articles during this process because they did not
provide relevant data for our systematic review analysis or contribute to answering our research
questions. Consequently, 37 articles were selected and included in our review.
Stage 4: Data Coding and Analysis
In this final stage, we analyzed the data based on specific categories. This structured approach
helped us to thoroughly examine and understand the findings. The categories we focused on
were:
1. Factors influencing learning problems in linear algebra.
2. Learning theories applied for diagnosing learning problems or implementing teaching
strategies in linear algebra.
3. Thematic contents within the subject of linear algebra that were the focus of the research.
4. Strategies implemented in teaching linear algebra.
5. Sizes of the samples used for validation or implementation in pilot tests.
This categorization facilitated a comprehensive analysis of the collected data, aligning it
closely with our research objectives.
3. RESULTS
Factors Influencing Linear Algebra Learning Problems
The factors identified as influencing learning problems in linear algebra are varied, as
observed in the systematic review of the research. Despite this diversity, there is a notable
consistency in the findings. This is apparent when we see that several factors recur across
multiple studies. In some instances, more than one factor is repeated between different
investigations, as detailed in Table 1. This repetition underscores the commonalities in challenges
faced by learners in linear algebra.
Table 1
Influential Factors in Linear Algebra Learning Problems Identified in Scholarly Research
This table compiles pivotal studies on linear algebra, listing the year of publication, authors, article
title, and the predominant factor influencing learning difficulties as identified in each piece of
research.
Year of Authors Title article Predominant Factor
publication
Increasing Reality and
Nishizawa et
2013 Educational Merits of a Virtual Abstract
al.
Game
2013 Parraguez The role of the body in the Abstract
construction of the concept of
Vector Space
2013 Rosso & A taxonomy of errors in Abstract, Language,
Barros learning vector spaces Various representations
University students’ solution
Abstract, Prior
2014 Birinci et al. processes in systems of linear
knowledge, Axiomatic
equation
Coordination of semiotic
Ramírez-
representation records in the
2014 Sandoval et Various representations
use of linear transformations
al.
in the plane
A teaching experience of Abstract
Salgado &
values, vectors and
2014 Trigueros
eigenspaces based on APOE
Gaisman
theory
Constructions and mental Abstract
Trigueros
mechanisms for learning the
2015 Gaisman et
matrix theorem associated
al.
with linear transformation
Berman & Definitions are important: the
2016 Formalism
Shvartsman case of linear algebra
Advanced mathematical
Marins & thinking manifested in tasks
2016 Formalism, Abstract
Pereira involving linear
transformations
A Teaching Proposal for the
Abstract, Formalism
2017 Beltrán et al. Study of Eigenvectors and
Eigenvalues.
2017 Costa & Teaching linear algebra in an
Abstract, Without
Rossignoli engineering school:
connection with other
Methodological and didactic
subjects, Language
aspects
From Practical to Theorical
2018 Pierri Thinking: The Impact of the Abstract, Formalism
Role-Play Activity.
2019 Álvarez- Teaching Linear Algebra in Epistemological
Macea & engineering courses: an component, didactic
Costa analysis of the process of schemes, Language
mathematical modeling within
the framework of the
Anthropological theory of
didactics
Teaching Linear Algebra Abstract, procedures
Aytekin &
2019 Supported by GeoGebra memorization, lack of
Kiymaz
Visualization Environment vinculation
2019 Gallo et al. Interpretation of linear
Formalism, Language,
transformations in the plane
Various representations
using GeoGebra
Linear algebra learning
García-
2019 focused on plausible reasoning Formalism
Hurtado et al.
in engineering programs
Teaching-Learning of Matrices Abstract, Formalism,
2019 Xavier et al.
in the civil Engineering Course Prior knowledge
Construction of the meanings
of vector space operations Abstract, Formalism
2020 Parraguez
through linearly
independent/dependent sets
A Didactic Sequence for
Teaching Linear
Transformation: Unification of Concept application
2020 Pizarro
Methods and Problems, conditions
Modeling and Explanation of
Learning
2021 Cárcamo et Hypothetical learning Abstract
al. trajectories: an example in a
linear algebra course
2021 Kariadinata Students Reflective Abstraction Abstract
Ability on Linear Algebra
Problem Solving and
Relationship with Prerequisite
Knowledge.
2021 Silva et al. Creation and uses of LineAlg Formalism
application as a learning object
in basic education
2021 Wibawa et al. Learning Effectiveness Abstract,
Through Video Presentations Demonstrations, Large
and WhatsApp Group (WAG) in number of operations
the Pandemic Time Covid-19 between variables
In our systematic review, we found that the most significant factor affecting linear algebra
learning, as identified by various authors, is the subject's level of abstraction and formalism (see
Figure 2). The high level of abstraction required by linear algebra itself poses a challenge for
students, demanding a substantial degree of abstract thinking for proper understanding (37). As
for formalism, it stems from the way linear algebra is presented, studied, and learned in the
literature, which heavily relies on the formalism of mathematical language (25).
Other key aspects impacting linear algebra learning difficulties include students' challenges in
differentiating between a concept and its various representations (29) and the use of diverse
languages when discussing vector spaces and linear transformations (8). Additionally, the
connection to the teacher's training emerges as a notable factor. If a teacher has a background in
mathematics or a related field, the issue often lies in not having the foundational structures in
place. Conversely, for engineering educators, the challenge is often linking the relevance and
applicability of linear algebra concepts to their specific field (28).
Incident factors in learning Linear Algebra
18
16
14
12
10
8
6
4
2
0
ct
ns
ism
ge
ge
rs
k
lin
ra
tio
ua
he
d
al
le
st
No
ng
Ot
ta
rm
w
Ab
en
no
Le
Fo
es
rk
pr
io
re
Pr
us
rio
Va
Figure 2: Prevalence of Factors Impacting Learning in Linear Algebra
This bar chart illustrates the frequency of various factors that influence the learning of linear
algebra, as identified in the reviewed research, including abstraction, formalism, language, and
others.
Learning Theories Applied in Linear Algebra
This section highlights the learning theories applied in diagnosing learning difficulties and
implementing didactic strategies in linear algebra. It also covers the tools used in the various
research projects analyzed. Additionally, we provide information about the countries where each
study was conducted, as detailed in Table 2.
Table 2
Learning Theories and Research Tools Utilized in Linear Algebra Studies.
This table enumerates the studies included in the systematic review, outlining the applied learning
theories, the research tools used, and the countries where the studies were conducted.
Autor y Año País Applied learning theory Tool used
(Parraguez,
Chile APOE Semi-structured interview
2013)
(Rosso &
Theory of didactic
Barros, Argentina Problems situations
situations and constructivism
2013)
(Parraguez
Questionnaire and
& Uzuriaga, Chile APOE
interviews
2014)
(Ramírez-
Theory of semiotic Interview with sequence of
Sandoval et México
represetations 5 activities
al., 2014)
(Salgado &
Trigueros Questionary and semi-
México APOE
Gaisman, structured interview
2014)
(Trigueros
Chile APOE Questionary and semi-
Gaisman et
structured interview
al., 2015)
(Murillo &
Beltrán, Spain APOE RGB color system
2016)
(González & Internalization of concrete
Colombia APOE
Roa, 2017) actions
(Roa-
Fuentes & Chile and
APOE Questionary
Parraguez, Colombia
2017)
(Costa, Argentina Anthropological Theory of
Study and research activity
2018) the Didactic
(Karrer, Theory of semiotic
Brazil Using GeoGebra
2018) represetations
(Rodríguez Questionnaire and
Chile APOE
et al., 2018) interviews
(Álvarez-
Anthropological Theory of
Macea & Colombia Study and research activity
the Didactic
Costa, 2019)
(Gallo et al., Theory of semiotic Series of computer activities
Argentina
2019) represetations using GeoGebra software
(Parraguez,
Chile APOE Written questionnaire
2020)
(Fortuny & Guía escrita, archivos de
Realistic mathematics
Fuentealba, Spain audio y video, entrevistas
education
2021) con algunos estudiantes.
(Betancur et Questionary and semi-
Colombia APOE
al., 2022) structured interview
In the systematic review, the APOE theory emerges as the most frequently applied learning
theory in the research works analyzed (see Figure 3). This theory has been predominantly used
to diagnose learning problems in linear algebra more accurately and deeply. It employs genetic
decomposition to develop mental schemes or structures that aid students in constructing
knowledge about specific concepts (30).
Regarding the theory of semiotic representations, the reviewed studies have utilized it to
support didactic strategies. These strategies involve varying representations of concepts, often
enhanced by computational tools for better graphic representation (16). The anthropological
theory of didacticism was applied to identify students' learning difficulties in linear algebra and
to back didactic strategies using modeling, incorporating technology such as mobile devices and
software (7).
The theory of didactic situations was employed to categorize common errors in learning the
topic of vector spaces (32). Additionally, the column labeled "others" in Figure 3 includes various
theories like the theory of didactic proposal situations and realistic mathematical education (10).
These theories have been instrumental in supporting didactic proposals for teaching linear
algebra.
Learning theories
12
10
8
6
4
2
0
APOE Semiotic representations Anthropology of the Others
didactic
Figure 3: Distribution of Learning Theories in Reviewed Research. This figure illustrates the
prevalence of different learning theories as applied in the research works reviewed. The APOE
theory leads in application, followed by semiotic representations, the anthropological theory of
the didactic, and other various theories.
In the systematic review, we noted the tools employed for conducting research. Prominent
among these are questionnaires and interviews, particularly in studies implementing the APOE
theory. The GeoGebra software stands out, along with the use of study guides on virtual platforms
and a variety of activities grounded in learning theories.
It is also worth noting the global reach of research in the field of linear algebra education.
Chile emerges as a leader in research production within Latin America. However, countries
outside the American continent, such as Spain, Turkey, and Indonesia, also contribute
significantly. This underscores the universal relevance of the challenges in teaching and learning
linear algebra, indicating that these difficulties are common in classrooms worldwide,
irrespective of location.
Regarding teaching strategies for linear algebra, the review also examined the specific subject
topics that have been the focus of research and the sample sizes used in these studies (refer to
Table 3).
Table 3
Overview of Teaching Models or Strategies, Topics, and Sample Sizes in Linear Algebra Research.
This table details the teaching models or strategies applied to linear algebra topics, specifying the
topics addressed and the sample sizes involved in each study.
Author and
Model or
year of Topics Sample size
strategy
publication
(Nishizawa et Digital Vectors in 3D 40 students
al., 2013) technology
(Yildiz Ulus, Digital Eigenvectors and
Not implementation
2013) technology eigenvalues
(Salgado &
Trigueros APOE-based 34 students on average
Eigenvectors and eigenvalues
Gaisman, activities per semester
2014)
Matrices and determinants,
(Petrov et Digital
Vector spaces, Eigenvectors 37 students
al., 2015) technology
and eigenvalues
(Gabriel Systems of linear equations,
Digital 35 teachers and 5
Vergara et Matrices, Eigenvectors and
technology students
al., 2016) eigenvalues
(Murillo &
Digital
Beltrán, Vector spaces Not implementation
technology.
2016)
Systems of linear equations,
(Torres et al., Digital Vector spaces, Matrices, Linear
Not implementation
2016) technology transformations, Eigenvectors
and eigenvalues
(Costa
Digital
&Rossignoli, Not specified Voluntaries 295 students
technology
2017)
(Meneu et Eigenvectors and
Activities Not implementation
al., 2017) eigenvalues
Digital
(Costa, 2018) Linear algebra with physics 50 students
technology
(Karrer, Digital
Linear transformations 2 students
2018) technology
(Kartika et Digital
Vectors 3D 69 students
al., 2018) technology
Digital Systems of linear equations,
(Pierri, 2018) 70 students
technology Matrices, Vector spaces
(Aytekin &
Digital
Kiymaz, Vector spaces 4 students
technology
2019)
(Gallo et al., Digital
Linear transformations Not implementation
2019) technology
(García- System of linear equations,
Mathematical 36 students
Hurtado et Matrices and determinants,
modeling
al., 2019) Vectors, Vector spaces
(Villalobos & Digital Vector operations 40 students
Ríos, 2019) technology
(Xavier et al.,
Activities Matrices Not implementation
2019)
Problem- 21 students and 21
(Nissa et al., Systems of linear equations,
based control group
2020) Matrices
learning
Didactic
engineering
(Pizarro, 17 students
and Linear transformations
2020)
Mathematical
modeling
(Fortuny & Hypothetical
7 students
Fuentealba, learning Vector spaces
2021) trajectories
(Silva et al., Digital Matrices, systems of linear
Not implementation
2021) technology equations
(Wibawa et Digital
Vector spaces 14 students
al., 2021) technology
This review reveals a strong emphasis on the use of digital technology in teaching the topics
discussed, with the specific tools and elements varying according to the research aims (Figure 4).
For instance, there is a focus on utilizing various mathematical software (35), knowledge
management platforms (26), web-based learning tools (17), virtual games (21), and virtual
evidence portfolios (36).
The systematic and thorough diagnosis of mental structures that underpin the understanding
of vector space concepts, linked to the design of proposed activities, was distinctly noted in the
study by (33). However, a common thread across many studies is that topics of higher complexity
and abstraction are most frequently addressed, both in diagnostic processes and in
methodological proposals for teaching and learning.
Notably, studies targeting instruction within the domain of engineering, particularly
mathematical modeling, are prominent (13). This aligns with the practical application
requirements characteristic of engineering curriculums.
Teaching Model or Strategy
14
12
10
8
6
4
2
0
Digital technology Mathematical Activities Others
modeling
Figure 4: Frequency of Different Teaching Models or Strategies Used
This bar graph illustrates the frequency with which various teaching models or strategies are
applied in linear algebra education, showcasing a predominant use of digital technology, followed
by mathematical modeling, diverse learning activities, and other strategies.
Research Focus on Linear Algebra Topics
The systematic review of research works revealed that most teaching strategies and diagnostic
efforts in linear algebra are focused on the more abstract concepts. Vector spaces (24), linear
transformations (30), and matrices are the topics most frequently addressed. Less commonly, but
still noteworthy, are studies on systems of linear equations (6) and eigenvalues and eigenvectors
(3). These findings align with the goal of the research: to develop tools that mitigate the factors
impacting the teaching and learning of complex linear algebra topics (4).
4. Discussion and Conclusions from the Systematic Review
The systematic review has led to several important conclusions regarding the factors that
hinder students' learning of linear algebra. High levels of abstraction (23), unfamiliar formalism
(18), language barriers (1), multiple representations of mathematical objects (12), lack of prior
knowledge (40), and weak connections in learning (18) are significant challenges. Additionally,
the complexity of new definitions, the quantity of operations between variables, and the subject's
epistemological and axiomatic characteristics are noted as less frequent but still impactful
factors.
In terms of learning theories, the review underscores the APOE theory as the predominant
framework for in-depth research on learning difficulties in linear algebra. The theory's popularity
suggests it effectively uncovers and addresses students' mental structures during knowledge
construction, as highlighted by Rodriguez et al. (31). Despite this, the APOE theory's main
application is in diagnosis, with other theories more commonly used to explore the results of
various teaching and learning strategies, except in the work of Salgado and Trigueros (33). This
review reveals a gap: the direct link between systematic diagnosis and strategy application is
often absent. This could be due to educational institutions' urgent need to produce quick results,
relying on authors' experience and conceptual understanding to design their approaches.
Digital technology's role is consistently significant in the research on teaching and learning
strategies. Mathematical software applications (16), (20), (41), web-based learning tools—
especially relevant during the COVID-19 pandemic for remote education (39), and virtual games
(38) are some examples that reflect the growing, irreversible trend of digital integration in
education. The main research focus in terms of content includes vector spaces (15) and linear
transformations (14), likely due to their complex and abstract nature requiring a deep
understanding.
Regarding sample sizes for statistical analysis in the reviewed studies, they ranged from 2 to
295 participants, with variations in application time and students' nationalities. This indicates a
need for further research with larger populations, leveraging digital technology for more
extensive validation and evaluation. The reviewed research, regardless of its focus, often bases
some methodological aspects on the authors' experiences, their conceptual understanding, and
sometimes the influence of a research community. The effectiveness of proposed solutions is
most significantly validated by the experiences of those who implement them.
Therefore, future research should aim to enhance the authors' experiences and perspectives
by developing methodologies that better connect with research communities and employing
digital technology. This approach could allow a broader student population to engage with and
benefit from the proposed methodologies in this review.
Acknowledgements
We would like to express our sincere gratitude to CONAHCYT for providing the scholarship that
supported the graduate studies enabling this research. Their generous assistance was invaluable
to the completion of this project.
References
[1] Álvarez-Macea, F., Costa, V. A. (2019). Enseñanza del Álgebra Lineal en Carreras de
Ingeniería : Un Análisis del Proceso de la Modelización Matemática en el Marco de la Teoría
Antropológica de lo Didáctico . Eco Matemático, 8231(2), 65–78.
[2] Aytekin, C., y Kiymaz, Y. (2019). Teaching Linear Algebra Supported by GeoGebra
Visualization Environment. Acta Didactica Napocensia, 12(2), 75–96.
https://doi.org/10.24193/adn.12.2.7
[3] Beltrán, María José, Murillo, Marina, Jordá, E. (2017). A Teaching Proposal for the Study of
Eigenvectors and Eigenvalues. Journal of Technology and Science Education, 5(3), 100–113.
[4] Berman, A., y Shvartsman, L. (2016). Definitions are important : the case of linear algebra.
European Journal of Science and Mathematics Education, 4(1), 26–32.
[5] Betancur, A., Roa Fuentes, S. y Parraguez Gonzá lez, M. (2022). Construcciones mentales
asociadas a los eigenvalores y eigenvectores: refinació n de un modelo cognitivo. AIEM -
Avances de investigación en educación matemática, 22, 23-46.
https://doi.org/10.35763/aiem22.4005
[6] Birinci, D. K., Delice, A., y Aydın, E. (2014). University Students’ Solution Processes in Systems
of Linear Equation. Procedia - Social and Behavioral Sciences, 152, 563–568.
https://doi.org/https://doi.org/10.1016/j.sbspro.2014.09.244
[7] Costa, Viviana Angélica. (2018). Uso de dispositivos móviles y de software matemático en la
enseñanza por investigación. Revista Electrónica de Enseñanza de Las Ciencias, 17, n, 626–
641.
[8] Costa, Viviana Angelica, y Rossignoli, R. (2017). Enseñanza del algebra lineal en una facultad
de ingeniería: Aspectos metodológicos y didácticos. Revista Educación En Ingeniería, 12(23),
49. https://doi.org/10.26507/rei.v12n23.734
[9] Dubinsky, E. y McDonald, M.A., APOS: A constructivist theory of learning in undergraduate
mathematics education research, the teaching and learning of mathematics at university
level, pp. 275-282, Springer Netherlands (2001)
[10] Fortuny, J. M., y Fuentealba, C. (2021). Las trayectorias hipotéticas de aprendizaje : un
ejemplo en un curso de álgebra lineal. ENSEÑANZA DE LAS CIENCIAS, 1, 45–64.
[11] Gabriel Vergara et al. 2016, Uso de Matlab como herramienta computacional para apoyar la
enseñanza y el aprendizaje del álgebra Lineal. Disponible en Revistas y Publicaciones de la
Universidad del Atlántico en
http://investigaciones.uniatlantico.edu.co/revistas/index.php/MATUA.
[12] Gallo, H. G., Verón, C., y Herrera, C. G. (2019). Interpretación de transformaciones lineales en
el plano utilizando GeoGebra. Revista Iberoamericana de Tecnología En Educación y
Educación En Tecnología, 24, e04. https://doi.org/10.24215/18509959.24.e0
[13] García-Hurtado, O., García-Pupo, M. M., y Poveda-Chaves, R. (2019). Linear algebra learning
focused on plausible reasoning in engineering programs. Visión Electrónica, 13(2), 322–330.
https://doi.org/10.14483/22484728.15164
[14] González, D., y Roa, S. (2017). Un esquema de transformación lineal: construcción de objetos
abstractos a partir de la interiorización de acción concretas. Enseñaza de Las Ciencias, 35(2),
89–107. https://ensciencias.uab.es/article/view/v35-n2-gonzalez-roa/2150-pdf-es
[15] Kariadinata, R., Info, A., Knowledge, P., Solving, P., y Abstraction, R. (2021). Students
Reflective Abstraction Ability on Linear Algebra Problem Solving and Relationship With
Prerequisite Knowledge. Infinity. Journal of Mathematics Education, 10(1), 1–16.
[16] Karrer, Mónica, M. R. (2018). Núcleo, Imagem e Composição de Transformações Lineares:
uma abordagem gráfica desenvolvida em ambiente computacional. Paper Knowledge .
Toward a Media History of Documents, 3409, 1–23.
[17] Kartika, H., Karawang, U. S., Barat, J., Mathematics, L., y Website, I. (2018). Teaching and
learning mathematics through web-based resource: an interactive approach. MaPan : Jurnal
Matematika Dan Pembelajaran, 6(1), 1–10.
[18] Marins, Alessandra, Pereira, A. M. (2016). Pensamento matemático avançado manifestado
em tarefas envolvendo transformações lineares. Ciencia Educación, Bauru., 22, 489–504.
[19] Meneu, M. J. B., Arcila, M. M., y Mora, E. J. (2017). A teaching proposal for the study of
eigenvectors and eigenvalues. Journal of Technology and Science Education, 7(1), 100–113.
https://doi.org/10.3926/jotse.260
[20] Murillo Arcila, M., y Beltrán Meneu, M. J. (2016). Coloreando el Álgebra Lineal. Modelling in
Science Education and Learning, 9(1), 25. https://doi.org/10.4995/msel.2016.3909
[21] Nishizawa, H., Shimada, K., Ohno, W., y Yoshioka, T. (2013). Increasing Reality and
Educational Merits of a Virtual Game. Procedia Computer Science, 25, 32–40.
https://doi.org/10.1016/j.procs.2013.11.005
[22] Nissa, Ita Chairun, Sukarma Ketut, S. (2020). Problem-based learning with role-playing: An
experiment on prospective mathematics teachers. Beta: Jurnal Tadris Matematika, 13(59),
14–32. https://doi.org/10.20414/betajtm.v13i1.370
[23] Parraguez González, M. (2013). El rol del cuerpo en la construcción del concepto Espacio
Vectorial. Educación Matemática, 25(1), 133–154.
[24] Parraguez González, M., y Uzuriaga López, V. (2014). Construcción y uso del concepto
combinación lineal de vectores. Scientia et Technica, 19(3), 328–334.
https://doi.org/10.22517/23447214.9303
[25] Parraguez, M. (2020). Construcción de significados de las operaciones del espacio vectorial
a través de conjuntos linealmente independientes/dependientes. Revista Chilena de
Educación Matemática, 60–70. https://doi.org/10.46219/rechiem.v12i2.22
[26] Petrov, P., Gyudzhenov, I., y Tuparova, D. (2015). Adapting Interactive Methods in the
Teaching of Linear Algebra – Results from Pilot Studies. Procedia - Social and Behavioral
Sciences, 191, 142–146. https://doi.org/https://doi.org/10.1016/j.sbspro.2015.04.579
[27] Pierri, A. (2018). From Practical to Theorical Thinking: The Impact of the Role-Play Activity.
Je-LKS. Journal of e-Learning and Knowledge Society., 14, 7–15.
https://doi.org/10.20368/1971-8829/1375
[28] Pizarro, S. P. (2020). Una Secuencia Didáctica para la Enseñanza de la Transformación Lineal:
Unificación de Métodos y Problemas, Modelización y Explicitación del Aprendizaje. Revista
Latinoamericana de Investigacion En Matematica Educativa, 23(3), 271–310.
https://doi.org/10.12802/relime.20.2331
[29] Ramírez-Sandoval, Osiel, Romero-Félix, César F., Oktac, A. (2014). Coordinación de Registros
de Representación Semiótica en el uso de Transformaciones Lineales en el Plano. Annales de
Didactique et de Sciences Cognitives., 19, 225–250.
[30] Roa-fuentes, S., y Parraguez, M. (2017). Estructuras Mentales que Modelan el Aprendizaje de
un Teorema del Álgebra Lineal : Un Estudio de Casos en el Contexto Universitario. 10(4), 15–
32. https://doi.org/10.4067/S0718-50062017000400003
[31] Rodríguez Jara, M., Parraguez González, M., y Trigueros Gaisman, M. (2018). Construcción
cognitiva del espacio vectorial R2. Revista Latinoamericana de Investigación En Matemática
Educativa, 21(1), 57–86. https://doi.org/10.12802/relime.18.2113
[32] Rosso, Ana, Barros, J. (2013). Una taxonomía de errores en el aprendizaje de espacios
vectoriales. Revista Iberoamericana de Educación., 63.
[33] Salgado, H., y Trigueros Gaisman, M. (2014). Una experiencia de enseñanza de los valores,
vectores y espacios propios basada en la teoría apoe. Educación Matemática, 26(3), 75–107.
[34] Sánchez-meca, J. (2010). Cómo realizar una revisión sistemática y un meta-análisis *. Aula
Abierta, 38, 53–63.
[35] Sánchez-Serrano, S., Pedraza-Navarro, I. y Donoso-González, M. (2022). ¿Cómo hacer una
revisión sistemática siguiendo el protocolo PRISMA? Usos y estrategias fundamentales para
su aplicación en el ámbito educativo a través de un caso práctico. Bordón, Revista de
Pedagogía, 74(3), 51-66. https://doi.org/10.13042/Bordon.2022.95090
[36] Silva, T. R. da, Santos, W. L. F. dos, Silva, E. R. B. da, & Arruda, A. W. A. de. (2021). Criação e
usos do aplicativo LineAlg como objeto de aprendizagem na Educação Básica. Diversitas
Journal, 6(1), 1415–1427. https://doi.org/10.17648/diversitas-journal-v6i1-1596
[37] Torres, J. T., García-Planas, M. I., & Domínguez-García, S. (2016). The use of e-portfolio in a
linear algebra course. Journal of Technology and Science Education, 6(1), 52–61.
https://doi.org/10.3926/jotse.181
[38] Trigueros Gaisman, M., Maturana Peña, I., Parraguez González, M., y Rodríguez, M. (2015).
Construcciones y mecanismos mentales para el aprendizaje del teorema matriz asociada a
una transformación lineal. Educación Matemática, 27(2), 95–124.
[39] Villalobos, G. M., y Ríos Herrera, J. F. (2019). Gamification as a learning strategy in the training
of engineering students. Estudios Pedagogicos, 45(3), 115–125.
https://doi.org/10.4067/S0718-07052019000300115
[40] Wibawa, K. A., Payadnya, I. P. A. A., Program, J., y Pendidikan, S. (2021). Learning
Effectiveness Through Video Presentations and Whatsapp Group (WAG) in the Pandemic
Time Covid-19. AKSIOMA: Jurnal Program Studi Pendidikan Matematika, 10(2), 710.
https://doi.org/10.24127/ajpm.v10i2.3451
[41] Xavier, R., Ibarra, C., y Reyes, M. E. (2019). Enseñanza-Aprendizaje de las Matrices en la
Carrera de Ingeniería Civil. Mikarimin. Revista Científica Multidisciplinaria., 51–60.
[42] Yildiz Ulus, A. (2013). Teaching the diagonalization concept in linear algebra with
technology: A case study at galatasaray university. Turkish Online Journal of Educational
Technology, 12(1), 119–130.