Teaching and Learning Computational Mathematics with Intensive

Application of the Virtual Campus

EDUARDO GAGO

Computer and Multidisciplinary Laboratory of Basic Sciences,

National Technological University - Rosario Regional Faculty,

Zeballos 1341,

ARGENTINA

Abstract: - Nowadays, the field of professional development in engineering requires appropriate and relevant

training for a satisfactory insertion in the labor market. This requires that educational institutions provide not

only the specific knowledge of the career being studied but also must complement it with other skills necessary

for adequate performance in the work environment. Based on these premises, this paper describes the design of

a classroom experience to be implemented in the subject of Advanced Calculus (AC), a third-level subject of

the Mechanical Engineering course. The proposed activities are developed in the Computer and

Multidisciplinary Laboratory of Basic Sciences available at the faculty and seek to encourage the use of the

virtual campus, where the mediation between the content of the subject and the concrete applications of simple

engineering models are the axis to develop the topic: Analytical Functions of Complex Variables (AFCV).

Key-Words: - Virtual Campus, Advanced Calculus, Laboratory, Complex Variables, Multidisciplinary,

Mathematics.

Received: June 29, 2022. Revised: August 9, 2023. Accepted: September 13, 2023. Published: October 13, 2023.

1 Introduction

Engineering is influenced by mathematical

modeling, i.e. increasingly complex elements

provided by mathematical science are becoming

more and more useful in concrete applications.

In engineering achievements, there is a rapid

transfer of theoretical results to the technological

field due to the explosive computational progress.

Technological advances in recent years have had

a strong impact on higher education, broadening

educational scenarios. Physical campuses are being

replaced or complemented by others of a virtual

nature, and relationships within the community are

developed preferably in a non-presential format and

not always synchronously.

The presence of physical campuses and virtual

campuses to develop teaching and learning

processes give rise to new educational

environments.

In these environments, there is not only a change

in space and time but also in responsibilities, tasks,

materials, activities, and evaluations.

In short, decisions about what, when, where,

how, and how much to teach transform the

traditional conceptions of educational environments.

Virtual workspaces, modeling systems, and

simulations complement real classroom work and

allow students to verify hypotheses, and explore

dynamic relationships to show the same element in

different contexts.

In the training of the Mechanical Engineer, basic

knowledge indispensable to approach the analysis

and resolution of complex systems, which involve

advanced mathematics, must be contemplated.

Students are expected to develop analytical,

qualitative, and numerical methods capable of

solving diverse mathematical models that describe

real-life systems or phenomena.

The modeling process is articulated by means of

a network of components of determined specificity,

where the student is subjected to constant decision-

making that induce him/her to systemic thinking.

Based on these premises, this paper describes the

design of a classroom experience to be implemented

in the subject of Advanced Calculus (AC), a third-

level subject of the Mechanical Engineering course.

The proposed activities are developed in the

Computer and Multidisciplinary Laboratory of

Basic Sciences available at the faculty and seek to

encourage the use of the virtual campus, where the

mediation between the content of the subject and the

concrete applications of simple engineering models

are the axis to develop the topic: Analytical

Functions of Complex Variables (AVCF).

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2023.20.11

Eduardo Gago

E-ISSN: 2224-3410

Volume 20, 2023

2 Laboratory as a Classroom

The class is held at the Mathematics

Multidisciplinary Computer Laboratory of the

faculty. The Laboratory is an area where activities

are carried out to support the Mathematics area

courses to fulfill the numerical, symbolic, and

graphic calculation aspects of the subjects of the

area.

The teaching team is involved in the

implementation of new methodological strategies

for the development of the curricular contents,

generating the necessary didactic material for the

realization of theoretical, practical, and

technological workshops.

From the line of educational innovation in

engineering, we continue working on new teaching

paradigms to achieve, in Higher Education, greater

development of intellectual capacities, acquisition of

skills, the substitution of obsolete techniques for

more efficient and faster means, and better

interaction in the teaching-learning process.

Based on this, and thanks to the inclusion of new

advances in computer science, we have

implemented the use of new pedagogical

methodologies and diverse didactic strategies,

aiming at the interdisciplinary work that is

considered necessary in the area of Engineering and

that must be approached from Mathematics.

From all angles, dimensions, perspectives of any

issue, problem, idea, or concept can be

contemplated from different disciplinary areas and

presented immediately through hypertext links and

search engines, [1].

Working in the laboratory offers vast

possibilities for incorporating new educational

technologies and online applications that propose a

more dynamic and participatory teaching-learning

environment.

The use of the virtual campus has created,

through activities, educational resources,

interaction, and communication, an expanded

educational space that transcends the classroom and

brings new forms of communication and expression

to the daily lives of both students and teachers.

3 Technology in the Classroom

The incorporation of information and

communication technologies (ICTs) into the

educational field is a central issue, which has gained

relevance in higher education centers and especially

in engineering careers, in which the disciplines must

present a line of work, research, or study dedicated

to digital technologies in their specific schedule.

Within the line of work proposed to be carried

out in the classroom intervention project, it is

essential to generate new work environments, apply

modern methodologies, and provide innovative

strategies in the course of mathematical subjects.

This is achieved with a computer-mediated

design and planning system, where a set of activities

and communicational expressions are created as

fundamental axes of the teaching process for the

achievement of independent learning.

The computer environment generated has the

characteristic of concentrating state-of-the-art

intercommunication technologies and highlights the

use of summary generation, immediate and

collaborative answers between students and

teachers; links to videoconferences and

presentations of topics with great deployment of

graphic teaching methods.

This solves the availability of material and

information, accessing it from different workstations

and solving the problems associated with learning

through a process of real-time feedback.

This virtual environment, implemented in

parallel with the lectures and laboratory activities,

favors and encourages self-management of learning

and efficient use of time, since it provides an

interactive medium that enriches the curricular

planning of the courses and interdisciplinary

activities.

4 Regulatory Context

Nowadays, the field of professional development in

engineering requires appropriate and relevant

training for a satisfactory insertion in the labor

market. This requires that educational institutions

provide not only the specific knowledge of the

career being studied but also must complement it

with other skills necessary for adequate performance

in the work environment.

It is necessary to increase intensive practice,

promote collaborative work, impose an environment

where the formation of groups and teams is natural,

and ensure that the student acquires a good

disposition towards the other members of his group,

[2], [3].

The new managerial lines in the working

environments aim at a special ductility and

conditions of adaptation to new technologies, a

tangible facility for the acquisition of their

operability, and a clear inclusion within the staff as

a whole.

The accreditation bodies of all countries echo

these new demands, so they urge universities to

promptly adapt their curricular contents to the

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E-ISSN: 2224-3410

Volume 20, 2023

competencies that engineers must possess, [3], [4].

The national commission for university

evaluation and Accreditation is a governmental

organization for the accreditation of engineering

careers that in its methodological proposal mentions

standards to be followed, among which it is

highlighted that the curriculum should include, [3]:

 Experimental laboratory, workshop, or field

training that trains the student in the specialty to

which the program refers.

 Engineering problem-solving activities, real or

hypothetical, in which knowledge of basic

sciences and technologies are applied.

 Engineering project and design activities,

contemplating significant experience in these

fields that require the integrated application of

fundamental concepts of basic sciences, basic

and applied technologies.

 Skills that stimulate the student's capacity for

analysis, synthesis, and critical spirit, awaken

their creative vocation, and train for teamwork

and evaluation of alternatives.

 Supervised instances of training in professional

practice, even if ideal, for all students.

The university needs to make educational spaces

expand and transcend the classroom, to get closer to

the everyday environments of the people involved,

and the Internet offers the possibility of using new

educational technologies to achieve this.

With the evolution and development of ICT, on

the one hand, and the need for better and more

effective online communication and collaboration

tools, on the other, technological platforms have

emerged that integrate a wide variety of

communication and collaboration resources and are

applied both for work and education and their use is

based on cooperative and collaborative work/study

methodologies, [5].

In this work, this sign of progress in the

teaching-learning process is considered and a site is

incorporated, the virtual classroom of AC, with the

characteristic of concentrating hypermedia

technologies to collaborate in the educational

process.

5 Organization of the Virtual Space

The university provides a virtual campus based on a

platform called Moodle that allows academic units

to manage online courses.

The word Moodle is an acronym for Module

Object-Oriented Dynamic Learning Environment.

This platform is a learning management system

installed on a server, which is used to manage,

distribute and control non-face-to-face training

activities.

Moodle is a free distribution tool, it is a global

project designed for the development of educational

environments in continuous growth and with the

possibility of incorporating contributions from the

national and international academic technological

environment, allowing exploring new ways or

generating their own, [3].

Within the virtual campus of the university, a

virtual space was built and configured for the

Computer and Multidisciplinary Laboratory of

Basic Sciences, among others, and also virtual

classrooms for the subjects AC, Mathematical

Analysis I, Algebra and Analytical Geometry,

Mathematical Analysis II, Computational

Mathematics and Engineering.

The author of this paper is an administrator in

several virtual classrooms. Figure 1 shows the

author's virtual classroom session with all the virtual

classes he is in charge of.

Fig. 1: Virtual classrooms

All the quality standards inherent to virtual

training sites were considered in their design, and

multiple educational resources were provided to

make the best possible use of the hypermedia tools

available and those offered by the platform.

Educational material in different formats is

included.

Figure 2 below shows the front page of the CA

virtual classroom page and some of the incorporated

resources currently in use.

6 Methodological Criteria

The basic problem of how to teach lies in creating

the conditions for the knowledge schemas

constructed by the learner to evolve in a given

direction.

The key question is not whether learning should

give priority to content or processes, but to ensure

that they are meaningful and functional.

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Considering that Mathematics influences all

aspects of human life and culture, it would be

desirable that any student, at any level, be able to

obtain the necessary skills to build his or her

knowledge. At the same time, it should be good that

teachers were able to provide with skills to promote

creative and meaningful teaching-learning situations

and activities that encourage students to learn, [6].

The learner needs to have sufficient prior

knowledge from which to approach the proposed

contents, to establish relationships between them as

complex and rich as possible that will allow him/her

to increase the meaning of his/her learning.

Therefore, it is convenient to help the student to

remember, reorder or assimilate the necessary

previous knowledge related to the proposed content,

to successfully address the programmed learning,

designing cognitive bridges between the new

content and the structure of knowledge that the

student has -previous organizers- and develop

appropriate strategies to put students in a favorable

situation to learn.

Fig. 2: CA virtual environment

This implies an intense activity on the part of the

student and a real commitment on the part of the

teacher in terms of direction, coordination, and

pedagogical assistance.

The aim is to get the student out of his passive

role, acquiring a memorizing capacity that does not

allow him to think on his own and create.

In addition, the widespread availability of new

interactive information and communication

technologies provides an immense amount of

possibilities that materialize with the development

of new models of teaching and learning, and given

that the work of engineering analysis is based on a

computer-mediated system and communication, it is

considered important to generate a space in which a

set of activities, exchanges, and communicative

relations are produced as the fundamental axis of the

educational process. The Cover of the AFCV theme

is presented in Figure 3.

The proposed didactic activities consist of

analyzing the students' previous ideas, emphasizing

a theoretical investigation by the students under the

guidance of the teachers, to make an analogy with

the physical parameters of the system under study,

and finally, with the information gathered, solve the

problematic situation proposed for the

conceptualization of the topic.

Fig. 3: Cover of the AFCV theme

7 Objectives

The programmed activities are aimed at:

 To carry out non-demonstrative laboratory

experiences that allow for the real participation

of small groups of students in the first years of

their careers and to facilitate the

conceptualization of the topics by designing

activities through the Moodle platform and with

digitalized didactic material specially prepared

for each topic.

 Transform the Mathematics class into an

experimental workshop, in which learning is

generated through techniques that shift the focus

from the teacher to the student, understanding

that the construction of concepts, skills, and

attitudes acquired by students should lead them

to achieve autonomy in the study of analytical

functions of complex variables, knowing their

mathematical basis and recognizing when and

where to apply them.

 The main objective of the implemented proposal

is to integrate computational mathematics with

technological areas in the Engineering

curriculum and to incorporate new work styles

based on organizing principles that allow linking

knowledge and giving it meaning, transforming

what is generated by disciplinary frontiers.

 To understand in depth the processes of change

initiated from the potentialities offered by the

different computational resources in the

construction of knowledge, being of utmost

importance to investigate the implementation of

classroom strategies that allow, from the

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mathematics-computing relationship, to

articulate the basic disciplines with the

technological ones.

 To facilitate the construction of the necessary

knowledge for the management of the complex:

to design and implement applications that

involve the resolution of engineering systems

and to use symbolic computation and advanced

graphic methods, different integrating proposals

were organized to present applications from the

department.

8 Classroom Intervention Project

8.1 Methodological Changes

The development of the AFCV subject is carried out

respecting the qualitative paradigm framed in what

is known as classroom experiences. We are working

with a population of approximately 25 students of

the Mechanical Engineering course within the

subject AC, a third-level subject of this course.

The purpose of this design is to make a

methodological change in the way of teaching the

subject, using the virtual classroom and the physical

space provided by the Computer and

Multidisciplinary Laboratory of Basic Sciences.

The experience corresponds to a class that does

not have the character of a traditional class, nor is it

a master class. The class is taught in a theoretical

and practical way, encouraging the active

participation of the students, and is oriented to the

understanding of the subject in an integrative way,

with isolated tools of symbolic calculus.

The development of the class is based on group

learning applying different dynamics: group

discussion, problem-solving technique, case

method, use of the virtual platform and the use of

computer resources will be encouraged whenever

possible and appropriate.

The teacher will assume the role of learning

facilitator, performing the functions of organizer,

stimulator, and supervisor of the task performed by

the group. The teacher interacts with the students at

all times to achieve the construction of knowledge.

The purpose of the teachers who carry out the

experience is to obtain active and critical

participation of the students, which will be achieved

by selecting and grading all classroom activities

according to their complexity and always providing

adequate modeling according to the problems of the

career or profession.

The teaching strategies are established from

different activities without neglecting the theoretical

foundation, respecting the interdisciplinary

approach and based on constructivist learning.

The methodology applied aims to create a

learning space where a set of experimental activities

are developed as a fundamental line of the

educational process.

If we focus on Higher Education, and more

specifically on Engineering, we can observe that

some authors propose that teacher should emphasize

that students develop capacities and skills, as well as

stimulate them to think, reason and deduce. In other

words, we should not only transmit concepts,

formulas, etc., but also provide them, from a

functionalist, utilitarian and practical approach, with

knowledge that allows them to develop in life, as

well as skills that improve their mathematical

culture and autonomy in learning [6].

8.2 Didactic Material

Students have at their disposal didactic material for

the development of the subject in the virtual space

of the subject AC.

The different files include videos with exercises,

texts with theory, solved exercises, self-assessments,

multiple-choice questionnaires with self-correction,

and space for forums and consultations.

Figure 4 shows the cover page of the material

available on the virtual campus.

Fig. 4: Materials available in the virtual classroom

8.3 Teaching Sequences

In the first instance, the aim is to motivate the

students with the presentation of an engineering

model of the Transport Phenomena branch of an

ideal fluid that corresponds to a case that in the

future they could study more exhaustively in the

upper cycle Fluid Mechanics course.

Motivation plays a fundamental role in making

the teaching-learning process functional and

meaningful.

Motivation is a commonly used concept when

pointing at individuals’ desire to do something. In,

[7] is described the motivation as the potential to

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direct behavior through the mechanisms that control

emotion or simplified as the inclination to do certain

things and avoid others.

Motivations are reasons individuals have for

behaving in a given manner in a given situation, [8].

This means that a motive is, therefore, something

that causes a person to act. These definitions

provide us a good starting point to consider

undergraduate engineering students’ motivation in a

mathematics course.

The motivation to do something, in this case to

learn mathematics, is commonly divided into two

distinct types according to the source of the

motivation. Intrinsic motivation describes the desire

to do academic tasks because one enjoys them. A

student who is intrinsically motivated is interested

in learning the content of a course. Extrinsic

motivation, in contrast, describes the desire to do

such tasks to earn rewards, such as credits, grades,

or simply approval, [9].

It has been observed that the incorporation of

practical cases in this course improves the overall

mechanical performance of the students. It is

explained by an inherent motivation for practical

applications that is useful to better understand the

course basis, [10].

For the training of engineers and other

professionals in the field of engineering, it is

attractive to use models and methods provided by

mathematics, since they are the ones who must offer

the most optimal solutions to the various models

proposed to achieve abstract thinking.

In the initial phase of learning, the aim is to

motivate the student so that the new learning

situation awakens curiosity in him/her. At this stage,

the learner must be mobilized and effectively

committed to the new activities to be performed,

[11].

For this purpose, a two-dimensional flow system

of an incompressible, non-viscous fluid

corresponding to an irrotational field moving in a

steady state for an expression of the complex

velocity potential is presented to the students for

analysis. The instructions are attached to the virtual

site of the course. The Practical work assignment is

presented in Figure 5.

Fig. 5: Practical work assignment

Another issue to take into account is that the

approach of a simple and real situation to introduce

the new content is to answer the constant question

that students repeat to Mathematics teachers: What

is this good for?

The choice of the motivating example is based

on the fact that there is a direct relationship between

the concepts of analytical functions of complex

variables with the velocity of an ideal fluid and

other physical parameters that will be used in the

planned model.

In the virtual classroom, there is a series of files

of different formats that will serve as a tutorial

guide to be able to face all the activities.

After the presentation of the case, the students

are informed that from the following day, they will

have available on the virtual campus a brief self-

evaluation with a google drive form, with questions

that they will have to answer through the multiple

choice selection system, immediately after finishing

it, they can see the answers online and thus have an

assessment of what their previous knowledge of the

subject is.

In all work sessions, both in the classroom and in

the virtual classroom, students work in small groups

of a maximum of three students.

Students themselves choose the pair to work in

the laboratory during all the sessions. The objective

is to enhance the discussion and to help each other

in the experimental setup. Working with groups

with large number of students may enhance the

discussion but each student will not have the chance

to experiment in the laboratory as much as if they

were only two. Another issue about the grouping is

the way that the pairs are done. If lecturers know in

advance the students, they could propose the pairs in

order to guarantee that they will cooperate and

collaborate during the experiment (and avoid

discussions between members of a group). Due to

the schedule, this never happens, and students

organize the groups themselves, [12].

Google Drive allows them to work

collaboratively on the same document and all at the

same time.

The day after the previous activity they will have

a questionnaire of previous ideas, more extensive

than the previous one, which they have to deliver

with a duly completed report that must be attached

in the virtual classroom, and they have an

inexorable deadline of 72 hours for its delivery. The

Previous ideas questionnaire is presented in Figure

Previous knowledge is organized in our mind in

the form of cognitive structures. A cognitive

structure is a set of already acquired knowledge that

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are interrelated with each other, and are those that

allow us not to make sense of any new knowledge,

[13].

Fig. 6: Previous Ideas questionnaire

The proposed questionnaire is a vehicle for the

teacher to detect the preconceptions that the students

have, and as these ideas manifest themselves it is

possible to construct and incorporate the new

knowledge.

Lots of skills related to research and presentation

of the results are worked in this activity, as well as

those more related to the theoretical concepts. It is

important to have a clear guideline to perform this

activity to help the students to focus on the different

parts of the experiment that they have to prepare,

[12].

In the following class, a discussion is organized

where students exchange opinions about the

preconceptions questionnaire. Then, in the same

class, after exposing the students’ preconceptions,

the aim is to establish, analyze, discuss, and

formalize the theoretical concepts inherent to the

subject under study. For this purpose, a series of

activities are designed, which will be complemented

by the previous ideas they have.

The student-student interaction intends that in all

class sessions, the students work collaboratively as

they will do in their future professional life, but

always under the supervision of the teachers who

carry out the experience, who assume the role of

guide of the teaching-learning process.

The students have at their disposal in the

Laboratory specific bibliography to be able to

investigate and analyze the topic to be taught,

besides, the Internet connectivity of all the PCs

helps them to explore and search for material on the

whole web.

All the groups of students will meet with the

same conformation that they had already chosen,

and with the suggested bibliographic material, the

articles that they search on the Internet and that will

be uploaded to the virtual campus to share them,

they will obtain a theoretical tutorial guide of the

concepts to be used in the work proposal. According

to our request, the conclusions reached by each

group will be listened to and reviewed to carry out a

debate on the concepts of the topic and then propose

a theoretical framework.

Table 1. Analogy between the definitions of the

derivative of real and complex functions

Derivative of a

real function of a

real variable

with

Derivative of the

function of

complex

variable with

As shown in Table 1, the analogy between the

definitions of the derivative of a function at a point

for a real function f of a real variable and the

derivative of a function at a point for a function of a

complex variable is used as a trigger for the topic.

From the analogy in both definitions, it must be

emphasized that they are different expressions,

because c is a real number, so it can be represented

by a point on the real line, while about the complex

variable and the functions dependent on them, it is

known that a plane is required to represent the

complex numbers, so w is a fixed point in the

Argand diagram, somewhere in the plane.

Students are then asked to develop the definition

in two different ways. From both developments,

students, with teacher guidance, can formalize the

theory of the topic.

After answering and discussing all the answers, a

joint document is written as a guide to continue

learning the topic.

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Solving a situation proposed by the teacher to

seek information from the student about their

knowledge of the topics covered in the strategy,

which will vary according to the type of situation:

bibliographic, experiments, teacher intervention,

and audiovisual, [14].

With the definitions, developments, and

conclusions that have been exposed, the students

can elaborate a written theoretical material that is

submitted to the virtual site of the subject to

continue with the activity.

Fig. 7: Theoretical material

Figure 7 shows an extract of a possible theory

uploaded to the virtual classroom by the students.

Fig. 8: Practical work assignments

After formalizing the theory, the students can

model the system under study, as requested by the

tutorial guide shown in Figure 8, trying to relate the

theoretical concepts with the variables that make up

the dynamic system.

From the formalization of the theory, the

students can model the system under study, as

requested in the tutorial guide that can be visualized

in Figure 8, trying to relate the theoretical concepts

with the variables that make up the dynamic system.

In Figure 9, we can see a possible modeling of

the system under study requested in the tutorial

guide to solve the practical work.

Fig. 9: Modeling the system under study

The students use software to carry out the

activity, the groups expose their answers and final

data. Throughout the process, the students can

consult with the teachers to clarify any doubts or

conflicts that may arise. Figure 10 shows the graph

of the system under study.

Fig. 10: Model graphic

The experience has not yet been evaluated.

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9 Conclusion

The use of the virtual space provides several

opportunities for students to participate in the

activities proposed in the Advanced Calculus

course, opening communication channels between

teachers and students.

The greater availability and variety of the material

promotes a better and faster understanding of the

topics, and encourage students to self-manage their

learning.

All the elements of the virtual classroom are

permanently available to the student, not depending

on schedules or fortuitous events, and the most

complex items are dealt with through different

media, which constitutes an enrichment of the

materials related to that topic and a deepening of its

approach.

The use of technological means invites the student

to operate with different applications and software

packages, to download special complements for

their operation, incorporating in parallel knowledge

on how to operate the different resources available.

The virtual classroom establishes better

communication, facilitating conceptualization in

areas of difficulty, and even the detection of

particular needs. Teachers can better organize help

for students who have some difficulty, both

conceptual and technical, and even explore and

propose new activities.

The space brings together a set of tools that evolve

according to the planned activities, to the needs of

the students, and to the detection of the most

frequent problems and the most marked

complexities.

The work environment and the virtual context

become a space for students to meet, and exchange

opinions, knowledge, and doubts that reinforce the

relationship between students and teachers, making

the activities more dynamic and motivating.

Acknowledgements:

The author would like to thank his teacher Ing. Mg.

Alicia María Tinnirello. Teacher, researcher, and

Director of Research Projects, her dedication, and

patience made possible the completion of this work.

Her teachings, advice, and precise corrections

enhanced my training as a researcher.

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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2023.20.11

Eduardo Gago

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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2023.20.11

Eduardo Gago

E-ISSN: 2224-3410

Volume 20, 2023