Computational Simulation: Multidisciplinary Teaching of Dynamic
Models from the Linear Algebra Perspective
EDUARDO A. GAGO, CAREN L. BRSTILO, NICOLÁS DE BRITO
Computer and Multidisciplinary Laboratory of Basic Sciences
Universidad Tecnológica Nacional – Facultad Regional Rosario
Zeballos 1341
ARGENTINA
Abstract: - The technological development has stablish a new learning perspective in the Linear Algebra
Concepts. In progress is being made in multidisciplinary teaching in engineering careers, since the proposal is
that students learn Math taking into account the benefits provided from the new technologies that analyse
different aspects of an engineering system. In this paper are presented a classroom experience in the Computer
and Multidisciplinary Laboratory of Basic Sciences where a fluid flow system is modelled when we want to
deal with the Eigenvalues and Eigenvectors topic, content that belongs to Algebra and Analytic Geometrical
program. The possibilities that the computational systems offers have unchained a reformulation of
methodological focus of educative programs, since they propose to intensify learning through the acquisition of
skills by students. The developed experience proposes a learning methodology that guarantees an academic
knowledge according to the new developments, and assert that Engineering students approach new
mathematical subjects trough adequate applications.
Key-Words: - Multidisciplinar Learning, Simulation, Eigenvectors, Modelling.
Received: August 24, 2021. Revised: May 16, 2022. Accepted: June 14, 2022. Published: July 3, 2022.
1 Introduction
In the Lineal Algebra courses, in the field of
engineering careers, the different reasons that
obstacle the teaching learning process in the basic
training of students.
Some of most relevant aspects that we can quote
are: The few knowledge levels from the students
bring from middle school, the difficult associated to
the abstraction process which is required to Lineal
Algebra study, and the lack of contextualization
from the subject content and its relation with
another Math or engineering courses.
Due to these issues, in the development of the
Mathematics class, the formulation of simple
models that address to the acquisition of new ways
of thinking and reasoning process is essential and
will be triggers of motivational situations.
Design a Mathematics class to Engineering
careers requires to establish an appropriate learning
strategy that allows a specific model situation
related to engineering application field.
The Lineal Algebra must provide to the students
the scientific basis for the management and use of
concepts in the approach and solution of problems.
In addition, it must provide them with the training of
mental skills, reasoning and mathematical
modelling. [1].
Mathematics teachers do not always have
incorporated this vision, which agrees with the
regulations agreed in the curriculum. In the actual
normative proposes that student must acquire skills
for diverse competences from basic subject cycle, to
obtain a significant learning process
The Engineering mathematics learning purpose
must find the correct equilibrium between
mathematical model formulation and skills that the
students need to solve the challenges that will arise
in the area of applied technologies and in their
future professional activity.
Motivation plays a fundamental role to achieve
that the Engineering learning teaching process will
be functional and relevant. For the engineers
training and others professionals from the
Engineering field are appealing use models and
methods that mathematics offers, since these
resources are the ones that must offer the most
optimal solutions to the different models proposed
to achieve the abstract thought.
The student must have a creative power that
allows to impulse the mechanisms that internalize
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DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
E-ISSN: 2224-3410
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knowledge and not limit it to a mere informative
aspect based on operational learning.
The use of Computational tools allows the
students to explore, deduce, make conjectures,
justify, test their arguments and thus build their own
knowledge independently of the teacher's
intervention.
These tools also make it possible for the teacher
to concentrate on stimulating and guiding learning,
but this new role requires greater activity from the
teacher, since constant creativity is necessary in
approaching the situations that arise in class.
2 Methodological criteria.
Computer and Multidisciplinary Laboratory of
Basic Sciences is the physical space where a
collaborative work field is generated, and for this
purpose, it is designed a workshop class which is
denominated as theoretical practical technological
class where it is realized a practical work with the
concepts of Eigenvalues and Eigenvectors subjects,
with engineering applications.
The class characteristic are to establish the
guidelines of an interactive engineering analysis
work that could manifest in a production activity of
new ideas.
The teaching learning methodology allows move
forward first with the theoretical subject research,
creating an environment where the student assume
an active role, putting aside the fact of being a
spectator, and permit to build the concepts by
experimentation, and elaboration of conclusions
The mathematics teaching in the first years in the
engineering career is needed for the student integral
education, but the depth of its study is addressing to
being limited.
It must give the knowledge with the goal of
prepare and educate the student so they could find
the tools that activate on a self-determined learning
process.
Actually, the Mathematical programs in the
University are oriented to the student, so they
dispose of previous knowledge enough to allow
build them a mental structures trending to
achieve a functional and relevant learning.
That implies that the student could establish
more complex relations in their learning
process.
The use of computational resources provides an
advantage in the teaching process, since the
observation in a workshop classroom allows
creating mental constructions trending to generate a
new knowledge.
In this situation, must be taking account that the
incorporation of computational tools are not limited
to the problem of counting with tools that conforms
those technologies: Equipment and computational
software, but the most important is build an
educational use and, in a strict sense, didactic of
them. [2].
The use of semiotic representations that implies
the management and conversion onto the
mathematical language produces a disarticulation in
thought that manifest in a relevant learning.
This approach to Mathematics is understood as a
linguistic resource to describe and discern the
processes that are seen in other disciplines, such as
physics or chemistry, where almost all of its laws
are stated with mathematical equations or with
procedures that derive from them [3].
More specifically, it is considered that
Mathematics education is the social, heterogeneous
and complex system in which it is necessary to
distinguish at least three components or fields:
a) The practical and reflexive action on the
Mathematics learning and teaching process.
b) The educational technology that proposes to
develop resources and materials, using the
available scientific knowledge
c) Scientific research, which tries to understand
the functioning of the teaching of mathematics
as a whole, as well as that of the specific
didactic systems (formed by the teacher, the
students and mathematical knowledge).
Those three fields are interested in the same
object: the functioning of the didactic systems, and
they even have a common ultimate goal: the
improvement of the teaching and learning of
Mathematics.
But the temporal perspective, the goals, the
available resources, the operating rules and
restrictions to which they are subjected, are
intrinsically different.
The world of practical action is the own teacher's
field, who is in charge of one or several groups of
students to whom he tries to teach mathematics [4].
Understanding an engineering problem means
converting this problem into a physical or chemical
problem and translating it into mathematical terms.
In the teaching of subjects such as Algebra and
Analytical Geometry, a certain level of concern is
distinguished by the scant interest of the students
regarding how the contents are presented in the class
and how the appropriation of the knowledge that
will be used in advanced courses is carried out.
Some students, however, state that this problem
is caused by the minimal understanding they have of
the concepts and the way in which they are
presented during classes. They also state that
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DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
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teacher’s exhibit concepts with a certain amount of
generalization and abstraction and this attitude do
not conform to student expectations.
In this paper, a class is designed with the purpose
of incorporating learning methodologies aimed at
generating intrinsic mechanisms in students that
allow them to discover knowledge and achieve
independence skills for reasoning and induction.
3 Objectives
The implemented actions to develop the
programmed activities have got as goal the
realization of a laboratory experience where the
students perform a self-managed and collaborative
project, to conceptualize the Eigenvalues and
Eigenvectors subject.
The task is developed by the design of an
engineering situation that study the flow of a fluid
when transit across two water tanks.
It pretends with the experience to convert the
Algebra classroom into a workshop class where the
student experiment a learning process generated by
interaction techniques between the teacher as
passive subject and the student as active protagonist
of knowledge,
This line of work allows the class to be
organized contemplating a transformation of
learning that leads the student to abandon the central
place that he has historically had within the
classroom to occupy another space in the class
dynamics; necessary space to interact with their
peers and with the work proposal [6].
Meanwhile, the first objective of a teacher is to
improve the learning of his students, so he will be
mainly interested in the action that can produce an
immediate effect on his teaching.
The second component, which we have called
technological (or applied research) is prescriptive,
since it is more involved with the elaboration of
devices for action and is the proper field of
curriculum designers, writers of school manuals and
teaching materials.
Finally, the scientific research (basic,
anaclitic and explanatory) is particularly
involved with the theory elaboration and it is
used usually at universities institutions [4].
In this experience, the use of a valuable tool such
as symbolic calculation is proposed, which is used
as a connector of mathematical functions applied to
real situations, which allows validating the student's
own skills for the development of basic capacities.
4 Pedagogical Bases in Classroom
In the actual paper is described a classroom
experience of Algebra and Analytical Geometry
course, in the Chemical Engineering career, where is
pretended that students integrate knowledge when
modelling a system of fluid flow in the development
of the Eigenvalues theme and Eigenvectors..
A theoretical practical technological class is
designed that takes place in a three-hour session
with different stages, with which the aim is to lead
students to learn the proposed topic.
The stages designed to carry out the class are:
assembly of work groups, presentation of a
problematic situation, review and subsequent
selection of bibliographic material on the subject,
review of the contents developed in the theory class,
modelling of the situation, and resolution of the
problem and conclusions.
Moving from a frontal class to one focused on
learning is probably a central task at the moment,
but this requires thinking about the use of ICT not as
a mere substitute for information.
Before, the information was entrusted to the
teacher (front class), now online programs are
developed that contain all the information that is
required. This vision changes the medium and
probably makes it a little more attractive, but it does
not fundamentally transform educational work.
In this sense, the ordering principle of the
learning task starts from the didactic foundations,
learning is built if it is taken into account that it
needs to be built as a personal work project, with the
effort that it implies on the part of the students, to
which is necessary to allow the previous knowledge
of the students, their notions, pre-notions and
prejudices to emerge, and at the same time open a
space for doubt, for questioning, so that the student
can build an enigma [3]..
The interconnection between Lineal Algebra
concepts, the real cases in the engineering field, and
the anticipation to knowledge that after will be more
complex in the courses that involve about basic and
applied technologies provide students with
versatility when dealing with increasingly
sophisticated models in the specific subjects of their
career.
5 Teaching Learning Sequence
Process.
Students work in the Laboratory class designing
the systems on their worksheet and doing the
calculations on the computer.
It is in this task where collaborative work,
together with the ability to build problem situations,
can allow the student to carry out information search
activities, analysis, and construction of their own
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DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
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Volume 19, 2022
responses. Ultimately, to reconstruct knowledge
from the conceptual structures that it already
possesses [3].
It is proposed to study a first order system
consisting of two interconnected tanks as shown in
Fig.1.
The interest in this type of system is formulate a
model that represents the outlet flow fluid in the
tank Nº2 in function of time, when it is applied in
the fluid entrance in tank N° 1, a variation of unitary
step of
type.
In the cross-sectional areas for the first and
second tank are and respectively, and the
hydraulic resistances for both tanks are
.
Being: Flow fluid that get into tank Nº1; Flow
fluid that get into tank N° 2 coming from the tank
N° 1; fluid flow coming out of tank Nº 3; level
reached in the tank N° 1, level reached in the tank
N° 2; is the hydraulic resistance of the fluid leaving
the tank N° 1 and is the hydraulic resistance of the
fluid leaving the tank N° 2 [5].
Fig.1 Studied System Scheme.
According to the data provided, the work of the
students is guided by the following instructions:
1) Which model rules the behaviour of the Fig.1
system?
2) Which differential equation system is
equivalent to the represented equation in the
proposed model?
3) Which are the Eigenvalues and Eigenvectors
linked to the differential equation system?
4) ¿Which is the fundamental Matrix associated to
the system?
5) Find the equation that determines the outlet
flow evolution in time function for the
indicated values in the question and graphic it.
6) How is the system behaviour?
7) Could the system represent another different
behaviours?
8) at what value the outlet flow in extremely long
times trends?
9) Is it exists some relative extreme, or inflexion
point in the graphic that could you infer as a
system characteristic?
Before starting to analyse Fig.1, the student
define the hydraulic resistance parameter, which
relate the liquid height on a tank with respect to the
outlet flow rate, as indicated in equation (1)
(1)
They also define the time constant, , expressed
in the equation (2), which is the product between
hydraulic resistance and the cross-sectional area in a
tank.
(2)
Then they perform the energy balance of the
dynamic system of Fig. 1. In tank Nº 1, the net flow
is the product of the cross-sectional area times the
velocity, it is indicated in equation (3)
(3)
With the relationships established in equation (1)
they can determine the speed in tank Nº1, which is
expressed by means of equation (4)
(4)
Resulting in the net flow for tank Nº 1, the one
identified by the first order differential equation (5)
(5)
Now if they apply a similar energy balance for
tank Nº2, the net flow in tank Nº 2 is that expressed
in equation (6)
(6)
According to the relationships defined above, in
equation (7), the velocity of the fluid in the tank is
indicated N°2
(7)
Equation (8) is the first order differential
equation, which indicates the net flow for tank N° 2
(8)
If and are time constant for
the tanks N° 1 and N° 2 respectively, it is designated
with and , so relating the
equations (5) and (8), may obtain the ordinary
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DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
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differential equation for constant coefficients that
determine the system model of Fig.1 which is
expressed in the equation (9) [5].
(9)
The starter conditions for the proposed model
are: outlet initial flow is null ; and the
initial outlet speed flow is also null
.
After discuss and propose solutions to the raised
questions, determine that Equation (9) can be
expressed by the differential system equation [7],
[8].
(10)
The Equation (10) can be represented by the
equation matrix (11)
(11)
According to the data supplied to the student, the
Equation (11) is transformed into Equation (12).
(12)
From Equation (12) we observe that system
matrix characteristic (10) is
(13)
Then, calculate the eigenvectors from the
expressed matrix on (13) by the use of characteristic
equation expressed on (14)
(14)
Resulting:
(15)
From Equation (15), the eigenvalues results:
and
, and if
(17)
is an eigenvector of corresponding to each of the
obtained , if only is a trivial solve of matrix
equations (18) and (20).
In the case of assuming that
the
eingenvector is a particular solution for equation
(18)
(18)
In this case,
(19)
Now if we replace by the eingenvector is
a particular solve from Equation (20)
(20)
In this case,
(21)
The expressed eingenvectors in expressions (19)
and (21) can be used to build a new Matrix, as
observed in (22)
(22)
Matrix (22) allows determining the Equation (9)
sol, resulting [7], [8]:
(23)
From the solution of Equation (9), and applying
the boundary conditions established in the particular
solution of equation (23), they obtained equation
(24) that represents the outlet flow
(24)
The students made Fig.2 that shows the graph
that determines the evolution of the outlet flow as a
function of time, they also noted that at the
beginning the system drains slowly, while in a time
of 3 hours the evacuation is significant and in 6
hours there is a meagre flow to be drained.
In a 9 hours lapse, practically all the fluid was
evacuated from tank No. 2. In addition, they
determined that the graph in Fig.2 does not have
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DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
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relative extremes. Instead, it has an inflection point
at the abscissa point 1.38 (represents 1 hour and 23
minutes from the beginning of the exit of the fluid
from tank No. 2), and at that moment only 25% of
the total flow was drained [9].
Fig.2 Outlet Flow
By system characteristics, and with the analysis
done in Fig.2 and Equation (7), the students
conclude that the system is hyper-damping, and the
outlet flow is bounded in 1
.
In Fig.2 graphic, the students indicated with a dot
line the asymptote that determines the outlet flow
value, they verified this result calculating with the
software
(25)
that allows prove the value for the amount outlet
flow from tank Nº2 is
.
From the analysis carried out, the students
determined that the expression:
(26)
is the discriminant of the characteristic polynomial
of Equation (9), and also verifies that
(27)
The students concluded that both sides of
Equation (26) are always positive, and therefore the
system is over damped. In addition, they stated that
it is impossible for the system to be under damped
since the relationship in equation (25) will never be
negative.
They confirmed that the system will not be
critically damped either because in the relationship
expressed in (25) the and constants would be
the same and that would imply a null output flow.
In Fig.3 they observed the behaviour of the
system taking different values of , with values of
constants and less than .
Fig.3 Outlet flow to (with a constant )
Analogously, in Fig.4 they show the inverse
situation, they analyse the behaviour of the system
taking different values, with constant values
less to .
Fig.4 outlet flow for (with constant)
In all cases, if they extend the domain in Fig.3 y
4 graphics, they check that when time trends to
infinite the outlet flow trends to
.
It is suggest as additional work to them analysing
another liquid flow system and taking the same
considerations given is this case. Also, it is suggest
to then to compare both models, to obtain
conclusions about it.
6 Conclusion
The collaborative environment developed in the
Basic Sciences Laboratory and the applied teaching
methodology provided students with a series of
strategies and skills to solve the proposed activity,
which resulted in increased interest and motivation.
The teachers who carry out the experience
facilitated the development and articulation of the
concepts through a didactic situation of little
complexity that offered the students to integrate the
knowledge in a functional and meaningful learning.
The visualization carried out with the support of
computational tools allows exploring the concepts
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
E-ISSN: 2224-3410
187
Volume 19, 2022
of the subject Eigenvalue and Eigenvectors, and
discovering the relationships that these concepts
have with others of the same subject and with the
model presented.
The use of technology made it possible to make
the role of the modes of representation more
explicit, in particular, the way in which the
complementarity between the graphic, the
numerical, the symbolic, the algebraic and the
simulation revealed by the software used, helped to
develop the processes of theoretical
conceptualization and the systems studied.
In this way, the description of the construction
process followed by the students has made it
possible to relate aspects of particular to reflective
abstraction derived from the modes of
representation in the construction of knowledge.
The proposal presented in class is based on
integrated and systemic learning. This experience is
different from the one presented in the rest of the
courses when the subject Eigenvalues and
Eigenvectors is treated, it is based on permanent
dialogue, the affinity of criteria and fundamentally
on the active participation of the main protagonist of
the educational fact that is the student.
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Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the Creative
Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_US
WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION
DOI: 10.37394/232010.2022.19.20
Eduardo A. Gago, Caren L. Brstilo,
Nicolás De Brito
E-ISSN: 2224-3410
188
Volume 19, 2022
