CASE STUDY: COMPLEMENTS OF MATHEMATICS AND
E-LEARNING
A. Hernández Encinas, A. Queiruga Dios, J. Martín Vaquero and J. L. Hernández Pastora
Department of Applied Mathematics, E.T.S.I.I. de Béjar, University of Salamanca
C/ Fernando Ballesteros, 37700-Béjar, Salamanca, Spain
Keywords: Mathematics, eLearning, Internet.
Abstract: In this study we present the experience of the work done during this course with students of the subject of
Complements of Mathematics. We used the computer in class every day and we have proposed to students
daily activities and a final paper related to any of the items on the subject. The use of eLearning platform
along with other tools is an improvement in the education system. This study has been developed
specifically with students from the last engineering courses and shows how Mathematics could help
engineer students to change their traditional point of view related to university studies and learning tools.
1 INTRODUCTION
Computers have been infesting all areas of our daily
activities, especially those related with information
management. The acquisition and processing of
news and information, mail and social contacts are
increasing by computer use. In addition and
simultaneously, the number of users that somehow
left out is unable to adapt to constant technological
renewal, is decreasing.
The continuous advances, unfortunately, not always
supposes an improvement in the person machine
interaction. At this time computers are more or less
“stupid” and lacking in initiative because people is
the responsible for initiating the interaction with
them. This will require knowledge of the
management of all the applications used. In the very
near future this will change because of software
agents. The emergence of these new tools will
facilitate what is known as indirect or delegated
administration. This will not remove the user of the
ability to interact directly with the computer.
However, it should always perform all processes, it
may delegate certain tasks.
Teachers are, every day, adopting computers as
teaching tools needed, and using them to carry out a
more effective teaching and greater quality levels.
The Information and Communication Technologies
(ICT) are an important element in the convergence
of universities to achieve a common European space
of higher education (Queiruga, 2008), which
represents a culture change associated with a change
in university studies.
This paper describes in detail some changes
related to the Complements of Mathematics subject
and the new methods and tools used for obtaining
the new educational system. We perform the
integration of new teaching methods with traditional,
using eLearning and others mathematical tools as a
model of further education (Mason, 1998).
As is well known mathematics are very close, in
general, with computational sciences. The area of
mathematics is therefore, one of the most adapted
for the incorporation of the new technologies, since
it is not only one purely formative subject, but is a
scientific tool for the students who must help them
to solve the problems that are generally throughout
their race and in its professional development.
Nowadays, the internet technologies and
software agents are becoming increasingly important
in online learning systems. Web-based training and
the use of agents offer the opportunities to enhance
traditional courses, encourage life-long learning and
enable more people to join the learning society.
Software agents prove to be the necessary tool in
improving the effectiveness and efficiency of
retrieval in such systems (Impagliazzo, 2004).
This paper is organized as follows: In section 2
we will show the Complements of Mathematics
subject structure, the work and tools used by
students and the works developed in classes are
154
Hernández Encinas A., Queiruga Dios A., Martín Vaquero J. and L. Hernández Pastora J. (2010).
CASE STUDY: COMPLEMENTS OF MATHEMATICS AND E-LEARNING.
In Proceedings of the 12th International Conference on Enterprise Information Systems - Software Agents and Internet Computing, pages 154-157
DOI: 10.5220/0002974701540157
Copyright
c
 SciTePress

detailed in Section 3, and in Section 4 the
conclusions of the study are presented.
2 COMPLEMENTS OF
MATHEMATICS SUBJECT
In this section we can differentiate what was the
subject so far and in what has become. The teaching
method we have proposed this course has changed
so that students can use the computer throughout the
duration of the course. We proposed at the beginning
of the course the use of Mathematica package
(Wolfram, 1999) for searching, analyzing and
developing any of the themes of the course.
The objectives proposed in the curricula of
Complements of Mathematics subject are the
following:
1. Modelling common scenarios in engineering
problems and applying appropriate techniques
to solve the problem.
2. Using exact and approximating mathematical
techniques.
3. Using appropriate tools to solve the problems.
The work plan in accordance with the existing
curriculum could be considered as follows: The
course is separated in lectures and problem classes,
with the schedule approved by the competent bodies.
Additionally, each teacher has student opening
hours. Since the course only lasts 4 months
(quarterly), it will not be a “classical” final exam on
the scheduled date.
The final assessment consists of carrying out the
proposed activities to make students understand all
the contents of the subject by focusing on one in
particular. All the classes are done in the computer
room and the practice session consists of different
problems to be solved in the Mathematica
environment.
The whole theoretic contents of the course are:
 Direct methods for solving linear systems:
Gauss, Cholesky and LU factorization.
 Iterative techniques in matrix algebra: Jacobi,
Gauss-Seidel relaxation.
 Approximation of eigenvalues power method,
deflation method, QR algorithm.
 Partial differential equations. Some methods of
integration of partial differential equations.
 Review of functions of complex variables.
Residue theorem. Applications of the calculus
of residues.
The methodology that we used for this course is
a virtual platform together with mathematical
software (Díaz, 2009).
Due to its inherently distributed nature, a
distributed-learning environment can be supported
and managed by a set of autonomously cooperation
software agents that communicate intelligently with
one another. They can interact with human users at
the right time, with the right information. Much
experimental research has shown that intelligent
software agents, such as interface agents,
information agents, and collaborative agents, have
great potential to reduce information workload and
to automatically perform many knowledge-an labor-
intensive tasks for users. For example, these agents,
with the functions of motivation, learning facilitation
and collaboration, and so forth, serve as students’
assistants, companions, and tutor (Oscar, 2005).
3 APPLICATION OF MATHS TO
REAL LIFE
An important task for engineers is to learn how it is
possible to model daily events and objects with
mathematical tools. It is considered that problem
solving is an important part of their education; by
interpreting such problems as a context within which
they can apply Mathematics to aspects of the “real
world”, they can learn how to make practical use of
their mathematical skills (Lantz-Andersson, 2008).
The School of Industrial Engineering aims to
prepare students for leadership in industry and to be
at the very centre of developments for addressing the
economic and social needs of the country. The
School has always been closely linked to the society
it serves.
One of the most used tools by students to
perform their tasks is the internet. Tutors are
responsible to make them see the dangers of the
Internet and the vast amount of data. Finding
information can be a very costly task. At this point
the software agents are useful because they help us
unravel the information saves time and effort. The
agents dealing with information use a series of
filtering mechanisms based on content. Filter selects
themes based on correlations between the content of
the topics and references of the user.
We will show in detail some of the activities and
final work that students made.
CASE STUDY: COMPLEMENTS OF MATHEMATICS AND E-LEARNING
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3.1 Application of Mathematics in a
Garden Sprinkler
In this case the working group uses the LU
factorization to explain how a sprinkler works. LU
decomposition is a matrix decomposition which
writes a matrix as the product of a lower triangular
matrix and an upper triangular matrix. This
decomposition is used in numerical analysis to solve
systems of linear equations.
3.1.1 Movement of the Sprinkler
Depending on the position of the output link of four-
bar mechanism, the sprinkler will water one way, on
the other or both ways.
The elements of the sprinkler are:
 Turbine and worm 1 and crown wheel 1: Water
affects some pressure on the blade by rotating
turbine. On the turbine shaft is located a screw
endlessly rotates the turbine. This screw
engages with a cogwheel. The mission of this
gear system is to reduce speed.
 Worm 2 and crown wheel 2: Again, another
mechanism of worm and crown wheel reduces
speed to reach the four-bar mechanism.
 Mechanism of four-bar linkage: The links are
connected by joints that allow the rotation of a
rigid body over another.
3.1.2 System of Linear Equations
The system of linear equations is obtained from
dynamic analysis of four-bar mechanism. For
resolution the Mathematica software is used, with
the implementation of some functions to get the LU
factorization, decomposing the matrix obtained with
real data, in two others. This is the way to calculate
the forces acting on the links.
3.2 Functions of Complex Variables
Before finalizing any design, should be checked how
it reacts with airflow to achieve a perfect
combination of aesthetics and performance. To
perform this check, used the so-called wind tunnel,
which is a compound which simulates actual air
flow. The components of this device are:
 Cavity to stabilize the flow establishment.
 The reduction cone compresses the air volume
increasing speed
 Test section where the model stands.
 Diffuser that reduces the wind speed.
Thinking about wind tunnel students quickly
relate it to the world of motoring, and particularly
with the automobile and aeronautics, but the reality
is that the wind tunnel has a very broad utility, from
skiers to missiles, through buildings or trains.
For general modelling the potential flow is
considered, excluding something as common as the
turbulence. The purpose of complex analysis is to
calculate the coefficient of wind resistance, and the
results of the flow lines. Due to the nature of certain
flows, longitudinal sections along the flow lines can
be considered for analysis. This remains then a two-
dimensional system that can be analyzed
mathematically using the complex plane where one
element is given by a complex expression.
Because of the complex analysis requires
millions of calculations, CFD (Computational Fluid
Dynamics) has been developed to analyze and solve
problems concerning fluid flows.
3.3 Partial Differential Equations
Any differential equation that contains partial
derivatives is called partial differential equation. In
any partial differential equation, the dependent
variable (unknown function) should be a function of
at least two independent variables since otherwise
would not appear partial derivatives.
The vibrations that occur during driving, flying,
travelling by train, or those produced by a machine
tool during cutting, which suffer the blades of a
turbine rotating at several thousand revolutions per
minute or vibration that supports a building during
an earthquake are usually governed by the wave
equation.
A membrane attached by the edge and under
uniform tension, the ideal patch of a tambourine,
vibrates according to a partial differential equation.
And also the heat equation, which describes how to
distribute the temperature versus time and space,
follows another type of partial differential equation.
3.4 QR-codes
In this issue we found the example of how the
amount of information available on the Internet is
not enough to find the solution to a problem. This
group of students wanted to find an application of
the QR decomposition, which writes a matrix as a
product of an orthogonal matrix and an upper
triangular matrix, and multiply the factors, and
iterate, but what they found were the QR codes, a
matrix code created by Denso-Wave corporation, to
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allow its contents to be decoded at high speed. The
meaning of “QR” is “Quick Response”.
4 CONCLUSIONS
This study has presented a case study in which
existing software agents are used, this is not about
making a new development related to agents or
similar education software, but to use what already
exists. We have detailed the subject of math
supplements, a subject of the second cycle of
industrial engineering. In it, students must perform
the proposed activities each day and make a final
work at the end of the course as application to real
life.
The internet is considered as supplement to
traditional teaching, we have seen, by the end of the
course a survey of students, that the online teaching-
learning system, has allowed its use to the maximum
use (Kearsley, 1998). Students and teachers can
access the work and activities at any time, students
have available and affordable all resources, and the
means for optimum use of the subject, and the
flexibility of the word wide web involves a
cooperative platform, rather than independently,
allowing students to interact and organize the course
as needed (Aliev, 2001).
ACKNOWLEDGEMENTS
This work has been supported by the Project FS/5-
2009 from “Fundación Memoria D. Samuel
Solórzano Barruso” (University of Salamanca).
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