EMBEDDING COMPUTER ALGEBRA CAPABILITIES IN AN ELEARNING PLATFORM A. I. Alonso, J.A. Calzada, J.M. Farto, A.B. González Dpto. Matemática Aplicada. E.I.I. Valladolid University (SPAIN)
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Abstract The aim of this work is to improve the interaction teacher-student in the technical education area. We provide an e-learning (Moodle based) environment to the user, who can tune the offered contents. In particular we intend to replace the static resolution of a specific scientific-technical problem, for a dynamic one. The student proposes the problem and the platform shows its resolution step by step. In this way, the students have a schema of the problem, as if were proposed in the classroom, but they can pose their own variations. Keywords: Moodle, e-learning, webMathematica.
1
INTRODUCTION
In this work we present an e-learning tool that came together Moodle and the web-based symbolic manipulator webMathematica (see Fig. 1.).
Fig. 1. Moodle environment Rapid developments in information and communication technologies (ICT) in recent years have resulted in significant changes in the way the world operates and communicates. This in turn has had an impact on the educational system, transforming the learning process. ICT can be used to develop course materials, deliver and share the course content, lectures and presentations, and facilitate communication among lecturers and students. An example of this is the Moodle platform. It provides a learning environment flexible and adaptable in which teachers and students can discuss without the limitations imposed by a face to face tutorial. A
Proceedings of EDULEARN12 Conference. 2nd-4th July 2012, Barcelona, Spain.
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ISBN: 978-84-695-3491-5
forum in Moodle allows participants to have asynchronous discussions which supplements the class time. Obviously these changes are not exempt from associated problems. In some cases it is difficult to replace satisfactorily the tutorial classes by an electronic consultation; especially if you need a technical language (for instance, written in LaTex, that a first grade student usually doesn't know). Moreover, it is easier to know if the student has understood quite well the subject in a face tutorial that with the exchange of several messages. This work aims to improve the interaction teacher-student in the technical education area. We provide an e-learning (Moodle based) environment to the user, who can tune the offered contents. In particular we intend to replace the static resolution of a specific scientific-technical problem, for a dynamic one. The student proposes the problem and the platform shows its resolution step by step. In this way, the students have a schema of the problem, as if were proposed in the classroom, but they can pose their own variations. To this end we add to Moodle the computing capacity that provides webMathematica. We must point out that the student does not need training time because no longer have to learn Mathematica, everything is done in the same language as the teacher uses in the classroom. Section 2 describes briefly what webMathematica is and the important features that it can offer. Section 3 discuss the implementation of an example applied to the ordinary differential equations, in particular, how to solve a second order ordinary differential equation by Frobenius method; showing the Moodle environment and the websites (with interactive calculations) built with Mathematica.
2
APPLICATION COMPONENTS
The basic components of the application that we present are Moodle, Mathematica and the set of technologies to work with them.
2.1
Moodle
Abbreviation for Modular Object-Oriented Dynamic Learning Environment, Moodle is a free source elearning software platform. We run Moodle on Linux, system that hold PHP and a database creating a course easy to manage and use ([3]-[7]).
2.2 2.2.1
webMathematica Description
webMathematica is a web application developed by Wolfram that allows you to access Mathematica through a web interface [8]. It is based on Java Servlet technology and JavaServer Pages (JSP). It is therefore a system centered in the server, where the client, through a web browser makes requests to a server running the service webMathematica in a Servlet container. The requests are processed by a Mathematica kernel running on the server or on another node and responses are sent to the client that displays in your browser. It also allows the integration of Web technologies as well as Javascript or Java applets. The user of a webMathematica site does not require programming skills Java, Javascript or JSP, and only the knowledge of Mathematica that the particular implementation of the required site. Something very different is the level of knowledge expected of a webMathematica site developer. To implement a webMathematica site, you need solid knowledge of HTML, JSP and Mathematica. If the site is moderately developed, it will be necessary to have some knowledge of at least Javascipt, CSS and XML.
2.2.2
Installation
The installation of WebMathematica on a Linux system (option we chose) is not straightforward. First you need a servlet container like Tomcat, operating perfectly. This is not a big problem with current Linux distribution. However, webMathematica has a number of peculiarities, some partially documented and undocumented others that cannot be deployed without further in Tomcat. First, with standard security settings for Tomcat 6, webMathematica does not start. We have tried to relax the rules to a minimum to work, but finally the only way to start webMathematica is disabling all
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Tomcat security protocols for this application, which we understand that suposses a risk to the server running. Second, webMathematica cannot be restarted with Tomcat administrative tools. Tomcat must be restarted whenever it is necessary to do the same with webMathematica. This would be a serious problem if the server used with webMathematica is used simultaneously with other web applications and would require a copy of Tomcat only for mathematical, which is an inefficient management of resources. Third, webMathematica needs to run a Mathematica kernel to perform the calculations and to return the results in certain formats, it also needs a frontend of Mathematica and this last requires imperatively a Graphical User Interface (GUI) running. We believe this is due to the special architecture of Mathematica, developed at the beginning, for other operating systems with a GUI running. The elegance of this solution may be discussed, but not the consequence on a Unix server running webMathematica. The Mathematica frontend for any Unix needs to connect to an X server (which provides the infrastructure for the GUI) to boot. However, a Linux server may not have installed a GUI or any of its components. For working webMathematica is necessary to have at least one X server running. The most economical and appropriate way is always running a VNC server (recommended by Wolfram), but it can suppose a security problem. Also, the necessary settings may not be as immediate as the webMathematica documentation suggests and require some knowledge of Unix systems. Finally, there are security problems arising from a possible execution of arbitrary Mathematica code by an attacker, which could expose sensitive information of the server unnecessarily. WebMathematica documentation explains how to avoid this problem and is something that every webMathematica site developer has to take into consideration and has to be careful. We must say that once the installation has been achieved, its performance is unlikely to present problems, although on a few times has presented excessive latency. It would be necessary to investigate its possible limitations. Our approach is the integration of some of the facilities of calculation and presentation of data provided in a Moodle site. Moodle is based on PHP, not JSP, so the calls to the server webMathematica will be internal, transparent to the user.
3
EXAMPLES
We are going to discuss the implementation of two examples: the first one applied to find a solution of an ordinary differential equation (ODE) applying the Frobenius method and the second one for the study the diagonalization of a real matrix.
3.1
Frobenius Method
Let us consider the second order linear homogeneous ODE
where P and Q are the analytic functions
We want to apply the Frobenius method ([9], [10]) to search for a solution in the neighborhood of the regular singular point x0. The Frobenius method extends the simple power series method. A Frobenius series (generalized Laurent series) of the form
can be used to solve the differential equation.
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The parameter r must be chosen so that when (3) is substituted into (1) the coefficient of the lowest power of (x-x0) is zero. This is called the indicial equation,
It gives two values ri, I =1, 2 for the index r. Next, a recursive equation for the coefficients is obtained by setting the coefficient of (x-x0) equal to zero.
consequently we have obtained the solution
Further considerations are needed to find the second solution. If the roots of the indicial equation are unequal and not differing by an integer, we get two independent solutions by substituting the two values of the parameter r into the series (3). The second solution will be
If m = r1-r2 = 0, we get the second solution as in the Euler equation, by substituting the value r2 into the series ∂y / ∂r
Finally, if m is an integer, the sought solution is
where the parameter α is
We have used Moodle to conduct a fully online course (see Fig. 1) where we present the learning resources and activities that will help the students to understand the subject. You can find, among others, an application to solve an ODE by Frobenius method: let’s see how it works. First, the student has to fill in a form specifying the values of x0 and the functions P and Q (keep in mind the notation used in the previous technical description of the Frobenius method). After that, a website is dynamically generated in which the student can see the problem statement and how to solve it step by step. In the example shown below, x0 = 0, P(x) = 2+x and Q(x) = x. The first calculations provide the coefficients pn and qn of the series expansions of the functions P and Q (see Fig. 2).
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Fig. 2. Problem statement and coeficcients Next, we calculate and solve the indicial equation. It is a quadratic equation whose roots, r1 and r2, are the possible values for the parameter r. These values determine in what case of the Frobenius method we are (see Fig. 3.).
Fig. 3. Indicial equation Next we show the general form of two independent solutions which form a basis for the solution space of the equation as well as its values for the particular case at hand. For each root ri, i =1, 2, recursive formulas are used to calculate the unknown coefficients (see Fig. 4.).
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Fig. 4. Indicial equation and recursive formulas for the coefficients. Finally we solve the recurrence relation for r=r1 and r=r2 to obtain y1 and y2 (here only the y1 case is shown). The result can be seen in Fig. 5.
Fig. 5. The y1 solution
3.2
Diagonalization of a Real Matrix
In this example the e-learning platform allows study the diagonalization of a matrix ([11], [12]). First we enter the matrix (see Fig. 6), which can be fully numeric or contain a parameter (a in our case).
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Fig. 6. Problem statement The answer begins by calculating the characteristic polynomial of the matrix pA(x) = det (A – λ In) and finding the eigenvalues that can be numeric or depending of the parameter (see Fig 7.).
Fig. 7. Characteristic polynomial and eigenvalues of the matrix For each relevant value of the parameter we have to find the eigensubspace associated to every distinct eigenvalue. This subspace is defined by V(λ) = Ker (A – λ In)
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Where Ker stands for the nullspace of the matrix. If the dimension of this subspace is equal to the multiplicity of the eigenvalue as a root of the characteristic polynomial then the matrix is diagonalizable (we assume that the diagonalization process is developed in C so all the roots of the characteristic polynomial are eigenvalues). Fig. 8 shows the cases for a=0 and a=-2 and a=1 of the problem. If the matrix is diagonalizable the program shows the final diagonal matrix wich can be obtained by similarity with matrix P.
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Fig. 8. Study cases for the problem Finally the platform shows the result for other values of parameter a (see Fig. 9).
Fig. 9. Other values of parameter a
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