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E-Learning for math class using Maple and MapleNet
Yasuyuki NAKAMURA
Graduate School of Information Science, Nagoya University
nakamura@nagoya-u.jp
Abstract
In a math class for students who major natural sciences, like physics, chemistry and so on, it is
important to lead students to use mathematics in several problems of sciences. In order to realize that
kind of education, we use mathematical software Maple and develop some teaching materials by
using Maple. Teaching materials consist of text document and simulation software. Furthermore the
teaching materials are accessible via web browser over the Internet with the aid of MapleNet.
1. Introduction
function and how discrete Fourier transform is
applied for time series analysis from the aspect
of frequency are mainly focused.
In order to realize the purpose, it seems to be
helpful to use teaching materials that appeal to
the eyes. Many groups have already been
developing that kind of teaching materials
innovating animations realized by Java and
Flush [2-4]. We also have developed some
simulation software so far by using Java and C
language [5-8]. Most of them display numerical
solutions of differential equations and only
some parameters and initial conditions can be
set. In order to make good the purpose of the
math class, however, solving differential
equations analytically and perform Fourier
expansion and Fourier transform are required.
Therefore there is a limitation in using
numerical calculation and it is critical to deal
with mathematical expressions, which needs
using mathematical software, like Maple,
Mathematica [9] and so on. Mathematical
software has its own document style, for
example, worksheet for Maple, notebook for
Mathematica and so on. Teachers use them in a
class for demonstration, but in order for students
to use them by themselves, it is effective to
prepare teaching materials with graphical user
interface (GUI) that does not require students to
learn how to use worksheet or notebook.
Furthermore we expect additional benefits if
those teaching materials are opened to the public
such as a Java applet and a Flush application and
students carry out them on the web browser
even at home.
We have a math class for students who major
natural sciences and teach mainly Differential
equation and Fourier analysis. We give a brief
introduction of existence-uniqueness theory for
differential equations and convergence of
Fourier series, but “solving and using”
Differential equations and Fourier analysis is
taught on a priority basis. In order to adopt the
policy, computer-aided teaching materials are
very helpful. We prepare some teaching
materials by using mathematical software
Maple [1]. In this paper, we introduce some
features of our teaching materials, summarize
problems in using them and discuss about future
plan.
2. What is required for teaching materials
Topics of the math class are as follows:
differential equations observed in natural and
social sciences, some solving methods for
differential equations, stability of equilibrium
points of differential equations, Fourier series
expansion, Fourier transform, discrete Fourier
transform, power spectrum, auto-correlation
functions, application to data analysis and so on.
In teaching differential equations, we emphasis
on how phenomena in natural and social science
are formulated by differential equations, how to
solve differential equations and understanding
how solutions of differential equations behave.
In teaching Fourier analysis, how periodic
functions are described by trigonometric
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3. Teaching materials developed by Maple
In order to meet the requirement mentioned
above, we adopted Maple as mathematical
software and have developed some teaching
materials by using it. One of the good features of
Maple is that there is a Maplets package.
Maplets is one of the packages by which GUI
tools such as button, text area, pull-down menu
and slider are customized on a window and
applications with GUI are developed. GUI is
realized by Swing API of Java, but we do not
have to know about Java. Therefore using
Maplets package is one of the effective ways to
develop GUI applications. As an example, we
show Maplet application for solving differential
equation in figure 1. We can set parameters of a
differential equation and the application solves
the equation analytically and displays the time
evolution of the solution graphically.
We can develop even applications with
animation. Figure 2 shows simulation software
for three-body problem [10]. When we develop
this kind of simulation software, ordinary or
partial differential equations have to be solved
numerically. C, Fortran, Java language and so
on are usually used for calculation and the
results are shown on the graph. We can get
solutions faster with those programming
language than using mathematical software.
However, there are some advantages to use
mathematical software in order to solve
differential equations [11]. Firstly it is easier to
build GUI component by using Maple than by
developing by Java and C language. Start, stop,
frame-forward and frame-backward are easily
implemented. In the Plotter area, even three
dimensional (3D) animation is possible. If we
develop a same kind of software by Java, Java
3D have to be introduced to realize it. Secondly
we can use sophisticated mathematical engine to
solve equations of motion numerically, which
helps us not make mistakes in programming
numerical calculation part of source code.
Therefore source code is usually very simple in
Maplet application
Fig. 1 Maplet application for solving differential
equation.
Fig. 2 Maplet application for Three-body
problem.
We can show a demonstration in a class with
those Maplet application and those applications
can be opened to the public by using MapleNet
[12] with the Java applet technology. Students
can carry out those applications even at home.
Two academic years have passed since the
math class started. In 2004, the first year, there
are only a few Maplet applications and we
mainly show a Maple worksheet for
demonstrations in a class. We think that it is
more effective when students carry out those
worksheet by themselves and they can
understand more deeply. However there were
not many chances for student to use those
worksheets because not all students know how
to use Maple. Based on the reflection, we
developed more Maplet applications and give
chances to use them on the web with the aid of
MapleNet. It is not necessary to know how to
use Maple when students use Maplet application.
Only they have to know is how to set parameters
and the meaning of what is displayed on the
application. However we could not say that
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students from using the teaching materials
without worrying about the outcome. They can
understand the behavior of the system
depending on the parameter set.
Figure 3 is just a snapshot of Maple worksheet
and the same teaching material can be opened to
the public on the web through Internet with the
aid of MapleNet. Since MapleNet 10 was
released, Maple worksheet is directly opened to
the public by just copying the worksheet on the
MapleNet server. The application is then
available to anyone who has access to the
MapleNet server through an Internet browser.
The analysis is carried out by Maple
mathematical engine on the server and neither
Maple nor special plug-in is not required on a
client computer. What is only required for client
computer is to install Java Runtime
Environment and the type of operating system is
basically no object. This feature is good for
e-learning because we do not know what kind of
computer students use and we do not have to
require extra expense for students.
students used applications actively. We cite the
following two reasons as a factor. Firstly, we did
not give a detailed instruction document. In our
class, we explained how to use applications and
the meaning of what is displayed, but minute
instructions like how to set mathematical
expressions were not clearly defined.
Furthermore, as relations between an
application and content in a textbook were not
clearly explained, students could not understand
the meaning of results obtained by applications.
Therefore it is important that we have to give a
detailed instruction and a main purpose of using
applications in the textbook. If applications are
coupled with a textbook, we expect more effects.
Secondly, it took time to load and carry out
applications on the web when a network
environment is not advanced. This loses
students’ will to use applications. It is difficult to
solve the problem completely, but we care for it
in part by strongly encouraging students to use
computers in a PC room.
4. Teaching materials coupled with textbook
After discussions in the previous section, we
developed some teaching materials coupled
with textbook. Figure 3 shows GUI applications
embedded in the textbook. This is used to study
the behavior of solutions of multiple differential
equations. As applications are coupled with the
textbook, students carry out the application
while reading the textbook and it seems to be
more user-friendly teaching materials. In this
teaching
material,
multiple differential
equations can be set as mathematical
expressions and a vector field and an orbit of
solution depending on initial conditions are
displayed. The complex vector field and the
orbit of solution can be easily drawn with the aid
of graphic functions of Maple. We can know the
stability of a fixed point solution of differential
equations by analyzing eigen values of Jacobean
matrix but it is more effective to display vector
field and to appeal to the eyes for understanding
the result of analytic discussion. This acts as a
bridge between theory and actual feeling.
Furthermore, parameter sets that students give
are recorded as a history and redrawing is
possible with any parameter set. This prevents
Fig. 3 Example of a new teaching material.
5. Summary and future plan
We introduced a teaching materials developed
by mathematical software, Maple, as an
effective teaching aid and discuss about what is
required on the results of former two academic
year. A huge variety of teaching materials can be
developed by using Maple, but it is necessary to
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prepare an instruction document in which how
to use the teaching materials and purpose of the
application should be clearly described.
Reflecting on the discussion, we have developed
teaching materials in which an application is
coupled with textbook. These materials can be
opened to the public and students can carry out
those materials on the web even at home.
In the future, we are planning to develop
teaching materials that can be carried out even
on the mobile phone or personal digital assistant
that are very popular among recent students.
Figure 4 shows an experimental sample of a
teaching material for PDA. This application is
carried out by JSP technology.
[4] ``Physical Mathematics'',
http://smith.cmt.phys.kyushu-u.ac.jp/~M.S
akurai/phys/physmath/
[5] Yasuyuki Nakamura and Hiroshi Nakano,
“Development of simulation programs for
clasical mechanics - Using UNIX system
-” (in Japanese), Computer & Education
vol.6, 1999, pp. 107-111.
[6] Yasuyuki Nakamura and Hiroshi Nakano,
“A Computer Network System for Physics
Education - Using Linux -”, Advanced
Research in Computers and
Communications in Education vol.2, eds. G.
Cumming, T. Okamoto and L. Gomez,
(IOP Press, 1999), 1999, pp. 821-822.
[7] Yasuyuki Nakamura, Hiroshi Nakano and
Keniichi Tokunaga, “Virtual Laboratory
for Physics Education”, Proc. International
Conference on Information Technology
Based Higher Education and Training
(ITHET2002), 2002, CD-ROM.
[8] Yasuyuki Nakamura and Hiroshi Nakano,
“Simulation Physics as a Physics
Experiment for Students and e-Learning”
(in Japanese), Computer and Education vol.
14, 2003, pp. 34-37.
[9] http://www.wolfram.com/products/mathe
matica/
[10] http://www.maplesoft.com/applications/ap
p_center_view.aspx?AID=1879&CID=2&
SCID=131
[11] Yasuyuki Nakamura, ”Simulation
Software Integrated with Java and
Symbolic Computation System”,
Proceedings of 3rd International
Conference on Emerging
Telecommunications Technologies and
Applications, 2004, pp.239-242
[12] http://www.maplesoft.com/products/maple
net/
Fig. 4 An example of a teaching material carried
out on PDA.
References
[1] http://www.maplesoft.com/products/
[2] ``Everyday Physics on Web'',
http://nkiso.u-tokai.ac.jp/phys/matsuura/
[3] ``Math, Physics, and Engineering Applets'',
http://www.falstad.com/mathphysics.html
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