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Abstract. After 5 years of experiments and
feedback we decided to continue the software development
on WebMathematics Interactive, a web-based e-learning
tool, rewriting it from scratch. The demonstration version
of WebMathematics Interactive 2 (WMI2) has been shown to
the expert audience on the CADGME conference. In this
article we summarize the development goals and results.
Key words and phrases: computer algebra systems,
computer aided education, distance learning, free
software, modularized knowledge, Maxima, gnuplot, PHP,
TeX/LaTeX, Linux.
ZDM Subject Classification: U55, U75.
In 2003 we wrote a first report about the start of the WMI project [VK03] at the University of Szeged. Now our second article shows that our continuous work of the last years has raised into an international consensus. E-learning experts and didactical advisors joined us to enhance the abilities of WMI1. The result is a much more valuable tool in self-assessment of the learning process in mathematics. Though, further effort is needed to fulfill all the requirements we planned to have a complete education tool.
In this article, after outlining the past of WMI1, in Section 2 we build up some expectations of the future versions of WMI. Section 3 shows the results of our work; finally Section 4 gives a full example of problem solving using the new user interface of WMI2.
The main goal of WMI1 was to make a complete e-learning system for students at the age of 14–21, containing basic services for solving routine problems (through CAS evaluation) and offering thematic modules of tutorials of selected areas of mathematics.
Our goal was partially achieved. WMI1 became a world-wide used frontend in several universities in Hungary. At University of Szeged we have been using this interactive web application during the every day teaching practise. In addition, scientific feedback was also recorded from the University of Pécs, College of Dunaújváros and College of Nyíregyháza. A dozen of introductory articles mentioned WMI1 during the last years, including both scientific and non-scientific papers ([ Bar05 ], [ Bla06 ], [ HKK04 ]). International feedbacks arrived and scientific contacts were established with Taiwanese, British, French and Spanish colleagues. Presentations about WMI1 in mathematics conferences during CAS sections in Spain
. Chu-Ching Huang (Chang-Gung University, Tao-yuan, Taiwan):
LiveTeXmacs: a platform for mathematical learning.
International Congress of Mathematics 2006, Madrid
. Michal Kaukic (University of Zilina, Slovakia): Moznosti a
perspektívy vyuzitia open source softvéru vo
výucbe a výskume na vysokých skolach.
Aplimat 2007, Bratislava
. Gábor Horváth, Antal Joós,
Bálint Nagy (College of Dunaújváros,
Hungary): WebMathematics Interactive in use. CADGME 2007,
Pécs
. Zoltán Kovács: WebMathematics Interactive 2.
Workshop, CADGME 2007, Pécs
. Chu-Ching Huang and Zoltán Kovács: Open
resources for mathematical education. Mathematical meeting
and annual meeting of the Mathemacial Society, Department of
Mathematics, National Taiwan Normal University, 2006, Taipei
Due to the statistics, our local installation of WMI1
Web based services are considered the most vibrating and modern way of
computer aided technologies. The Google boom, i.e. the rapid growth of
a freely available quality search engine, shows that
“usability” usually means “nearness”. If you
find the information fast enough
One of our main goal was always to focus on the ease of use. Much experience was collected during the years to enhance usability and make WMI more intuitive than before. In parallel, a second goal was to extend WMI to be a complete system to help in both self-learning and self-assessment.
During the last years many remarkable new software has appeared on the market and many among them have been freely available and useful. Indeed, a few goals of ours have been apparently implemented by other projects, some features of our implementation proved itself to be old fashioned, compared to the new kind of solutions. Our limited resources forced us to focus on such features of WMI1 which are still leading technologies world-wide, and we had to decide what kind of future developments are feasible for WMI at all. This is how we finally found out to concentrate on CAS based evaluation of routine problems and omit tutorial sections and questionnaires for some time. This means that WMI2 has less kind of built-in opportunities for its 2.0 release than for 1.0; hopefully for the next releases (2.1 and above) we shall focus on the missed features again.
From this point of view we listed all problems which could be solved within a limited time. In this section we summarize all such known problems of WMI1 and their planned solutions for WMI2.
Problem 1: Incomplete knowledge base. There were too few areas of mathematics covered by WMI1. The basic services were limited to introductory calculus and algebra.
The source of this problem was the lack of technical knowledge of didactical designers. WMI1 was extremely difficult to expand due to the large amount of needed technical skills.
Problem 2: Ambigous usage. The javascript based calculator was unable to start in Microsoft Internet Explorer and a wide range of users complained that it is not self-explanatory.
Problem 3: Circuitous work. The output formulas were not able to re-use in a following context. Many formulas had to be re-typed to continue the work with them in a similar, but not the same problem situation.
Problem 4: Slowness. There was too much idle time between raising the question and getting the answer.
Problems 2, 3 and 4 could have been solved by skilled programmer technicians. Solving problem 2 seemed to be rather a didactical challenge than a programming skill.
Problem 5: Difficulties in extending. Problem 1 is very important from the teacher's view, too. To gain additional colleagues to encourage them to contribute their own ideas, exercises and solution methods, it is necessary to give an easy-to-use development tool for the teacher as well, preferably a web based editor.
Problem 6: Math insertion. Entering math is a headache for most teachers. Both TeX and MathML (or simple HTML) seems to be infeasible to learn for elementary and grammar school teachers. Indeed, TeX was designed for typesetting technicians, and MathML for mathematical web designers. A much more convenient way should be found for “mortal people”.
Problem 7: Customization. Flexible reorganization of the available learning material. Some teachers have their own logic of building up a theory or a part of mathematics. This demand should also be supported by web editor tools.
A big project always have the danger that too much expectations should
be fulfilled with limited resources. “Less is more”, motto
of minimalists, may make us understand that it can be more valuable to
develop a tool which is useful for only one purpose but functioning
perfectly, better than to have a multifunctional tool which is full of
faults. In other words, if a well-known working tool already exists
for a subproblem, then it should be checked if it is possible to merge
with the existing system or not
After finishing WMI1 in May 2005 and patching its bugs until July 2006, it became clear for us to draw bright lines around the planning of the second version: what do we want to do for WMI2, and which features are more or less important. For that we set up a web server with a MediaWiki database and collected all ideas together: what to do and what to forget. This time interval for planning took almost 6 months. In December 2006 we started to build up the underlying database structure and fixate how WMI2 will work.
A very short version of the final decision of our judging committee follows:
Enhancement 1: Problem database. A Google-like searching machinery is needed to classify mathematical problems. If it is possible, the classified problems must be able to be parameterized.
Enhancement 2: Solution charts. Thematic modules should be redesigned from scratch allowing teachers to design solution methods using flow charts.
Enhancement 3: Assessment interface. Assessment techniques should co-operate with already working, existing, specialized systems.
During the kick-off of WMI1 development we defined the key properties for WMI. This has not been changed in our continous work. Here we summarize the longterm goals for the WMI development process:
In this section we summarize how Problems 1–7 and Enhancements 1–3 were solved in the development process.
During the implementation a clear picture was starting to be outlined that WMI2 should be a web based interface for freely available CAS and geometry software providing a textual worksheet containing the problems and their solutions in a savable and printable HTML format. Surprisingly, we found that other developer groups are also committed to similar goals. SAGE and Axiom have both a web interface for scientific problem solution and TeXmacs [vdH+07], a native application for Unix-like operating systems has also a very convenient interface to get on-the-fly answer from CAS, inside a user friendly environment. wxMaxima [Sou07], a platform independent approach for such a system is also committed for international support of CAS usage.
The uniquity of WMI2 is that it focuses on education, not research or typesetting. It can be reached from everywhere on the internet independently of the used operating system and the web browser software. In addition, the intuitive graphical user interface is designed especially for young people at the age of 14–21 with its colorful theme and challenging layouts.
To solve Problem 1 we created a database of 21 tables instead of the existing 10 ones in WMI1. All modifiable information is put down in a database. To enable web based editing, we used a commercial underlying database engine, Wekker, which is confirmed to be freely available for everyone to use. (The source code is not available.) Now all data is editable through a web interface, including creating new calculator layouts which can cover different mathematical fields.
We rewrote the javascript based calculator (Problem 2) to enable immediate check of the displayed result of a one-dimensional formula. We used bleeding edge Ajax technology [ZMF06] to connect formconv [KB07], LaTeX, Maxima [SA+07] and ImageMagick [Sal06] to get a WYSIWYG formula editor, based on pure Javascript/DOM technology (no client-side Flash or Java applet for portability, compatibility and speed reasons). These programming challenges shortly solved Problem 3 and 4, too.
Problem 5 was also solved using the Wekker Database Engine.
To solve Problem 6 we started to use the Xinha editor [BSM+07] which offers Javascript/MathML based on-the-fly formula editing. However, to harness the power of its equation editor, ASCIIMathML [Jip05], the web browser should be carefully configured on a large variety of workstations. Usually a
To properly display the MathML on this page you need to install the following fonts...
message is sent within the browser, if configuration is not perfectly set up. That's why this problem cannot be considered to be solved completely. (The problem is that definition standard of the MathML language is still a work-in-progress.)
Problem 7 is also solved using the hierarchy editor of the Wekker Database Engine, however, latest version of WMI2 has no underlying database of a learning material tree yet.
Enhancements 1 and 2 are in planning stage yet. Enhancement 3 is expected to be solved via an on-line connection with Moodle [Ric06].
Finalizing WMI 2.0, we concluded the following plans for continuing the project:
The following table shows the differences between WMI1 and WMI2 and there is also a roadmap plan for versions 2.1, 2.2 and 2.3. (2.0 has been released in September 2007.)
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Table 1. Comparison of WMI1 and WMI2. (LAPP
stands for Linux + Apache + PostgreSQL + PHP, see [Wik07a]
for details.)
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The following figures show a typical web page of WMI1 and WMI2.
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Figure 2. WMI2, calculus layout, calculating
and plotting the derivative of the function
f:R + →R,
f(x) = sin x2 + cos
ln x
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In this section we describe how WMI2 works from the user's point of view.
You need a recent web browser, Mozilla Firefox 2.0 is suggested,
however Internet Explorer 6.0 and 7.0 also works properly. On the left
side of the window of the web page there is a calculator which is
designed for entering the formula of a problem, e.g.
x2 - sin x. The formula can be
entered by typing it in a one dimensional “intuitive” form
(i.e. x^2-sin(x) or x^2-sinx) or
clicking on the following calculator buttons:
,
,
,
,
. The two dimensional formula will be immediately shown
on the right side of the window in red, displayed by the
Ajax/PHP/formconv/LaTeX/dvips/ImageMagick machinery.
Now we can work on this formula, applying one of the 30 currently available basic services:
The CAS evaluation is powered by Maxima, plots are generated by Gnuplot [Gav07]. The answer is shown in blue color. Both the question formula and the answer are put into a worksheet on the right side. Both formula can be re-used for further calculations by clicking on them. It is also allowed to use a former formula as a part of a new input. (The figure above shows an example of using WMI2 only by a mouse. The calculations illustrate the possibilities of WMI2 in an introductory calculus course.)
WMI2 also offers different calculator layouts for different type of students. Currently we offer 5 types of calculator layouts:
Each layout has 5 buttons in 8 rows. These 40 buttons are usually not enough to list all available opportunities. To solve this problem, we put some similar types of buttons into the same button group, i.e. some buttons open up a set of new buttons available for further use.
Besides the author, the main concept of WMI was initiated by Róbert Vajda (University of Szeged). The biggest part of programming was made by Árpád Fekete (University of Szeged) as his master thesis [Fek07] in computer science. Here we would like to thank our colleagues for their continuous encouragement and help. In addition, our Taiwanese colleague, Chu-Ching Huang (Chang-Gung University, Tao-yuan, Taiwan) was also very kind to us in his full support during the development.
We also thank Christopher Sangwin (University of Birmingham), primary author of the STACK self-assessment system, for his worthy comments, including suggestion of considering Moodle as an interface for students (instead of using a native solution).
WMI2 was put together by dozens of mathematicians and programmers. We cannot give a full list of them here, but a complete enumeration of the contributors is available on http://wmi.sf.net, primary page of WebMathematics Interactive.
[Bar05] T. Barik. LaTeX equations and graphics in PHP. Linux Journal, 2005. http://www.linuxjournal.com/article/7870.
[BBD+04] M. Bronstein, W. Burge, T. Daly, J. Davenport, M. Dewar, M. Dunstan, A. Fortenbacher, P. Gianni, J. Grabmeier, J. Guidry, R. Jenks, L. Lambe, M. Monagan, S. Morrison, W. Sit, J. Steinbach, R. Sutor, B. Trager, S. Watt, J. Wen, and C. Williamson. The Axiom Book. The 30 Year Horizon. 2004. http://wiki.axiom-developer.org/public/book2.pdf.
[Bla06] I. Blahota. Maxima: an open source CAS solution (in Hungarian). Linuxvilág, pages 36–39, September 2006.
[BSM+07] M. Bazon, J. Sleeman, R. Meyer, et al. Xinha: The community-built open source online HTML editor, 2002–2007. http://xinha.webfactional.com.
[Buc90] B. Buchberger. Should students learn integration rules? ACM SIGSAM Bulletin, 24(1):10–17, January 1990.
[EMX07] R. Eixarch, D. Marquès, and S. Xambó. Wiris, 1999–2007. http://www.wiris.com.
[Fek07] Á. Fekete. WebMathematics Interactive 2 (master thesis, University of Szeged, Department of Computer Science), 2007.
[Gav07] H. P. Gavin. Gnuplot 4.0
[HKK04] J. Hubička, Z. Kovács, and Z. Kovács. Visualizations on the complex plane. Computer Algebra Systems and Dynamic Geometry Systems in Mathematics Teaching. Proceedings of Sprout-Selecting Conference, Pécs, Hungary, pages 12–27, 2004.
[Jip05] P. Jipsen. AsciiMathML, 2005. http://asciimathml.sourceforge.net/.
[JS+04] D. Joyner, W. Stein, et al. SAGE Tutorial. 2004. http://www.sagemath.org/doc/html/tut/index.html.
[KB07] Z. Kovács and G. Bakos. Formula Converter, 2003–2007. http://formconv.sf.net.
[Ric06] W. Rice. Moodle E-Learning Course Development. Packt Publishing, 2006.
[SA+07] W. Schelter, J. Amundson, et al. Maxima Book. 1968–2007. http://maxima.sourceforge.net/docs/manual/en/maxima-5_13.pdf.
[Sal06] S. Salehi. ImageMagick Tricks. Packt Publishing, 2006.
[Sou07] SourceForge.net. wxMaxima, 2004–2007. http://wxmaxima.sf.net.
[vdH+07] J. van der Hoeven et al. TeXmacs, 1999–2007. http://www.texmacs.org.
[VK03] R. Vajda and Z. Kovács. Interactive web portals in mathematics. Teaching Mathematics and Computer Science, Debrecen, Hungary, 1(2):347–361, 2003.
[Wik07a] Wikipedia. LAMP
(software bundle)
[Wik07b] Wikipedia. Unix
philosophy
[ZMF06] N. C. Zakas, J. McPeak, and J. Fawcett. Professional Ajax (Programmer to Programmer). Wrox, 2006.