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Mathdox in Education

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Mathdox in Education MathDox in education Hans Cuypers Jan Willem Knopper Rikko Verrijzer Department of Mathematics and Computer Science Eindhoven University of Technology [email protected] ABSTRACT MathDox is a system for presenting highly interactive mathematical documents over the world wide web. It provides easy connections to Computer Algebra Systems, dynamic geometry systems and other mathematical services. MathDox shows its potential when demonstrating the workings of an algorithm or explaining new concepts with dynamic on-screen calculations. Moreover, within MathDox we have developed an exercise system providing parameterized exercises to students and offering intelligent feedback. We will describe the MathDox-system, its applications and use in concrete situations. INTRODUCTION Recently developed E-learning tools like Learning Management Systems (LMS) are very well suited for distance education and for delivery of online educational materials to students. They also provide various new ways of communication. However, these systems only offer a good learning environment for eduction via the web. The success of good education is still very dependent on the learning material itself. In the area of Mathematics we see that quite recently various new tools have been created that offer students a rich environment for practicing mathematics. These systems offer parameterized exercises with open questions that are automatically graded. Indeed, Maple TA (www.maplesoft.com), , Aleks (www.aleks.com), MyMathLab (www.mymathlab.com) en WebAssign (www.webassign.net) are commercial systems offering this type of facilities, while STACK (www.stack.bham.ac.uk), WIMS(wims.unice.fr) and WebWORK (webwork.math.rochester.edu) are some open source initiatives in this direction. The feedback that these systems provide, however, is usually very restricted. The DWO (www.fi.uu.nl/dwo) and the LeActiveMath system (www.leactivemath.org) are also two systems for mathematical exercises offering more intelligent feedback. Here we report on our own initiatives to create a system, called MathDox, offering highly interactive mathematical documents and exercises. MathDox is a system serving dynamic and interactive web pages with high quality rendering of mathematics and easy access to mathematical services like computer algebra systems or dynamic geometry software. MathDox shows its potential when demonstrating the workings of an algorithm or explaining new concepts with dynamic on-screen calculations. In this chapter we focus on some features of the MathDox system and in particular, we explain how these features have been used to create a rich environment for parameterized mathematical web pages and interactive exercises providing easy access to all kinds of mathematical (web)services, taking care of good presentation of mathematics, offering easy and natural input of mathematics and of facilities to provide meaningful feedback to users. The MathDox software is open source and can be found at www.mathdox.org. INTERACTIVE MATHEMATICAL DOCUMENTS Many of today's web pages are dynamic and interactive in the sense that they adapt to the user and offer the user some control over the contents of the page. Examples of such web pages are e-Banking, auction, and e-Learning sites. The dynamics and interactivity of these sites are a real asset when information gathering is at stake at both the user and the server end. When the user consults his/her account at an e- Banking site, the pages presented to the users are based on their latest data and are able to execute money transfer orders. Without dynamic and interactive websites, e-Banking would not be possible. Web pages about mathematics are often static and do not invite the user to experiment with and experience the material. We see three main reasons for this. First of all, interactivity in mathematical content often requires non-trivial computations that are not easily performed without the assistance of specialized mathematical software, e.g. computer algebra systems. Second, displaying mathematics properly on a web page can be challenging. There are a number of methods to render mathematics on a web page, such as MathML, CSS, or by images. Unfortunately, all of these are not intuitive to write by an author. Moreover, MathML is not yet natively supported by all major browsers, CSS style sheets are not always accurate and uniform on different browsers, images are slow to create and cannot or are hard be generated on the run. Third, the input of mathematical expressions is difficult, both for the user as well as for the author. It is a lot of work for an author to enter complex or deeply nested formulas into a web page with the standard set of HTML elements. Without any aid --for instance of pallets with mathematical symbols-- it is nearly impossible for an user to enter a complex mathematical expressions. Web pages that do contain interactive mathematics, often include various applets, see for example www.cut-the-knot.org or www.fi.uu.nl/wisweb/en, or flash applications as demonstrated by web sites as www.intmath.com or www.aplusmath.com/flashcards/. The various applets and flash items tend to be and remain separate items and do not come together and from a consistent interactive mathematical document. They also tend to focus on graphical aspects of the mathematics involved. To address these issues the MathDox research group at TU/e has developed the MathDox system, see (Cuypers et al., 2009). It consists of the MathDox format and a set of tools including the MathDox Player. The MathDox format is an XML based language for interactive mathematical documents. As such MathDox documents can be interpreted and transformed into web pages by the MathDox Player, software designed for this sole purpose. These resulting web pages are dynamic and interactive, support rendering of mathematics, offer easy access to computer algebra systems, and are equipped with a convenient mathematical input system. MathDox shows its potential when demonstrating the workings of an algorithm, testing readers' skills with exercises, or explaining new concepts with dynamic on-screen calculations. However, it can also be used for publishing static mathematical documents on paper or on the web When developing MathDox we have the following requirements in mind. Interactivity. A user needs means to interact with the mathematical content of an interactive document. For example, answering a question or to supply data for an example and to study the effects. In short the user needs ways to change parameters and the format needs to be able to adjust to those parameters. Non mathematical interactive pages use techniques like JSP, PHP, and ASP to this purpose. Mathematical applets or flash applications are interactive but often lack the opportunity to communicate and cooperate and form a consistent page with the other elements. Usable in multiple formats. An interactive mathematical document is best served in an electronic environment, such as a web browser. However the need may arise to view such a document on different media, such as a mobile phone or more traditional on paper. We consider it important that such formats are supported from the same set of source files. Meaning no changes to the source code are required merely a change of the program responsible for the transformation, in favor of another that is capable of producing the desired presentation format. Being adaptable. Mathematics has a heterogeneous field of applications. The usability of the MathDox format is greatly enhanced if it allows for adaptations to the format and the MathDox Player that would extend the functionality and make them suitable for other applications of the MathDox system. Representation of mathematics. The representation of mathematics occurring in the document has to include sufficient semantic information, such that the mathematics can be translated into several different formats for use by mathematical back engines or various forms of rendering. Ease of usage. The success of a format is mostly given by the ratio of richness of the format and the ease of usage. A format that does not offer much but is still easy to use will not gain much of popularity. The same goes for a rich format that is not easy to use. The users involved here are both the authors and the readers of the documents. For instance, users should not have to install any extra programs or plugins in order to read mathematical documents. At the same time authors should not have to much difficulty creating the documents. Creating an interactive web page is a programming task that many authors have trouble with. Therefore an interactive mathematical format should be designed so as to make this task as easy as possible for authors. Use of open standards. By following existing standards, a format benefits from other tried and proven technologies. It also helps to lower the learning curve considerably as compared to a completely new format. Which again directly assists in the previously discussed ease of usage of the format. Also the format itself should be open, so that every author will have direct access to all the specifics of the format used and is free to make adaptations if necessary. This will further expand the opportunities the format offers. THE MATHDOX FORMAT MathDox sources combine in a modular way various existing XML formats best suited for our purposes. Each format contributes a useful facet for interactive documents. The main XML formats used in MathDox are: DocBook, for structuring documents OpenMath, for semantic encoding of mathematics. Xforms Jelly, a programming and scripting format. These formats, their use and purpose, will be discussed. DocBook DocBook is a well-known documentation standard, see (Walsh, 2004). Its logical structure facilitates searching and parsing of specific elements, enabling easy translation into other formats including HTML and PDF. This makes DocBook suitable for our needs. We have extended DocBook with some structuring for mathematical statements such as assertions, proofs, examples and the like.
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