Some Solutions for Representing Mathematics on the Internet

Matti Lehmuskero and Martti E. Pesonen
University of Joensuu

    Abstract

    An important goal of our project "Exploiting Information Technology in Mathematics Instruction" is to find convenient and practical solutions for representing mathematics on the WWW. For example, we require reasonable richness in notation and extendability towards interactive and lively documents. Standard and new technologies have been explored and some applied in classroom teaching. As an application, we shall demonstrate the architecture of a Web based (classroom and) distance learning environment that we are creating.

    Introduction

    The rapid development of Internet techniques enables totally new possibilities for teaching and learning. A web browser can be seen as a tool that can be used to combine different methods to produce teaching material. The subject itself may have existed as such a long time but the new ways to represent give new possibilities for teaching it. In Chapter 3 the different types of materials are discussed. In Chapter 4 we will take a look at the technical aspects that are involved in the development of Internet based teaching environments combining different materials to a single unit.

    Material offered to students

      Text material

      The material given in textual format will give the basic knowledge of the subject. They are based on lecture notes of a traditional course held at the university. The material contains all the theory involved in the subject and serves also as a reference material for the other components that are used in the teaching.

      The text is offered in two formats. The HTML document is intended to be read in the browser while studying on-line. The material is linked so that a click of a mouse to a reference changes the position to the corresponding label. The intention is that while the student does exercises in the browser one can easily look for help and references from the text.

      The text material in HTML document alone is not sufficient because one can not access the document all the time. That is why also printed material may be needed and for this purposes a PDF document of the same lecture note is provided. The printing quality of an HTML document containing mathematics is not very high. On the other hand the PDF document is not that convenient to be read on-line, despite the support of plugin-programs. One solution is to provide documents that are best suited for different purposes.

      Exercises

      Another part of the teaching are the exercises. The traditional way of exercises is to hand them out and to correct them in weekly demonstrations. In distance learning these phases change because there is little possibility of reaching all students at the same time. One possibility is to deliver the exercises and then ask the students to send their answers in written form. Then the answers will be corrected by the tutor and the answers are given as a web page. The problem is how to collect the answers. HTML provides form fields that can be used to collect some answers, but not all. For example proofs are not easy to write just by using the basic alphabets and digits.

      Using web also provides a new form of exercises, namely the kind that the system can check automatically. This kind of exercises can not be too complicated because the correcting is done by a computer which is not capable of recognizing answers full of rich notations. They will be more like multiple choice questions and identifying and connecting the corresponding items. These kinds of exercises are easy to check with simple programs. They do not provide any information to the tutor but act as supporting material for the students.

      Active components

      The real innovation of web based teaching is the possibility to use different kinds of active components in the teaching. In traditional lecturing illustration and the use of motion and sounds are limited. In web teaching all the techonology is already available.

      One form of active components are the animations. When using blackboard it is difficult to illustrate for example how a complex exponential function maps an area to another. In a web document one can easily link an animation that shows the process in few seconds. The animation can be repeated as many times as needed to invoke the desired mental construction in the student's minds.

      Animations offer a way of representing motion in the document, but there will be no user interaction with the movement. To add also the interactivity there exists a great deal of Java-applets. Java, a programming language that all modern browsers support, can be used to produce almost any kind of applications (applets) that can be used in the browser. It supports graphical user interfaces so that students can for example manipulate some parameters using a mouse and see the effects immeadiately. The applets offer more possibilities than animations but are much more difficult to produce because they require a good amount of programming skills. The lack of programming skills does not prevent authors from using Java as a part of the teaching material. There are lots of applets that can be downloaded and used freely in educational purposes. Some of them are very specific to a certain subject but there are also more general purpose applets that can be further programmed to do wanted things. For example dynamic geometry programs have their counterparts in Java.

      Helping students

      Another important thing is to provide support and aid for the student at a distance. A natural way of doing this is e-mail. This can be done either as a personal contact or as a mailing list that reaches all of the students at the same time. Although the learning environment may not require any e-mail address it should be recommended that all the students participating the course will have one.

      Another way of promoting communication between students and the tutor is using discussion boards. Discussion board is a web page where users can send messages either as a new subject or as a reply to some other's message. This way students can interact with each other and also the students not participating the actual conversation can follow it.

    Technical aspects

      Ways to represent mathematical formulas

      Currently, there are no standards for the representation of mathematics on the Internet. There exist many systems that can be used in combination to produce satisfactory results, though none of them is much better than the other. Usually the quality of appearance and the ease of use are the dominant factors, but both of them are not that easy to reach.

      Perhaps the oldest and most widely used method of mathematical representation is to use images for formulas and equations. For example, most of the difficult equations are converted to images (e.g. gif-files) straight from the DVI-file created by the LaTeX-compiler. This solution does not restrict the availability of mathematical symbols: Anything you type to a TeX-file is viewed in the web-browser. Though very versatile, this method is not very efficient in the use: the formulas appear quite coarse and there may be a lot of images in a single page meaning that the loading of the page is time consuming.

      Other solution is to use more of the browser's capabilities and present the symbols using fonts that are drawn to the screen by the browser itself. Usually the system fonts include much of the symbols needed for the mathematical representation, for example integral and partial derivative symbols and brackets and braces. This way the document to be viewed consists only of the HTML-document, with images only for graphs etc. Now the document can be viewed with quality in different screen resolutions without distortion and also the printing quality is better.

      The two methods described above can be considered as low-level solutions: both of them rely directly on the browser and use only the capabilities that are easy to use. But there are also more advanced systems. One can use for example programs that co-operate with the browser. These external plugin-programs extend the capabilities of the browsers in many ways, but this is also a weaknesses. Usually they do not come with the browser package but have to be downloaded by the user, which means that everyone does not have the required programs.

      In the future there may be a solution for most of the problems on the presentation of mathematics. World Wide Web Consortium has already developed MathML, a recommendation of a Mathematical Markup Language that will be used to represent mathematics directly on a web document and then rendered to the screen. Whereas now we have to do tricks to get the appearance we want, in the future the browser will produce a good output from mathematical markup instructions written by the author. At the moment none of the popular browsers support MathML, but the support at least in Netscape is under development.

      Compatibility issues

      The choice between different representation methods has many consequences. Most notably, it raises the problem of differences between computers and operating systems. If one plans a learning environment that can be accessed anywhere, it will be used in different computer environments. All of them handle things differently: a document that apperas nicely in MS Windows may not do that in UNIX. Also, if one uses external programs to assist the browser, there may not be versions for different operating systems.

      There are also different web browsers that do not support exactly the same things. Also there exists newer versions, which have more features than the old ones. These things have to be taken into account in the development and the material should be tested also with different browsers. On the other hand, if the system is made "too compatible", some features that may be essential will be lost. Because new versions of the browsers are easily available, we can put requirements for browser capabilities.

      Extendability

      We can also require the system to be extendable. For example, if a new version of a browser enables new features that make things simpler, the system should be designed so that it can adopt these new capabilities with ease. At the moment we are using HTML for almost everything but as the support for MathML becomes a reality the transition to MathML will become actual.

      The extendability does not concern just the browser part. As we take new features in use there may also be a matter of rewriting the material so that it uses the new system. For example, if the original textual material is written directly as HTML document, there is no easy way to convert it to MathML. On the other hand, if the material was originally as a LaTeX-file, it will be easy to recompile it to a HTML document that also uses MathML encoding.