CyberMath

It is well known that the current state of mathematics education is problematic in many countires. The Interactive Learning Environments group at CID is developing an avatar-based shared virtual environment called CyberMath, aimed at improving this situation through the presentation of mathematics in a new and exciting way. CyberMath is suitable for exploring and teaching mathematics in situations where both the teacher and the students are co-present and physically separated. Our first prototype environment was built on top of DIVE, a toolkit for building interactive shared distributed virtual environments that support multiple simultaneous participants. However, due to problems with instability and usability, we are now migrating to an in-house platform, Wasa.

In the CyberMath environment, people (represented by avatars) can gather and share their experience of mathematical objects. When a person points to an object with the mouse, a red beam running from his avatar to the object appears in the 3D environment, similar to a laser pointer. Objects can easily be manipulated (rotated and translated) using the mouse. Since live audio is distributed as well, a person can point, act and talk - much as he/she would do in real reality - as if the mathematical objects were hanging there in front of him/her. Hence, mathematics teachers are provided with a tool that integrates the best of both the virtual and the real world: virtual (mathematical) objects can be manipulated and discussed in a realistic way.

An avatar is using the laser pointer to point to a geometric object in the DIVE version of CyberMath.

CyberMath is built like a museum with a virtual lecture hall in its center. Special care has been taken to make the environment as inviting as possible. The lecture hall contains a virtual slide projector and can be used to hold traditional PowerPoint-based presentations. The lecture hall is surrounded by a number of different exhibition galleries, each of which contains a collection of mathematical components expressing a common theme. In the lecture hall the overall theory can be presented and then the participants can move into the surrounding exhibition halls and study the mathematical objects in practice.

The lecture hall in CyberMath can be used for traditional PowerPoint-based presentations (DIVE version).

At this time, we have completed four exhibition halls with a content that is concerned with the differential geometry of curves and surfaces and with transformations. Three-dimensional Mathematica objects and animations can be converted for exhibition in CyberMath. The transformation exhibit allows dynamic exploration of mathematical transformations. Here, an arbitrary transformation (from R³ to R³) can be specified and the effects of this transformation can be studied interactively by manipulating different objects in the domain and observing what happens in the image. Together, the lecture hall and exhibits support a wide variety of teaching styles, ranging from traditional one-to-many lectures to teacher-supported interaction and individual off-line exploration. We are not aware of any other group that is exploring the possibilities of performing live teaching in environments like ours.

The transformations exhibit (Wasa version) can be used to examine many interesting mathematical concepts.

Two usability studies of the DIVE version of CyberMath have been performed, one small-scale test in August '00 with 2 subjects (both teacher and students located in Stockholm) and a larger study in October '00 with 13 subjects (teacher located in Stockholm and the students in Uppsala). These studies were aimed at evaluating the DIVE version of CyberMath and provide feedback for further enhancement. We can also run CyberMath in the six-walled CAVE at KTH and we are investigating the possibility of connecting different CAVE-based CyberMath sites through the Internet. We believe that CyberMath in a networked cave environment holds the potential to provide a high-tech front end which is interesting enough to create public interest and contribute to a more positive attitude towards mathematics - especially among young people. It could also provide a useful platform for developing various forms of interactive problem solving games - where the present violence-orientation of the gaming industry can be shifted towards an emphasis on cooperative problem solving skills, leading to future knowledge games that are designed to stimulate the learning interest of the gamers.

The exhibit hall (Wasa version).

CyberMath was introduced at SIGGRAPH 2000, New Orleans as part of the Geometric Algebra Course. It was also presented at SIGGRAPH 2001, Los Angeles, as an Educators Program Paper.

• Gustav Taxén and Ambjörn Naeve. A system for exploring open issues in VR-Based education. Computers and Graphics, 26 (2002), 593-598.
• Gustav Taxén and Ambjörn Naeve. CyberMath: Exploring Open Issues in VR-Based Learning. SIGGRAPH 2001 Educators Program. In SIGGRAPH 2001 Conference Abstracts and Applications, 49-51.

System requirements:

• Pentium III PC (450 MHz recommended)
• Windows 98/2000/ME
• OpenGL 1.2-compliant graphics card (or OpenGL 1.1 with the GL_ARB_multitexture extension)
• 4 Mb free hard disk space.

• Cybermath 1.0 Public Demo. (~600 Kb ZIP archive)
• Cybermath 1.0 Public Demo README file. (Text file)

• An exhibit hall containing surfaes illustrating the differential geometry of curves and surfaces.
• The laser pointer.
• The lecture hall.
• The transformations exhibit.
• An exhibit hall containing generalized cylinders and a point focus animation (on the far right).
• An exhibit hall containing focal surfaces.
• An object imported from Mathematica into CyberMath. The LiveGraphics3D applet is used to display the same object in Netscape. Netscape can be pointed to different webpages by clicking on objects in CyberMath.

• The exhibit hall.
• The cylindrical mirrors exhibit.
• Another view of the exhibit hall.
• The transformations exhibit.

• Quicktime movie gallery

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