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Communications of the ACM

Virtual extension

Constructive Function-Based Modeling in Multilevel Education

It is a digital age, especially for children and students who can be called the world's first truly digital generation. Accordingly a new generation education technology with a particular emphasis on visual thinking and specific computer-based notions and means is emerging. This is a new challenge for computer graphics which is a wide discipline dealing with creating visual images and devising their underlying models.

There have been two major paradigms in computer graphics, and shape modeling as its part, for a certain period of time: namely, approximation and discretization. Their purpose is to simplify ideal complex shapes to make it possible to deal with them using limited capabilities of hardware and software. The approximation paradigm includes 2D vector graphics, 3D polygonal meshes, and later approximations by free-form curves and surfaces. The discretization paradigm originated raster graphics, then volume graphics based on 3D grid samples, and recently point-based graphics employing clouds of scanned or otherwise generated surface points. The problems of the both paradigms are obvious: loss of precise shape and visual property definitions, growing memory consumption, limited complexity, and others. Surface and volumetric meshes, lying in the foundation of modern industrial computer graphics systems, are so cumbersome that it is difficult to create, handle, and even understand them.

The need in compact precise models with unlimited complexity has lead to the newly emerging paradigm of procedural modeling and rendering. One of the possibilities to represent an object procedurally is to evaluate a real function representing the shape and other real functions representing object properties at the given point. Our research group proposed in a constructive approach to creation of such function evaluation procedures for geometric shapes and in extended the approach to the case of point attribute functions representing object properties. The main idea of this approach is the creation of complex models from simple ones using operations similar to a model assembly from elementary pieces in LEGO.

In terms of educational technology, such an approach is very much in the spirit of a constructionism theory by Seimur Papert. The main principle of this theory is active learning when learners gain knowledge actively constructing artifacts external to themselves. Applications of this theory coupled with modern computer technologies are emerging although a relationship with educational practice is not always easy. It is known that constructive thinking lying in the heart of LEGO games enable children to learn notions that were previously considered as too complex for them. There was research at the MIT Media Laboratory that led to the LEGO MindStorms robotics kits allowing children build their own robots using "programmable bricks" with electronics embedded inside. We have been developing not physical but virtual modeling and graphics tools that make it possible to use an extensible suite of "bricks" (see illustration in Figure 1) with a possibility to deform and modify them on the fly.

Such an approach assumes mastering the basic mathematical concepts, initial programming in a simple language with subsequent creating an underlying model, generating its images and finally fabricating a real object of that model. We believe it is of interest as an educational technology for not only children and students but also for researchers, artists, and designers. It is important that learners interacting with a created virtual world acquire knowledge not just about mathematics and programming but also about structures and processes of the real world.

We found soon after the introduction of our approach to modeling in the mid-90s that none of existing modeling systems or languages support this paradigm. Another necessity was to start preparation of qualified students to be involved in the R&

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