VRMath: A 3D Microworld for Learning 3D Geometry
Andy Yeh and Rod Nason
Centre for Mathematics, Science and Technology Education
Queensland University of Technology
Brisbane, Queensland, Australia
Abstract: This paper reports on the design of a prototype VRLE (Virtual Reality Learning
Environment) named VRMath. VRMath is an online application that utilis VR (Virtual Reality)
technology combined with the power of a Logo-like programming language and hypermedia and
the Internet to facilitate the learning of 3-Dimensional (3D) geometry concepts and process.
VRMath is being designed within the framework of a design-experiment (The Design-Bad
Rearch Collective, 2003) during which VRMath will evolve through a ries of iteration cycles of
design enactment reflection redesign into an educational tool that will provide
mathematics educators with new and more powerful ways of facilitating the construction of 3D
geometry knowledge. During the iteration cycles, in addition to many new ways about thinking
and doing mathematics being identified, implications to inform the future design of this and other
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VRLEs also will emerge.
Background
There are many powerful ICT (Information Communication Technology) tools such as Logo (Clements & Sarama, 1997; Papert, 1993), Cabri-Géomètre (Laborde, 2000) and Geometer’s Sketchpad (Jackiw, 1995) that have been designed to facilitate the learning of geometry, yet most of them operate within 2D environments. Therefore, it can be argued that the number and types of investigations of 3D shape, position and orientation concepts and process that can be investigated in the environments are rather limited. This is in marked contrast to the number and types of 3D g
eometry investigations that can be carried out in Virtual Reality (VR) environments. Becau of the enhanced capabilities of VR environments for 3D exploration (Hedberg & Alexander, 1994; Winn & Jackson, 1999), it has been suggested that VR learning environments (VRLE) are worthy of investigation as tools for facilitating the teaching and learning of 3D geometry knowledge (Kwon, Kim, & Kim, 2002). Also, the currently prevailing hypermedia and Internet have much potential in education (Dede & Palumbo, 1991). Therefore, it is now appropriate to integrate the powerful computational tools to form a better learning environment or microworld empowered by VR technology for the learning of 3D geometry.
Papert (1996) commented that 99% of the software for “learning math” in the market was teaching “junk math” becau its idea about what mathematics is, and why the kids should learn it are “flimsy”. Therefore, a technology-rich learning environment needs to be well designed. Further, a good design is informed by theory (Wilson, 1999). In arching for a sound theoretical framework, the rearchers conducted an extensive literature review and identified three fields of literature that have implications for the design of a virtual reality learning environment (VRLE) for the learning of 3D geometry:
1.The fallibilist philosophy about the nature of mathematics (Ernest, 1994) which argues that mathe
matical
knowledge is fallible, subject to revision, and a construction of human kind. Therefore, the teaching of
mathematics should focus on open-ended investigations of mathematics.
2.The constructivist/constructionist perspective about learning (Kafai & Resnick, 1996) which argues that:
(a) knowledge is personally and socially constructed, (b) learning is learner centred, situated, authentic,亲子课教案
and is achieved by designing and making personally meaningful artefacts, and (c) multiple perspectives
and reprentations of knowledge should be encouraged during learning.
3.The miotic perspective about mathematics as a meaning-making endeavour (Cunningham, 1992; Lemke,
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2001) that argues that: (a) any single sign (e.g., icon, index, or symbol) is an incomplete reprentation of
the object (e.g., abstract concept or knowledge), thus multiple miotic resources including topological,
typological, and social-actional resources must be employed for meaning-making, and (b) mathematical meanings must be constantly challenged.
The key notions from the three fields of literature (that actually resonate cloly with each other) were integrated into a theoretical framework to inform the design of the VRLE. Informed by this framework, the rearchers developed a VRLE named VRMath, which:
1. is an open and generative learning environment that enables learning by making and designing personally meaningful artefacts;
2. employs rich miotic resources that enable multiple perspectives and reprentations for mathematical meaning-making; and
3. features many facilities such as Logo-like programming language and online discussion forum that
enable the expression of mathematical ideas and collaborative learning in online community.
The Three Components
VRMath consists of three components namely topological, typological, and social-actional components (Fig.
1) which will be discusd in turn.
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Figure 1: Three Components of VRMath
glut4The topological component : The main part of this component is the virtual reality interface that provides rich reprentations in colours, textures, geometric objects and behaviours, and allows for real-time navigation within the 3D virtual space. This component refers to any meaning by degree or continuous reprentations of geometry such as visual graphics, images and 3D shapes and space.
The typological component : The main part of this component is the Logo-like programming interface that engages students in logical procedural thinking and the tutee mode of computer using (Taylor, 1980), and links to topological component while the students are programming to manipulate and build within the 3D virtual reality microworld. This component refers to any meaning by kind or discrete reprentations of geometry such as language, texts, numbers, icons and buttons.
The social-actional component : The main part of this component is the hypermedia and forum interface that aggregates information and scaffolds discour. Students thus can contribute ideas, arch for information and ask for help from more knowledgeable peers. This component refers to many facilities such as online discour that stimulate thinking, and the provision and sharing of information.
VR Interface
The VR interface is the incarnation of the topological component. The initial state of this interface contains a sunny background with a 3D turtle placed in the centre of the VR space. VR is 3D in nature, but is more than just 3D
real time computer graphics. The VR interface allows urs to navigate within the 3D virtual space in
a manner similar爱如什么
闺蜜蜜语to our real life. Urs can walk as a person, fly as a jet plane, or examine by rotating the whole virtual world within VR space by using the mou and the keyboard. Urs can also build geometrical objects and manipulate tho objects within VR space by using the mou and the keyboard. Thus, urs can perceive the dynamic nature of the world from many possible viewpoints in two ways: navigation and manipulation of objects within 3D virtual worlds. This relates to the two main categories of human spatial ability identified by McGee (1979): spatial visualisation which is the ability to mentally rotate, manipulate, and twist two- and three-dimensional stimulus objects, and spatial orientation which is the comprehension of the arrangement of elements within a visual stimulus pattern; the aptitude for remaining unconfud by the changing orientations in which a figure may be prented; and the ability to determine spatial relation with respect to one’s body. Therefore, this VR interface has the potential to foster the development of both types of spatial abilities in urs.
However, the u of 2D input devices such as mou and keyboard to directly manipulate within a 3D space has a fundamental problem of dimensionality mismatch (Leach, Al-Qaimari, Grieve, Jinks, & Makay, 1997), which may cau difficulties in 3D navigation and confusion about directionality with
in VR space. The rearchers thus developed some navigation aids to ea and overcome this problem. The navigation aids are integrated into a VR toolbar placed above the VR space. Currently designed navigation aids include:
fly, and examine are the three modes of navigating in VR space: The three modes of navigation were defined in the VRML specification (Web3D Consortium, 2003). Walk and fly modes
reflect the navigation in real world. The examine mode, which is not possible to perform in the real world
when the target scene is large (e.g., a building), is advantageous in virtual reality becau it enables urs
to examine 3D objects from many different viewpoints. The icons enable urs to switch the navigation
modes to whichever is the best one to get to the viewpoint they wish.
Rotation centre ts the turtle’s current position as the rotation centre for Examine mode: Examine mode of navigation has a centre point to rotate the whole scene. It is envisaged that urs can examine the
virtual world easier with this function.
Horizontally straightens the view to be parallel to the ground: In fly or examine mode of navigation, urs can easily lo the horizon. This function can help restore the horizon.
Restore viewpoint returns to the initial view: Becau of the easy disorientation in VR space, this function enables the urs to restore the initial viewpoint. Thus, urs can always restart the navigation
whenever they feel lost in VR space.
Full screen enlarges the VR space to full screen: The 3D VR space occupies only about a third of the 2D screen. This function can enlarge the 3D space to full screen and thus it enables urs to have a better
examination and exploration of the virtual space.
Head light switches the light on and off: The light in VR space is an environmental cue that influences the colouring model and rendering of the geometrical objects within the virtual space. This function
enables urs to experience the 3D stereo visualisation when the headlight is on and 2D picture effect
when the headlight is off.
Collide switches the collision with objects facility on and off: This function is ud in conjunction with walk mode of navigation. In walk mode, there is collision detection of geometrical objects. For
example, urs will be stopped when navigating onto a wall (face). This function makes it possible to
walk through the wall.
Compass switches the compass on and off to show the absolute direction (e.g., east, west, north, and south): The compass can provide a constant message to remind urs about direction. This also helps西游记笔记
urs from getting disoriented.
Wire frame and solid turns 3D objects into wire frames or solid shapes: The wire frame mode makes solid shapes transparent. Thus urs can have both inside and outside perspectives of solid geometrical objects.
Grid, chessboard, and sand ground toggle the grid line, chessboard, and ground on and off: The grid line is 40 by 40 square meters with 1 meter interval, which provides a n of distance in the 3D space. Urs can also get a reference point from grid, chessboard, and sand ground to improve their 3D n in VR space.
Show or hide the turtle in 3D space: This function can temporarily turn the turtle invisible for better obrvation of the construction of geometrical objects. This function can also be done by giving command ST (SHOWTURTLE) and HT (HIDETURTLE).
LABEL objects; and the Quick Command facilitates the u of basic commands by mou clicking that is an alternative to using commands by typing keyboard.
Figure 3: GUI Tools in VRMath
The programming GUI also integrates many functions such as project management, in which urs can create, edit, save, and delete projects. Urs can also make their projects public so other urs can share their projects. Projects are stored in an online databa, which increas the accessibility for projects from many computers over the Internet.
Hypermedia and Forum Interface
The hypermedia and forum interface is the incarnation of the social-actional component. This interface acts as the information centre that enables information arching, contributing, sharing, and linking to world-wide resources. There are two main elements in this interface: the hypermedia documents and online discussion forum named VRMath Forum (Fig. 4).
Figure 4: VRMath Forum
The hypermedia documents contain many informative web pages that include multimedia and hyperlinks. The informative web pages such as introduction, tutorial, command library, 3D navigation, and help index pages etc. provide rich information for lf-paced learning within the VRMath system and the construction of 3D microworlds. The online discussion forum provides opportunities for members within forum community to interact and collaborate.
