Institute for Mathematical Behavioral Sciences

Cognitive scientists have emphasized, in recent years, the importance of our embodiment in the world for skill acquisition and cognitive processing. In mathematical education, this emphasis has led to exploration of the way people use their bodies to learn and to exploit mathematical concepts. Following this line of thinking, we reasoned that immersion in an interactive 4D virtual environment (VE) might be a far better way for most people to learn about 4D space and objects than gazing at diagrams, pictures or rotating hypercubes. We are happy to report positive results with such methods. Greg Seyranian, Philippe Colantoni, Barb Krug and I developed software for presenting VEs with four spatial dimensions in a manner similar to that in which 3D VEs are presented in action games. The software renders a 3D cross-section of a 4D VE and, in real-time, changes the position and orientation of the 3D cross-section in response to user input. In a first experiment, we tested whether people can learn to move efficiently from one location to some remote location in a rich, immersive 4D VE. All participants in this experiment improved their search and navigation skills dramatically. Yet it was clear that they used landmarks to select efficient routes, a navigation technique common in the real world (“turn left at the Shell station”), rather than some high-dimensional map of the environment. In a second experiment, we used non-immersive, maze-like VEs to test whether people can learn to point towards an unseen, remote location in 4D space and to estimate its straight-line distance. Ability to perform such tasks is often taken as evidence, when working in 2D and 3D environments, for use of a more global, map-like representation of space. Results depended on the individual participants. While all improved their performance in what are initially extremely difficult tasks, at the end of training only some were able to point immediately and accurately towards a remote location. Interestingly, certain participants in these experiments, particularly the first, immersive one, reported a strong feeling , when walking about afterwards, that the real world was but a 3D cross-section of a 4D one and that they should have been able to move their bodies to explore the missing dimension.