Experiencing 3d Simulated Space Through Different Perspectives

The contribution of technology to the teaching and learning of geometry is perceived to be strongly linked with interactivity, multiple interlinked representations, including symbolic ones, dynamic manipulations, and dynamic visualizations (Laborde et al. 2006). However, relatively little research has been carried out on the way the above distinct characteristics of digital media can be exploited so as to engage students in meaningful 3d geometry investigations. Aiming to understand the way in which students’ intuitions and ideas concerning spatial visualization and thinking (Presmeg 2006; Arcavi 2003) are challenged in 3d digital media, we developed a set of microworlds and a set of activities adopting a constructionist theoretical perspective (Kafai and Resnick 1996). A distinct feature of the microworlds was that they were “halfbaked” (Kynigos 2007), i.e., incomplete or buggy digital artifacts that students had to investigate how they work and to change and fix them. Our pedagogical aim was to engage the students in navigating a moving entity, the turtle, to construct graphical digital objects through Logo programming and the dynamic manipulation of procedure variable values in a 3d simulated space. Research seems to conclude that carefully designed Logo-based microworlds are an effective medium in offering rich mathematical experiences and encouraging the construction of meaning in 2d through the turtle metaphor (Clements and Sarama 1997; Kynigos 1992). Navigating the turtle requires the formation of essentially novel methods of spatial orientation, where the reference point is not the position of the user’s body but the turtle’s body, relative to which the entire system of orientation may change. In this framework, body-syntonicity is a critical concept in 2d Turtle Geometry (Papert 1980) that refers: (a) to navigating the turtle by coordinating one’s body-posture, Experiencing 3d Simulated Space Through Different Perspectives