ABSTRACT
This chapter considers the students' thinking and learning to prior considerations of history and epistemology. De Villiers suggests that teaching approaches for developing students' understanding be constructive, and that a progression can be considered from visual to economical. The technological functionalities of dynamic geometry software (DGS) can provide an additional set of operators for students to use as they solve problems. DGS provides rich learning opportunities related to the conception of figure as construction of small objects in the microspace. From the point of view of conceptions of geometric figure, the use of DGS with 3D shapes is an interesting area for research on students' thinking and learning. Learning trajectories have three parts: a mathematical goal, a developmental path, and a set of instructional activities. In elementary schools, children's learning is likely to be based on their practical, hands-on experience such as navigating, drawing, constructing, comparing or describing geometrical objects in macro-, meso- and micro-space.
