In 2010 a course called MathsLab was designed and implemented in a Johannesburg secondary school, aimed at Grade 9 learners, with the objective of using technology to explore and develop mathematical concepts. One module of the course used Geometer’s Sketchpad to explore concepts in Euclidean geometry. This research report investigates whether technology can result in progression in the zone of proximal development as described by Vygotsky. Progression was measured through the use of a pre- and post-test designed to allocate Van Hiele Levels of geometric thought to individual learners. Changes in the Van Hiele Levels could then verify movement through the zone of proximal development. The results of the pre- and post-tests showed a definite change in learners’ Van Hiele Levels, specifically from Van Hiele Level 1 (visualisation) to Van Hiele Level 2 (analysis). This observation is in line with research that places learners of this age predominantly at these levels. Some learners showed progression to Van Hiele Level 3 (ordering) but this was not the norm. The value of using technology in an appropriate and effective manner in mathematics education is clear and is worthy of further research.

This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.