The aim of this study was to investigate the relationship between teachers' instructional practices in terms of specific areas of focus pertaining to the teaching and learning of geometry described in literature and, their learners' levels of geometry thinking as elaborated in the Van Hiele theory. A review of literature on the development of geometry understanding was conducted to frame what is meant by 'teachers' instructional practices' as they pertain to the teaching and learning of geometry in this study. These instructional practices are understood to include the appropriate allocation of time for the facilitation of geometry concept development, the use of concrete apparatus, the use of relevant and level appropriate language as well as the use of level appropriate geometry activities. The structure of the curriculum in terms of its content and opportunity for conceptual progression was also considered. Literature reveals continuing discourse regarding the levels of thinking described in the Van Hiele theory, and even though there is no consensus regarding the nature of the levels and that assessing learners' levels of thinking remains difficult and inconclusive, it is generally accepted that the Van Hiele test is a reliable measure in assessing learners' levels of geometry thinking. An exploratory case study design was chosen for this study. The phenomenon being explored is the teaching and learning of geometry in the Foundation and Intermediate Phases of a particular private school. In order to do this, the teachers' timetables and Work Schedules were analysed to determine how much time was allocated to the instruction of Mathematics in general and for the instruction of geometry in particular. These documents also yielded data regarding the type of geometry experiences included in the implemented curriculum. The learners' level of geometry understanding according to the Van Hiele theory was assessed using an instrument designed by Usiskin (1982). This assessment was facilitated by the researcher in the learners' home class and happened in June after six months of instruction in a particular grade level. Data regarding the teachers' perception of geometry and the best method to facilitate the learning of geometry was gathered through a teacher's questionnaire. The teachers were requested to facilitate geometry lessons, which were digitally recorded by the researcher. Each grade level (0-5) was regarded as a sub-unit and analysed as the case for that grade level. The data was then assimilated to present the case of geometry teaching and learning in the Foundation and Intermediate Phases in the school. The findings report that when juxtaposed alongside research, geometry instructional practices in this school, compare favourably with regards to the teachers' professed and observed practice of using concrete aids and tasks that engage the learners actively in developing geometry insight. There is also evidence that these instructional practices support progression through the levels however the shortfall of time allocated to facilitating this progression and the lack of conclusive data regarding the language used and the types of experiences may justify further research into whether this progression is satisfactory. Copyright
Duvenhage, Rocco de Villiers(Institute of Physics, 2014-01)
We study noncommutative Ricci flow in a finite-dimensional representation of
a noncommutative torus. It is shown that the flow exists and converges to the
flat metric. We also consider the evolution of entropy and a ...
Duvenhage, Rocco de Villiers; Van Staden, Wernd; Wuzyk, Jan(Elsevier, 2018-02)
We study a first variation formula for the eigenvalues of the Laplacian evolving under the Ricci flow in a simple example of a noncommutative matrix geometry, namely a finite dimensional representation of a noncommutative ...