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The purpose of this paper is to investigate some of the properties of convex sets in the plane through synthetic geometry.

Call number: LD2668 .T4 1930 E85

The unsatisfactory performance of learners m geometry specifically and mathematics in general, as revealed in SACMEQ and TIMMS reports, is an indication of the difficulties experienced by learner~ in dealing with geometry. In response to many years of learners' challenges with geometry, the South African curriculum must advocate reforms in teaching strategies applied by educators to teach geometry. The purpose of this study is to understand the challenges that educators face in the teaching of geometry in order to explore and recommend strategies of teaching geometry that will assist educators to improve learners' understanding of geometry in the senior phase. Grade six and nine mathematics results indicate that learners' performance in geometry is of poor standard. This study was conducted in Moses Kotane West Area Office of the Not1h West Province of South Africa. A total number of 35 mathematics educators in 27 middle schools participated in the study by filling in questionnaires. Data were also collected through observation and interview. The constructivism theory provided a theoretical framework for the study. Data collected were analysed by using the Statistical Package for Social Sciences 17.0 (SPSS 17.0) programme and conclusions were drawn. The study revealed that educators still used trad itional methods of teaching in the era of Outcomes - Based Education (OBE) when teaching geometry to learners. Amongst others, teachers do not use the OBE principles such as learner-centred approach, teacher facilitation and integration of knowledge in assisting learners to understand geometry. For most of the participating schools, educators sti ll follow teaching by telling rather than engaging learners in their teach in g. Reasons advanced by teachers in sticking to using methods used before the introduction of OBE include lack of pedagogical content knowledge and learner-centred instructional methods. The teachers attribute the use of the traditional teaching methods to lack of proper training and also resistance to change. The study further revealed that the majority of educators' workshops are concentrated on algebra content knowledge and less on strategies that educators should use to teach geometry. Tn view of the findings in this study, some interactive teaching methods such as problem solving, visualisation and co-operative learning are recommended and could be used for further studies in order to achieve better application of teaching strategies for geometry in our schools. / Thesis (M. Ed) North-West University, Mafikeng Campus, 2011

In this thesis, we study the open book decompositions in high dimensional contact manifolds. We focus on the results about open book decomposition of manifolds and their relationship with contact geometry.

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Given a subgroup B of a finitely-generated abelian group A, the saturation B of B is defined to be the largest subgroup of A containing B with finite index. In this thesis we consider a crucial step in the determination of the Mordell-Weil group of an elliptic curve, E(K). Methods such as Descent may produce subgroups H of E(K) with [H:H] > 1. We have determined an algorithm for calculating H given H, and hence for completing the process of finding the Mordell-Weil group. Our method has been implemented in MAGMA with two versions of the programs; one for general number fields K and the other for Q. It builds upon previous work by S. Siksek. Our problem splits into two. First we can use geometry of numbers arguments to establish an upper bound N for the index [H:H]. Second for each remaining prime p < N we seek to prove either that H is p-saturated, i.e. p|[H:H], or to enlarge H by index p. To solve the first problem, 1. We have devised and implemented an algorithm that searches for points on E(K) up to a specified naive height bound. 2. We have devised and implemented an algorithm that calculates the subgroup Egr(K) of points with good reduction at specified valuations. 3. We have implemented joint work with S. Siksek and J. Cremona to calculate an upper bound on the difference of the canonical and naive height of points on an elliptic curve. 4. We have helped to devise and have implemented joint work with S. Siksek and J. Cremona to calculate a lower bound on the canonical heights of non-torsion points on E(K) with K a totally real field. To solve the second problem, 1. As in earlier work by Siksek, we use homomorphisms to prove p-saturation for primes p. We however use the Tate-Lichtenbaum pairing, and we show that, using this pairing, our method will always prove H is p-saturated if that is the case. 2. We show that Siksek's original method will fail for some curves.