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The teaching and learning of Euclidean geometry in black South African schools is characterised by teachers' poor knowledge of Euclidean geometry and the use of traditional taaching methods that encourage rote learning. In the light of this, the study investigated the extent to which learners perform in Process-Based Instruction. Process-Based Instruction is charactarised by the gradual transfer of instruction from the teacher to the learner. In Process-Based Instruction learners are expected to communicate thought,form concepts and master theorems by drawing, 'showed with and used hands', talking and writing. Learners' performance is assessed by rubrics and is analysed using graphs. The findings of the empirical investigation revealed that learners of both grades 8 and 9 cannot communicate thought by any media (talking, writing, 'showed with and used hands' and drawing). A few learners formed a few concepts and mastered theorems. On the basis of the findings and the shortcomings of the study recommendations are made. / Educational studies / M. Ed. (Didactics)

Text in English, abstract in English and Afrikaans / The teaching and learning of Euclidean geometry in black South African schools is characterised by teachers' poor knowledge of Euclidean geometry and the use of traditional teaching methods that encourage rote learning. In the light of this, the study investigated the extent to which learners perform in Process-Based Instruction. Process-Based Instruction is characterised by the gradual transfer of instruction from the teacher to the learner. In Process-Based Instruction learners are expected to communicate thought, form concepts and master theorems by drawing, 'showed with and used hands', talking and writing. Learners' performance is assessed by rubrics and is analysed using graphs. The findings of the empirical investigation revealed that learners of both grades 8 and 9 cannot communicate thought by any media (talking, writing 'showed with and used hands' and drawing). A few learners formed a few concepts and mastered theorems. On the basis of the findings and the shortcomings of the study recommendations are made. / Die onderrig van Euclediaanse Meetkunde in swart Suid-Afrikaanse skole word gekenmerk aan onderwysers se swak kennis van die vak, asook die gebruik van onderrigmetodes wat op die leerder se geheue staatmaak. Hierdie studie ondersoek die sukses van leerders se prestasie in proses-gebasseerde ondenig. Proses-gebaseerde onderig word gekenmerk aan die geleidelike oordra van kennis, van die onderwyser na die leerder. Daar word van leerders verwag om gedagtes oor te dra, konsepte (begrippe) te verstaan en stellings onder die knie te kry deur middel van sketse, die gebruik van gebare, verbale verduideliking en skriftelike werk. Leerders se prestasie word geevalueer deur die gebruik van klassifisering in kolomme (rubrics) en word ge-analiseer deur gralieke. Die bevinding van die empieriese ondersoek het getoon dat leerders van sowel graad 8 as graad 9 nie gedagtes kan oordra deur enige van die gemelde metodes nie. 'n Paar leerders het wel enkele begrippe verstaan en kon stellings bemeester. Na aanleiding van die bevindings van die studie en die tekortkominge wat daardeur opgespoor is, word sekere voorstelle gemaak. / Didactics / M. Ed. (Didactics)

Orientador: Claudina Izepe Rodrigues / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T10:14:12Z (GMT). No. of bitstreams: 1 Oliveira_VivianneTassoPeruginide_M.pdf: 42677277 bytes, checksum: e029738b1504da7dbb6995d59c3b35f5 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho apresentamos o estudo sobre a Geometria do Táxi, uma Geometria não-Euclidiana de fácil compreensão e muito próxima do cotidiano das pessoas, uma vez que tem uma ampla gama de aplicações em situações relacionadas à geografia urbana. A Geometria do Táxi é uma geometria muito semelhante à Geometria Euclidiana, diferindo desta apenas pela definição de distância. Enquanto que, na Geometria Euclidiana, a distância entre dois pontos é o comprimento do segmento de reta que os une, podendo ser obtida com o auxílio do Teorema de Pitágoras, na Geometria do Táxi, a distância entre dois pontos é o comprimento do menor caminho percorrido por linhas horizontais e verticais de um ponto a outro. Esse pequeno detalhe sob o ponto de vista matemático, apresenta grandes diferenças, principalmente nas figuras geométricas que estão relacionadas à distância. Abordamos esse aspecto sob a forma de exemplos e apresentamos no final do trabalho uma sugestão de atividades pedagógicas para serem trabalhadas em sala de aula / Abstract: In this paper we present the study of the Taxicab Geometry, a non-Euclidean Geometry of easy understanding and very close to people's daily lives, as it has a wide range of applications in situations related to urban geography. The Taxicab Geometry is a geometry very similar to Euclidian Geometry, differing only by the definition of distance. While in Euclidean Geometry the distance between two points is the length of the line that unites them, which can be obtained with the help of the Pythagorean Theorem, in the Taxicab Geometry the distance between two points is the length of the shortest path travelled by horizontal and vertical lines from one point to another. This small detail, from the mathematical point of view, presents major differences, particularly in the geometric figures that are related to distance. We cover this aspect in the form of examples and present in the end of the work a suggestion of pedagogical activities to be used in class / Mestrado / Matemática em Rede Nacional / Mestra em Matemática em Rede Nacional

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A Thurston map whose postcritical set consists of exactly four points and for which the local degree at each of its critical points is 2 is called textit{nearly Euclidean}. These maps were specified to parse Thurston's combinatorial characterization of rational functions. We determine an extension of the half-space theorem which provides an open hyperbolic half-space such that the negative reciprocal of any fixed slope value is excluded from the boundary of the half-space. / Master of Science / Thurston proved necessary and sufficient conditions under which a certain class of mappings defined topologically are equivalent, in a precise sense which can be considered less strict than topological conjugacy, to a rational map. The conditions presented in the proof of this theorem are not ones for which computational algorithms are easily admitted in all settings. Nearly Euclidean Thurston maps are a sub-class of the maps to which this theorem is applicable and for which an abundance of information is algorithmically attainable. We extend a theorem in this setting. One main example which speaks to the utility of this extension is in determining when certain rational maps arise as matings of polynomials.

In the usual definition of a Euclidean domain, a ring has a norm function whose codomain is the positive integers. It was noticed by Motzkin in 1949 that the codomain could be replaced by any well-ordered set. This motivated the study of transfinite Euclidean domains in which the codomain of the norm function is replaced by the class of ordinals. We prove that there exists a (transfinitely valued) Euclidean Domain with Euclidean order type for every indecomposable ordinal. Modifying the construction, we prove that there exists a Euclidean Domain with no multiplicative norm. Following a definition of Clark and Murty, we define a set of admissible primes. We develop an algorithm that can be used to find sets of admissible primes in the ring of integers of quadratic extensions of the rationals and provide some examples.

This dissertation deals with aspects of axiomatization, intuition and visualization in thehistory of mathematics. Particular focus is put on the end of the 19th century, before DavidHilbert's (1862–1943) work on the axiomatization of Euclidean geometry. The thesis consistsof three papers. In the first paper the Swedish mathematician Torsten Brodén (1857–1931)and his work on the foundations of Euclidean geometry from 1890 and 1912, is studied. Athorough analysis of his foundational work is made as well as an investigation into his generalview on science and mathematics. Furthermore, his thoughts on geometry and its nature andwhat consequences his view has for how he proceeds in developing the axiomatic system, isstudied. In the second paper different aspects of visualizations in mathematics areinvestigated. In particular, it is argued that the meaning of a visualization is not revealed bythe visualization and that a visualization can be problematic to a person if this person, due to alimited knowledge or limited experience, has a simplified view of what the picture represents.A historical study considers the discussion on the role of intuition in mathematics whichfollowed in the wake of Karl Weierstrass' (1815–1897) construction of a nowheredifferentiable function in 1872. In the third paper certain aspects of the thinking of the twoscientists Felix Klein (1849–1925) and Heinrich Hertz (1857–1894) are studied. It isinvestigated how Klein and Hertz related to the idea of naïve images and visual thinkingshortly before the development of modern axiomatics. Klein in several of his writingsemphasized his belief that intuition plays an important part in mathematics. Hertz argued thatwe form images in our mind when we experience the world, but these images may containelements that do not exist in nature.

This thesis presents the research work regarding the solution of a localization problem in indoor WLANs by introducing a distributive time division multiplexed localization technique based on the convex semidefinite programming. Convex optimizations have proven to give promising results but have limitations of computational complexity for a larger problem size. In the case of localization problem the size is determined depending on the number of nodes to be localized. Thus a convex localization technique could not be applied to real time tracking of mobile nodes within the WLANs that are already providing computationally intensive real time multimedia services. Here we have developed a distributive technique to circumvent this problem such that we divide a larger network into computationally manageable smaller subnets. The division of a larger network is based on the mobility levels of the nodes. There are two types of nodes in a network; mobile, and stationery. We have placed the mobile nodes into separate subnets which are tagged as mobile whereas the stationary nodes are placed into subnets tagged as stationary. The purpose of this classification of networks into subnets is to achieve a priority-based localization with a higher priority given to mobile subnets. Then the classified subnets are localized by scheduling them in a time division multiplexed way. For this purpose a time-frame is defined consisting of finite number of fixed duration time-slots such that within the slot duration a subnet could be localized. The subnets are scheduled within the frames with a 1:n ratio pattern that is within n number of frames each mobile subnet is localized n times while each stationary subnet consisting of stationary nodes is localized once. By using this priority-based scheduling we have achieved a real time tracking of mobile node positions by using the computationally intensive convex optimization technique. In addition, we present that the resultant distributive technique can be applied to a network having diverse node density that is a network with its nodes varying from very few to large numbers can be localized by increasing frame duration. This results in a scalable technique. In addition to computational complexity, another problem that arises while formulating the distance based localization as a convex optimization problem is the high-rank solution. We have also developed the solution based on virtual nodes to circumvent this problem. Virtual nodes are not real nodes but these are nodes that are only added within the network to achieve low rank realization. Finally, we developed a distributive 3D real-time localization technique that exploited the mobile user behaviour within the multi-storey indoor environments. The estimates of heights by using this technique were found to be coarse. Therefore, it can only be used to identify floors in which a node is located.

This thesis is an investigation of some of the properties of polynomial rings, unique factorization domains, Euclidean domains, and principal ideal domains. The nature of some of the relationships between each of the above systems is also developed in this paper.