The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)

There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter. / Mathematics Education / M. Sc. (Mathematics)

[pt] Consideramos duas formas de complexidade em geometria euclidiana plana.Na primeira, problemas são descritos algebricamente, e a complexidade é cotada essencialmente pelo grau de um polinômio. Como consequência, mostramos que vários resultados gerais e familiares em geometria podem ser demonstrados a partir da simples verificação de dois ou três casos particulares. A segunda forma faz uso da descrição sintática dos teoremas, que permite uma quantificação da complexidade em termos lógicos (número de quantificadores e átomos de uma fórmula). Inspirados por esta última abordagem, são descritos alguns procedimentos de demonstração automática. Alguns grupos habituais de operções em geometria são apresentados com a intenção de simplificar as duas abordagens.Através do estudo de técnicas mais avançadas em matemática trazemos novos pontos de vista a assuntos estudados no ensino médio. / [en] Two forms of complexity in Euclidean plane geometry are considered. In the first one, problems are described algebraically, and the complexity level is measured essentially by the degree of a polynomial. As a consequence, many familiar and general results in geometry can be proved by inspecting two or three special cases. The second form uses the syntactic description of a theorem allowing for a quanti.cation of the complexity in logic terms (number of quantifiers and atoms in the formula). Inspired by this approach, some procedures in mechanized proofs are described. We also present some traditional groups of operations in geometry which simplify the two approaches. The study of more advanced techniques in mathematics sheds new light on standard high school topics.

Made available in DSpace on 2016-04-27T16:57:55Z (GMT). No. of bitstreams: 1 dissertacao_eliane_cabariti.pdf: 4199946 bytes, checksum: 7b4a1cc8c562d90ec0b98ff672a71a8b (MD5) Previous issue date: 2004-09-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The main aim of this work is to contribute to the process of teaching and learning of geometry, in particular the non-Euclidean geometries, seeking to support the implementation of proposals associated with the introduction of a hyperbolic model, with the help of a computational tool, in mathematics teacher education courses. To this end, we conducted an experimental study to investigate the possible relations that teacher educators of Euclidean geometry establish when asked to solve situations involving notions of hyperbolic geometry, using the software Cabri-géomètre. The activities developed for the experimental study were inspired by the principals for the development of thought-revealing tasks, described by Lesh et al. (2000). Our analyses were based on two aspects: the dynamics behind movements between the geometrical domains Euclidean geometry and hyperbolic geometry as well as interactions between the spatio-graphical and theoretical fields (Laborde, 1999) and the role of Cabri as a tool for construction, exploration and validation, especially with respect to its dynamic aspects and the different possible drag modes (Olivero, 2002). Through our analysis of teachers' interactions with these situations, we confirmed the importance of the use of the hyperbolic menu of Cabri, fundamental for access to representations of hyperbolic objects favouring the understanding of concepts, properties and relations involved in this domain. The results of this study enabled us to reconsider some choices, leading to the re-design of the activities included in our initial proposal, in particular with reference to the makeup and use of the tools available in Cabri-géomètre. As a consequence, we were able to present a new pedagogic proposal consistent with the original aims / Este trabalho tem como objetivo principal contribuir para o processo de ensino e aprendizagem de Geometria, em particular das Geometrias não Euclidianas, procurando subsidiar a implementação de propostas que visam a introdução de um modelo hiperbólico, com o auxílio de uma ferramenta computacional, em cursos de formação de professores de Matemática. Para nos auxiliar no delineamento dessa proposta, realizamos um estudo experimental que teve como intuito investigar as possíveis relações que professores-formadores de Geometria Euclidiana, estabelecem quando solicitados a resolver situações envolvendo noções de Geometria Hiperbólica, com o auxílio do software Cabri-géomètre. As atividades desenvolvidas para o estudo experimental foram inspiradas nos princípios para o desenvolvimento de tarefas thought revealing descritos por Lesh et al. (2000). Nossas análises foram baseadas em dois aspectos: a dinâmica das trocas entre os domínios geométricos geometria Euclidiana e Hiperbólica além das interações entre os campos espaço-gráfico e teórico (Laborde, 1999) e o papel do Cabri como ferramenta de construção, exploração e verificação, especialmente relacionadas ao seu aspecto dinâmico, nos diferentes modos de arrastar (Olivero, 2002). Por meio das interações dos professores nessas situações, confirmamos a importância do uso da barra do menu hiperbólico do Cabri, fundamental para o acesso às representações de objetos hiperbólicos favorecendo a compreensão de conceitos, propriedades e relações envolvidos nesse domínio. Os resultados desse estudo permitiram-nos reconsiderar algumas escolhas, levando-nos à reelaboração das atividades de nossa proposta inicial, em particular no que se refere à constituição e utilização das ferramentas disponibilizadas no Cabri-géomètre. Consolidamos assim, uma nova proposta pedagógica com os mesmos objetivos iniciais

Tento projekt se věnuje problematice zobrazení velkých hierarchických struktur, zejména možnostem vizualizace stromových grafů. Cílem je implementace hyperbolického prohlížeče ve webovém prostředí, který využívá potenciálu neeukleidovské geometrie k promítnutí stromu na hyperbolickou rovinu. Velký důraz je kladen na uživatelsky přívětivou manipulaci se zobrazovaným modelem a snadnou orientaci.

Euclidean Geometry is a challenging topic for most of the learners in the secondary schools. A qualitative case study explores the reasoning skills of ninth graders in the proving of congruent triangles in their natural environment. A class of thirty-two learners was conveniently selected to participate in the classroom observations. Two groups of six learners each were purposefully selected from the same class of thirty-two learners to participate in focus group interviews. The teaching documents were analysed. The Van Hiele’s levels of geometric thinking were used to reflect on the reasoning skills of the learners. The findings show that the majority of the learners operated at level 2 of Van Hiele’s geometric thinking. The use of visual aids in the teaching of geometry is important. About 30% of the learners were still operating at level 1 of Van Hiele theory. The analysed books showed that investigation help learners to discover the intended knowledge on their own. Learners need quality experience in order to move from a lower to a higher level of Van Hiele’s geometry thinking levels. The study brings about unique findings which may not be generalised. The results can only provide an insight into the reasoning skills of ninth graders in proving of congruent triangles. I recommend that future researchers should focus on proving of congruent triangles with a bigger sample of learners from different environmental settings. / Mathematics Education / M. Ed. (Mathematics Education)

This study examined “problem solving skills in circle geometry concepts in Euclidean Geometry. This study was necessitated by learners’ inability to perform well with regards to Euclidean Geometry in general and Circle Geometry in particular. The use of naturalistic observation case study research (NOCSR) study was employed as the research design for the study. The intervention used for the study was the teaching of circle geometry with Polya problem solving instructional approach coupled with social constructivist instructional approach. A High School in the Northern Cape Province was used for the study. 61 mathematics learners (grade 11) in the school served as participants for the first year of the study, while 45 mathematics learners, also in grade 11, served as participants for the second year of the study. Data was collected for two consecutive years: 2018 and 2019. All learners who served as participants for the study did so willingly without been coerced in any way. Parental consent of all participants were also obtained. The following data were collected for each year of the research intervention: classroom teaching proceedings’ video recordings, photograph of learners class exercises (CE), field notes and the end-of-the- Intervention Test (EIT). Direct interpretations, categorical aggregation and a problem solving rubric were used for the analysis of data. Performance analysis and solution appraisal were also used to analyse some of the collected data. It emerged from the study that the research intervention evoked learners’ desire and interest to learn circle geometry. Also, the research intervention improved the study participants’ performance and problem solving skills in circle geometry concepts. Hence, it is recommended from this study that there is the need for South African schools to adopt the instructional approach for the intervention: Polya problem solving instructional approach coupled with social constructivist instructional approach, for the teaching and learning of Euclidean geometry concepts. / Mathematics Education / M. Sc. (Mathematics Education)

Researchers conceptually recommend inquiry-based learning as a necessary means to alleviate the problems of learning but this study has embarked on practical implementation of inquiry-based facilitation and learning in Euclidean Geometry. Inquiry-based learning is student-centred. Therefore, the teaching or monitoring of inquiry-based learning in this study is referred to as inquiry-based facilitation. The null hypothesis discarded in this study explains that there is no difference between inquiry-based facilitation and traditional axiomatic approach in teaching Euclidean Geometry, that is, H0: μinquiry-based facilitation = μtraditional axiomatic approach. This study emphasises a pragmatist view that constructivism is fundamental to realism, that is, inductive inquiry supplements deductive inquiry in teaching and learning. Participants in this study comprise schools in Tshwane North district that served as experimental group and Tshwane West district schools classified as comparison group. The two districts are in the Gauteng Province of South Africa. The total number of students who participated is 166, that is, 97 students in the experimental group and 69 students in the comparison group. Convenient sampling applied and three experimental and three comparison group schools were sampled. Embedded mixed-method methodology was employed. Quantitative and qualitative methodologies are integrated in collecting data; analysis and interpretation of data. Inquiry-based-facilitation occurred in experimental group when the facilitator probed asking students to research, weigh evidence, explore, share discoveries, allow students to display authentic knowledge and skills and guiding students to apply knowledge and skills to solve problems for the classroom and for the world out of the classroom. In response to inquiry-based facilitation, students engaged in cooperative learning, exploration, self-centred and self-regulated learning in order to acquire knowledge and skills. In the comparison group, teaching progressed as usual. Quantitative data revealed that on average, participant that received intervention through inquiry-based facilitation acquired inquiry-based learning skills and improved (M= -7.773, SE= 0.7146) than those who did not receive intervention (M= -0.221, SE = 0.4429). This difference (-7.547), 95% CI (-8.08, 5.69), was significant at t (10.88), p = 0.0001, p<0.05 and represented a large effect size of 0.55. The large effect size emphasises that inquiry-based facilitation contributed significantly towards improvement in inquiry-based learning and that the framework contributed by this study can be considered as a framework of inquiry-based facilitation in Euclidean Geometry. This study has shown that the traditional axiomatic approach promotes rote learning; passive, deductive and algorithmic learning that obstructs application of knowledge in problem-solving. Therefore, this study asserts that the application of Inquiry-based facilitation to implement inquiry-based learning promotes deeper, authentic, non-algorithmic, self-regulated learning that enhances problem-solving skills in Euclidean Geometry. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)