A dissertation submitted to the Faculty of Education in partial fulfilment of the requirements for the degree of Masters of Education in the Department of Mathematics, Science and Technology Education at the University Of Zululand, South Africa, 2015 / This research aims to establish the level of students’ cognitive skills using GeoGebra, and investigates whether GeoGebra as a technological tool helps in improving poor performance in respect of Euclidean geometry or geometry of the circle. Students’ interests, in learning about circle geometry in mathematics, are also being tested. GeoGebra is an innovative, dynamic mathematics software which integrates algebra, geometry and calculus to aid students during the learning process. The specific sample in this research consists of 112 Grade 11 secondary school learners within the UMkhanyakude district, Hlabisa circuit, under the Empembeni and Ezibayeni wards. During this research, GeoGebra and the concept of circle geometry were introduced to students. Afterwards, students had to answer several geometry of the circle questions, entailing key theorems as prescribed by the National Mathematics pacesetter for Grade 11 and Grade 12. As students answered the above questions, they solved problems and conducted discussions among themselves. At the end, students were individually required to answer questionnaires which consisted of 15 closed items relating to views on GeoGebra and its impact on Euclidean geometry and mathematics, as well as three open-ended questions which asked learners about their reflections on the application of GeoGebra. The above methods provided a strong base to explore whether GeoGebra as a tool helps students in the learning process. The results showed that students endorsed the use of GeoGebra as a technological tool in the teaching of Euclidean geometry. Some students even suggested that GeoGebra be used in other mathematical topics. Students overall enjoyed the use of GeoGebra, finding it user-friendly and a highly significant learning motivator.

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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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.

The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.

Made available in DSpace on 2016-05-17T16:51:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-08-24. Added 1 bitstream(s) on 2016-05-17T16:54:14Z : No. of bitstreams: 1 000864552.pdf: 940388 bytes, checksum: 9434dc3881bd56d95761d7780a369193 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho e apresentar um estudo sobre o c alculo de areas de pol gonos atrav es das coordenadas de seus v ertices. Faremos isto utilizando determinantes de ordem 2 e conceitos b asicos de Geometria Euclidiana Plana / The aim of this work is to present a study on areas of polygons through their vertex coordinates. We treat the subject using determinants of order 2 and basic Euclidean Geometry

Orientador: Jamil Viana Pereira / Banca: Thiago de Melo / Banca: Márcio de Jesus Soares / Resumo: Esta dissertação tem por finalidade oferecer um método alternativo para o ensino de Geometria Euclidiana para estudantes dos Ensinos Fundamental e Médio. Este método proporciona: momentos de descontração, melhora da concentração, aprimoramento das funções motoras e da performance dos estudantes, incorporando novos elementos a linguagem formal da Matemática / Abstract: This work aims to offer an alternative method for teaching Euclidean Geometry for Middle and High School students. This method includes: relaxation techniques, enhancement of concentration, improving motor function and academic performance of students, incorporating new elements to the formal language of Mathematics / Mestre

Made available in DSpace on 2015-12-10T14:22:13Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-04-24. Added 1 bitstream(s) on 2015-12-10T14:28:14Z : No. of bitstreams: 1 000853640.pdf: 937744 bytes, checksum: b52102f7a14c810b40c740190c225af1 (MD5) / Esta pesquisa apresenta o ponto de vista e as opiniões que um grupo de professores formadores considera como deve ser abordada a Geometria Euclidiana em um curso de Licenciatura em Matemática para que o professor tenha uma formação geométrica adequada ao exercício da docência na Educação Básica. A coleta de dados se deu por meio de entrevistas semiestruturadas, procurando identificar as considerações dos entrevistados em relação a temas como o nível de rigor com que trabalham a axiomatização e formalização da geometria, das estratégias e da importância do uso da régua e do compasso e, no contexto atual das tecnologias, destacamos suas opiniões acerca da introdução dos softwares de geometria dinâmica na Licenciatura em Matemática, especificamente no caso da geometria. Identificamos, também, qual a literatura de geometria que é por eles utilizada ou considerada adequada na formação inicial do professor de matemática e apontamos suas opiniões sobre quais os motivos do abandono da geometria nas salas de aula da Educação Básica. Os entrevistados apontam que a Geometria Euclidiana deve ser trabalhada de forma axiomática e com formalização rigorosa, de modo que em uma demonstração a figura é um recurso didático, sendo as justificativas decorrentes de resultados e teoremas já demonstrados. Devido à imaturidade do aluno para entender o sistema axiomático formal, os depoentes sugerem que a geometria pode ser trabalhada mais para o final do curso, proporcionando melhor entendimento e ganho em relação ao conteúdo. Régua e compasso são considerados essenciais, e o software de geometria dinâmica, importante como recurso didático que facilita a visualização e movimentação. A bibliografia nacional existente seja complementada com obras estrangeiras, o que nos faz concluir que existe uma carência nesse setor em nosso país. Falta de conhecimento específico e... / This research presents the view of a group of lecturers of undergraduate courses on how Euclidian Geometry should be approached in a Mathematics Degree Program, so that the graduating teacher has knowledge of geometry adequate for work on Middle and High School Education. The data collection method was questionnaire and interview, in which it is tried to identify the interviewee take on: rigor level of how axiomatization and formalization of geometry are presented, teaching strategies and how important the use of a ruler and compass is to the undergraduate formation of teachers. Taking into consideration modern technologies, it was also intended to highlight their opinions on the introduction to dynamic geometry softwares in Undergraduate Mathematics Education, specifically for the study of Geometry. The research sought to identify geometry textbooks that they consider adequate for the instruction of math teachers and pinpoint the reasons why there has been a neglect of geometry in Middle and High School classrooms. The interviewees shows that Euclidian Geometry must be presented in a strict axiomatic and formal way. In one demonstration the figure is an important teaching aid, being the demonstration justified by results and theorems already proven. Because of students lack of ability to comprehend the formal axiomatic system, it is suggested that geometry be studied closer to the end of courses to provide better understanding and knowledge retention. Ruler and Compass are found to be essential. Combined with important use of dynamic geometry sotware, these teachings aids will improve visualization and movement. The existing Brazilian Bibliography must be supplemented by foreign works, concluding that there is a need of reference works in this área in our country. The lack of specific knowledge and teaching tools is pointed as the reason for the absence of geometry in the Middle and High School classroom, problem originated in the ...

Made available in DSpace on 2014-08-13T14:50:59Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-19Bitstream added on 2014-08-13T17:59:48Z : No. of bitstreams: 1 000773293.pdf: 913947 bytes, checksum: fb0d144787f3b41ace2288e4ccd24157 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Esta dissertação tem por finalidade oferecer um método alternativo para o ensino de Geometria Euclidiana para estudantes dos Ensinos Fundamental e Médio. Este método proporciona: momentos de descontração, melhora da concentração, aprimoramento das funções motoras e da performance dos estudantes, incorporando novos elementos a linguagem formal da Matemática / This work aims to offer an alternative method for teaching Euclidean Geometry for Middle and High School students. This method includes: relaxation techniques, enhancement of concentration, improving motor function and academic performance of students, incorporating new elements to the formal language of Mathematics

Made available in DSpace on 2014-11-10T11:09:47Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-06-27Bitstream added on 2014-11-10T11:58:05Z : No. of bitstreams: 1 000790270.pdf: 586851 bytes, checksum: a0198a8e85a2177bac5159890b67523b (MD5) / O objetivo desta pesquisa é analisar o desenvolvimento histórico da obra clássica de geometria, Os Elementos, de Euclides e os fundamentos da geometria proposto por David Hilbert em seu livro Grundlangen der Geometrie (Fundamentos da Geometria), estudando a estrutura axiomática da geometria abordada por cada autor. O rigor dedutivo utilizado por Euclides, apoiado na lógica clássica de Aristóteles, recebeu diversas críticas de matemáticos modernos no que tange a lacunas no seu sistema dedutivo. As diversas incertezas em relação ao sistema axiomático ameaçavam seu desenvolvimento lógico e especificamente, tratando-se da geometria, surgiram muitas discussões sobre a aceitação do quinto postulado de Euclides. Somente no final do século XIX os sistemas axiomáticos alcançavam níveis profundos nos fundamentos da geometria e, na tentativa de completar a axiomática da geometria, Hilbert publica os Grundlangen der Geometrie, abordagem axiomática mais amplamente adotada na geometria euclidiana. Neste contexto, discutimos as diferentes concepções dos sistemas axiomáticos clássicos e modernos, estudando seus significados lógicos e suas relações com os objetos da geometria. Como parte das reflexões finais, o presente trabalho destaca algumas considerações sobre o conceito de movimento em geometria e uma possível abordagem axiomática da mesma / The objective of this research is to analyze the historical development of the classical work of geometry named The Elements and written by Euclid and the foundations of geometry Grundlangen der Geometrie (Foundations of Geometry) written by David Hilbert by studying the axiomatic structure of geometry dealt with by each author. The deductive rigor used by Euclid, which is based on the classical logic of Aristotle, has received several criticisms from modern mathematicians with regard to the gaps in its mathematical deductive system. The various uncertainties regarding the axiomatic system threatened its logical development and in the specific case of geometry, many discussions arose on the acceptance of the Euclid's fifth postulate. Only in the late nineteenth century, axiomatic systems reached deeper levels in the foundations of geometry and, in an attempt to complete the axiomatic geometry, Hilbert publishes “Grundlangen der Geometrie”, which is the axiomatic approach more widely adopted in the Euclidean geometry. In this context, we discuss the different concepts of classical and modern axiomatic systems , studying their logical meanings and its relations with the objects of geometry . As part of the final thoughts , this paper highlights some considerations on the concept of motion in geometry and a possible axiomatic approach to it