Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 91-93). Also available in electronic format on the Internet.

Made available in DSpace on 2016-04-01T17:54:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-10-15. Added 1 bitstream(s) on 2016-04-01T18:00:12Z : No. of bitstreams: 1 000858313.pdf: 4395831 bytes, checksum: d54570b3a003adccfa38587bd387fc33 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O objetivo desta pesquisa é compreender o papel das tecnologias digitais (TD) nos processos educativos associados a disciplinas de Álgebra Linear de quatro cursos de Licenciatura em Matemática a distância vinculados à Universidade Aberta do Brasil (UAB), no contexto de seus Ambientes Virtuais de Aprendizagem (AVA). Trata-se de uma pesquisa qualitativa. Perspectivas associadas aos modos de descrição em Álgebra Linear, à noção de seres-humanos-com-mídias, à distância transacional e ao papel da interação e da colaboração na modalidade a distância fornecem sustentação teórica ao estudo e ao processo analítico. A Teoria Enraizada é utilizada para conduzir a análise de dados, que foram produzidos a partir de quatro fontes: observação em ambientes virtuais de aprendizagem, entrevistas, projetos político pedagógicos e notas da pesquisadora. Os resultados permitem inferir que há dois papéis em evidência, cada um analisado em uma das categorias, intituladas TD como promotoras de variedade comunicacional e TD na construção de materiais didáticos digitais. Da primeira categoria foram exploradas quatro propriedades: conteúdo, agentes, temporalidade e avaliação. Da segunda, três: conteúdo, natureza e recursos envolvidos. Posteriormente, as duas categorias foram integradas em uma categoria central que sugere que as TD, a internet e o uso do AVA podem transformar esse último em Material Didático Digital Interativo (MDDI) a partir do registro automático das interações. Esta transformação pode se dar via diferentes linguagens, como a textual, a audiovisual ou a multimodal. Do ponto de vista da disciplina de Álgebra Linear, notou-se um desequilíbrio em termos de abordagem dos modos de descrição (formal, algébrico e geométrico) e destacou-se a necessidade de estimular o movimento entre eles, que pode ser favorecido pelas possibilidades que se abrem com a presença das TD. O modelo construído foi... / The objective of this research is to understand the role of digital technologies (DT) in educational processes associated with Linear Algebra in four distance teacher education in mathematics courses linked to the Open University of Brazil (UAB), in the context of its Virtual Learning Environments (VLE). It is a qualitative study. The study and the analysis are guided by theoretical perspectives related to modes of description in Linear Algebra, the notion of humans-with-media, transactional distance and the role of interaction and collaboration in distance learning. Grounded Theory is used to analyze the data, which was produced from four sources: observation in virtual learning environments, interviews, educational projects and researcher's notes. Results show that two roles are evident (each treated as an analytic category): DT as promoting communication range and DT in building digital learning materials. Four properties of the first category were explored: content, agents, temporality and evaluation. For the second category, three properties were analyzed: content, nature and resources involved. The two categories were then integrated into one central category, which suggested that DT, the internet, and the use of VLE can turn the latter into Interactive Digital Didactic Material (IDDM) through the automatic registration of interactions. This transformation can take place via different languages, such as textual, audio-visual or multimodal. With respect to the linear algebra course, an imbalance in description mode approaches (formal, algebraic, and geometric) was noted, which stressed the need to encourage movement among them. This movement can be encouraged by the possibilities introduced by DT. The model constructed was used to identify patterns of use of technologies in the institutions as a way to validate the analysis and provoke reflections about the consistency between observed practices and institutional goals. In this... / FAPESP: 2012/12176-3

Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with non-discrete topology. The only local fields (up to isomorphism) are the real, complex, and p-adic numbers, finite extensions of the p-adic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, the special orthogonal Lie algebras, and the special unitary Lie algebras is proved.

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg. December 2014. / The National Curriculum and Assessment Policy Statement document (CAPS) (Department of Basic Education, 2012: 4) outlines some of the salient skills related to the modelling of word problems that Mathematics learners are expected to acquire during the course of their learning. One such critical skill is the learners' ability to represent and describe situations in algebraic language, formulae and expressions. In line with this broad objective, the present study investigated learners' linguistic knowledge and skills in translating from word to linear algebraic representations. Particularly focussing on errors learners made during the translation process, a class of 40 Grade 11 learners at one school in Gauteng West District were selected and given a written test on word problems. To gain access to learner misunderstandings leading to the observed errors and their sources, five learners were selected for interviews that were audio recorded. A Mixed-method Sequential Explanatory Design was used in the study. The major finding of this present study was that learners committed more semantic errors than syntactic and schematic errors suggesting that they (learners) had challenges in understanding the language used in the word problems. The major sources of semantic errors were found to be learners' lack of vocabulary knowledge, inexperience in interacting with expository text structures and their lack of syntax awareness. In addition to the above finding, learners exhibited limited metacognitive skills essential in word problem solving as well as their inability to reflect on the appropriateness of their written algebraic representations for all categories of word problems. The study concludes by providing recommendations on how errors committed in word problem solving can be attended to

CAPES / O estudo de matrizes no ensino médio é quase sempre centrado no ensino de técnicas operatórias, mesmo foco dado pelos materiais didáticos que abordam esse assunto nesse nível de ensino. O texto propõe uma visão diferenciada a essa prática de ensino, mostrando de forma simples, clara e de fácil compreensão para um aluno do ensino médio algumas aplicações de matrizes, usando para isso as transformações geométricas de translação, rotação e transformação de escala. / The study of matrices in high school is almost always centered on the teaching of operative techniques, same focus given by textbooks that address this issue at this level of education. This text proposes a different approach, showing in a clear and simple way some applications as translation, rotation and scaling.

CAPES / O estudo de matrizes no ensino médio é quase sempre centrado no ensino de técnicas operatórias, mesmo foco dado pelos materiais didáticos que abordam esse assunto nesse nível de ensino. O texto propõe uma visão diferenciada a essa prática de ensino, mostrando de forma simples, clara e de fácil compreensão para um aluno do ensino médio algumas aplicações de matrizes, usando para isso as transformações geométricas de translação, rotação e transformação de escala. / The study of matrices in high school is almost always centered on the teaching of operative techniques, same focus given by textbooks that address this issue at this level of education. This text proposes a different approach, showing in a clear and simple way some applications as translation, rotation and scaling.

Indiana University-Purdue University Indianapolis (IUPUI) / Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent.

One major challenge facing mathematics education in South Africa in general and Limpopo in particular, is learners’ underachievement and lack of motivation to learn the subject. Some studies have shown that one of the topics that learners dread is linear functions. Many teachers also find it difficult to teach the topic effectively. Studies in other parts of the world have advocated the integration of graphing software with the teaching and learning of functions to enhance learners’ learning of mathematics. This study therefore investigated the effect of integrating GeoGebra graphing software into the teaching of linear functions on the achievement of Grade 9 learners. The study was guided by APOS theory which, in accordance with constructivist theories, posits that an individual needs to construct the necessary cognitive structures in order to make sense of mathematical concepts. A total of 127 Grade 9 learners from four schools in a circuit in Mopani district of Limpopo Province participated in the study which followed a pretestpost- test quasi-experimental study design. Two schools, namely B (35 learners) and D (33 learners) formed the experimental groups while school A (31 learners) and school C (28 learners) were the control groups. Data were collected using an achievement test and analyzed using descriptive and inferential statistics. The pretest results showed that the groups were of comparable cognitive abilities. The post-test results showed that there was a significant difference between the mean scores of the experimental groups and control groups. There were also statistically significant differences between group treatment means (p < .05). Bonferroni post-hoc test results showed that there were no statistically significant differences between treatments A and C. The results showed that the learners in the two control groups were of comparable cognitive abilities. The implications of the findings are discussed and recommendations made. / Mathematics Education / M. Sc. (Mathematics Education)

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é apresentar o método dos mínimos quadrados para resolver sistemas lineares sobredeterminados, ou seja, sistemas da forma Ax=b, em que A m×n , com m>n. Neste sentido, veremos como a resolução destes sistemas estão relacionados com encontrar a projeção ortogonal b sobre o subespaço gerado pelas colunas de A . Este tipo de sistema é usado ainda para modelar um problema de geoposicionamento, cujo objetivo é determinar a posição de um receptor que recebe o sinal de vários satélites. / The main goal of this work is to present the least squares method to solve overdetermined linear systems, that is, systems of the form Ax = b , where A m×n , with m > n . In this sense, we showed that the resolution of these systems is related to the orthogonal projection problem of b on the subspace generated by the columns of A. This type of system is used to model a problem of geo-positioning, whose objective is to determine the position of a receiver that receives the signal from several satellites.

Abstracts in English, Zulu and Afrikaans / The worldwide growth of open distance and e-Learning cannot be denied, nor can it remain unnoticed. In the South African setting, which is the focus of this study, the increasingly high demand for higher education has also led to the development of distance education. However, the throughput rate for this mode of education has been consistently unimpressive for over a decade and this has led to high dropout rates. There is cause for concern regarding how best to assist students and improve throughput. Therefore, this research has focused on identifying strategies that can be incorporated to support sustained learning in South African ODeL. It considers issues such as the social inequalities, previous disadvantages, and ICT affordances that characterise South African ODeL students. This research took a qualitative research approach, using a case study that was conducted as a dissertation of limited scope. The aim was to unveil the best approaches that could be incorporated in South African ODeL in order to support sustained learning. The study attempted to understand such strategies, based on students' perceptions and reflections on their experiences while studying first-level modules in distance education. The case for this research study was the University of South Africa (UNISA), where eight student participants were interviewed, using semi-structured interviews to collect data. The findings of this study revealed that there are some readily available resources that the institution and instructors can use to promote sustained learning and improve student throughput. This research recommends frequent student surveys to discover student needs and the challenges that they experience. It also recommends improved communication with students, provision of financial aid, development of open educational resources (OERs), ensuring rapid delivery of study material, and the use of different methods of delivering learning content. It is important to note that the findings of this dissertation are of limited scope and cannot be generalised to the issues affecting the majority of South African ODeL students – this because the study was carried out at one institution and involved just one module and eight participants. However, these findings have the potential to enlighten ODeL stakeholders as to matters affecting some of their students. The findings could also provide valuable information for future large-scale research. / Ukudlondlobala kohlelo kohlelo lwemfundo evulekile nokufunda kude kanye nohlelo lwe-inthanethi akunakuphikiswa, futhi angeke kwathathwa sengathi akunakiwe. Ngokwesizinda seNingizimu Afrika, esiyindikimba yalolu cwaningo, izinga elikhulayo lokufuneka kwemfundo ephakeme nakho sekuholele ekukhuleni kwemfundo yokufunda ukude.Yize-kunjalo, izinga legalelo lalolu hlelo lwemfundo kudala lubonakala lungeluhle esikhathini esingaphezu kweminyaka eyishumi kanti lokhu sekuholele ekutheni kwehle izinga labafundi abayeka ukufunda. Kunomnako mayelana nendlela nokuthi abafundi bangancedwa kangcono kangakanani ukuze bathuthukise izinga labo lokusebenza. Ngakho-ke, lolu cwaningo selugxile ekutholeni amasu lawo angasetshenziswa ukuxhasa imfundo esimelele ohlelweni lwe-ODeL lwaseNingizimu Afrika. Lokhu kufaka izindaba ezinjengokungalingani komphakathi, ukuncishwa kwamathuba esikhathini esedlule, kanye nokwazi ukukhokhela uhlelo lwe-ICT okuwuhlelo oluyinsika kubafundi bohlelo lwe-ODeL eNingizimu Afrika. Lolu cwaningo lulandele indlela yocwaningo eyencike kwizinga lengxoxo (qualitative), ngokusebenzisa ucwaningo lotho olwenziwa njengedezetheshini enobukhulu obunqunyiwe. Inhloso kwaye kuwukuveza izindlela ezingcono ebezingafakwa kwi-ODeL yaseNingizimu Afrika, ukuxhasa imfundo imfundo esimelele. Ucwaningo beluzama ukuzwisisa amasu anjalo, asuselwa phezu kwemiqondo yabafundi kanye nokubheka ulwazi lwabo kube ngakolunye uhlangothi bafunda omojuli besigaba sokuqala sohlelo lwemfundo yokufunda kude. Ucwaningo lotho lwalesi sifundo lwenziwa eNyuvesi yaseNingizimu Afrika (UNISA), lapho abafundi abayisishiyagalombili befakwa imibuzo yenhlolovo, ngokusebenzisa inhlolovo yesakhiwo sikanxambili ngenhloso yokuqoqa idatha. Ulwazi olutholwe yilolu cwaningo luye lwaveza ukuthi kunemithombo elungisiwe esesimeni esifanele engasetshenziswa yiziko kanye nabayaleli ukuthuthukisa uhlelo lokufunda olusimelele kanye nokuthuthukisa umsebenzi wabafundi. Lolu cwaningo luncoma amasaveyi enziwa njalo abafundi ukubona izidingo zabafundi kanye nezinselelo abafundi abahlangabezana nazo. Ucwaningo luphinda futhi luncoma uhlelo oluhle lokuxhumana nabafundi, ukuhlinzekwa ngosizo lwezimali, ukuthuthukiswa kwemithombo yemfundo evulekile (OER), ukuqinisekisa kohlelo olusheshayo lokuhlinzekwa. ngomatheriyeli wokufunda, kanye nokusebenzisa izindlela ezahlukene zokwethula indikimba yokufundwayo. Kubalulekile ukukhumbula ukuthi ulwazi olufunyenwe lwedizetheshini wulwazi oluncane olubekelwe imingcele kanti lolu lwazi angeke lwafaniswa nodaba oluthinta iningi labafundi base-ODeL yaseNingizimu Afrika – lokhu kungenxa yokuthi ucwaningo lwenziwa kwiziko elilodwa futhi lwaxuba umojuli owodwa kanye nabadlalindima abayisishiyagalombili. Yize-kunjalo, lolu lwazi olutholakele lunethuba lokukhanyisela abasebenzisani njengezinto ezithinta abanye babafundi. Ulwazi olutholakele belunganikeza ulwazi olusemqoka mayelana nocwaningo olubanzi olungenziwa esikhathini esizayo. / Wêreldwyd het oopafstand- en e-leer groei ervaar wat nie ontken kan word of ongesiens kan bly nie. In die Suid-Afrikaanse konteks, wat die fokus van hierdie studie is, het die toenemende vraag na hoër onderwys ook tot die ontwikkeling van afstandsonderrig gelei. Die slaagsyfer vir hierdie vorm van onderrig is egter al vir langer as ʼn dekade deurlopend onindrukwekkend, en dit het tot hoë uitsaksyfers gelei. Daar is rede tot kommer rakende die beste maniere om studente by te staan en die slaagsyfer te verbeter. Die fokus van hierdie navorsing was daarom op die identifisering van strategieë wat geïnkorporeer kan word om volgehoue leer in Suid-Afrikaanse oopafstand- en e-leer te ondersteun. Die volgende kwessies is ondersoek: die sosiale ongelykhede, benadeling in die verlede, en IKT-status wat met Suid-Afrikaanse oopafstand- en e-leer-studente geassosieer word. ʼn Kwalitatiewe navorsingsbenadering is gevolg; ʼn gevallestudie wat uitgevoer is, is in die verhandeling van beperkte omvang gebruik. Die doel was om die beste benaderings te bepaal wat in Suid-Afrikaanse oopafstand- en e-leer geïnkorporeer kan word om volgehoue leer te ondersteun. Tydens die studie is daar gepoog om sodanige strategieë te verstaan op grond van studente se persepsies en hul refleksie op hul ervarings terwyl hulle eerstejaarmodules deur afstandonderrig studeer het. Die geval wat in hierdie navorsing bestudeer is, is dié van die Universiteit van Suid-Afrika (Unisa), waar daar semi-gestruktureerde onderhoude met agt studentedeelnemers gevoer is om data in te samel. Die bevindinge van hierdie studie het getoon dat daar geredelik beskikbare hulpbronne is wat deur die instelling en die onderriggewers ingespan kan word om volgehoue leer te bevorder en die studente se slaagsyfers te verbeter. Op grond van hierdie navorsing word aanbeveel dat opnames gereeld onder studente gemaak word om hulle behoeftes en uitdagings te bepaal. Verdere aanbevelings sluit in verbeterde kommunikasie met studente, voorsiening van finansiële steun, ontwikkeling van oop opvoedkundige hulpbronne, versekering van spoedige lewering van studiemateriaal, en die gebruik van verskillende metodes om leerinhoud te lewer. Dit is belangrik om te besef dat die bevindinge van hierdie verhandeling beperk in omvang is en nie veralgemeen kan word om die kwessies in te sluit wat die meerderheid van Suid-Afrikaanse oopafstand- en e-leer-studente raak nie – aangesien die studie by een instelling uitgevoer is en slegs een module en agt deelnemers behels het. Hierdie bevindinge het egter die potensiaal om vir oopafstand- en e-leer-belanghebbers lig te werp op kwessies wat sommige van hul studente raak. Die bevindinge kan ook waardevolle inligting vir toekomstige grootskaalse navorsing oplewer. / Educational Studies / M. Ed. (Open Distance Learning)