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This thesis is concerned with the development of novel numerical methods for solving nondifferentiable convex optimization problems in real Hilbert spaces and with the investigation of their asymptotic behavior. To this end, we are also making use of monotone operator theory as some of the provided algorithms are originally designed to solve monotone inclusion problems. After introducing basic notations and preliminary results in convex analysis, we derive two numerical methods based on different smoothing strategies for solving nondifferentiable convex optimization problems. The first approach, known as the double smoothing technique, solves the optimization problem with some given a priori accuracy by applying two regularizations to its conjugate dual problem. A special fast gradient method then solves the regularized dual problem such that an approximate primal solution can be reconstructed from it. The second approach affects the primal optimization problem directly by applying a single regularization to it and is capable of using variable smoothing parameters which lead to a more accurate approximation of the original problem as the iteration counter increases. We then derive and investigate different primal-dual methods in real Hilbert spaces. In general, one considerable advantage of primal-dual algorithms is that they are providing a complete splitting philosophy in that the resolvents, which arise in the iterative process, are only taken separately from each maximally monotone operator occurring in the problem description. We firstly analyze the forward-backward-forward algorithm of Combettes and Pesquet in terms of its convergence rate for the objective of a nondifferentiable convex optimization problem. Additionally, we propose accelerations of this method under the additional assumption that certain monotone operators occurring in the problem formulation are strongly monotone. Subsequently, we derive two Douglas–Rachford type primal-dual methods for solving monotone inclusion problems involving finite sums of linearly composed parallel sum type monotone operators. To prove their asymptotic convergence, we use a common product Hilbert space strategy by reformulating the corresponding inclusion problem reasonably such that the Douglas–Rachford algorithm can be applied to it. Finally, we propose two primal-dual algorithms relying on forward-backward and forward-backward-forward approaches for solving monotone inclusion problems involving parallel sums of linearly composed monotone operators. The last part of this thesis deals with different numerical experiments where we intend to compare our methods against algorithms from the literature. The problems which arise in this part are manifold and they reflect the importance of this field of research as convex optimization problems appear in lots of applications of interest.

We have many developmental theories contributing to our understanding of children as they meander steadfastly toward maturation. Yet, none have reported on how young children interpret the qualitative meaning and importance of their own music-making experiences. Music created by average, not prodigious, young children is perceived by adults as “play” music rather than “real” music. But do young children take the same view as adults? When Piaget speaks of the young child’s qualitatively unique view and experience of the world (Ginsberg & Opper, 1988), can we assume that his statement encompasses young children’s predispositions related to music-making? Music is understood to occur when people act intentionally to produce and organize sound into rhythm and form. The guiding questions for this study are, What evidence is there to show that, when following an adult music leader, young children can engage in authentic music-making behavior and produce identifiable musical structures that move beyond random sounds or ‘noise’? What evidence is there to show that children's music-making behavior develops according to developmental stages? trek This qualitative field study observed and videotaped over 100 children between 2 and 7 years old who chose to engage in music-making behavior in a socially-rich school environment during structured activities guided by an adult “music leader.” The data gathered from this study suggest that young children’s motivation to make music derive from predispositions unrelated to notions of cultural and artistic expression thereby differing from adult musical needs and are instead based on more primary responses to their own developmental needs and their social environment. Functioning as “music leader,” the PI appeared to serve as an indispensable interface for assuring authenticity in the children’s music-making at all stages of development. The older children did not introduce any novel behavior specifically related to making music. However, due to the progression of cognitive and social maturity across the range of ages, new extra-musical behavior (EMB) slowly emerged at each developmental stage always seeming to enrich the experience relative to a particular group.

The purpose of this study was to evaluate a programme aimed at equipping adult mine workers in the North-West Province in computer-based skills. The study describes how the use of computers for learning has transformed some of the mine workers in an Own-Time learning environment, where computers were integrated in their learning activities, compared to the Full-Time adult learning environment, which did not present a similar opportunity. The perspective of using a combination of two teaching methodologies in the Adult Education and Training (AET) curriculum provided at this mine sought to minimise the conventional didactic instructional approaches, whereby the facilitator does most of the talking while learners become passive listeners. Activity theory (AT), in tandem with the reviewed literature, was well suited as a theoretical framework for this study, since it takes a developmental view of minds and technological interaction in context. The activity system was used to map how existing work-related practices fitted into each component, which includes the subject, object, rule, community and division of labour. A case study was applied as a guiding design in the study in which seventeen mine workers participated. The computer activity was illustrated from multiple perspectives, using a qualitative approach and associated epistemologies and beliefs regarding the engagement with mediational tools. Boundary crossings, the zone of proximal development, and the eight-step-model from activity theory were used to scrutinise the learners’ beliefs, attitudes, knowledge, practices and contradictions. These important aspects assisted in obtaining data through participant observation, interviews and field notes. To discover patterns, concepts, themes and meanings from the notes and transcripts, data analysis was blended with these aspects. This study demonstrated the use of the above-mentioned methods at appropriate times. Without ideal and effective research methods, the quality of evidence of this study would have suffered and elucidations of connections would have been inadequate. The findings revealed that the AET programme made an impact on the company and on the individual participants, in their homes, their place of work, and their communities. Change in the workers’ performance after attending AET computer-based programmes was noticeable especially regarding their literate and numerate knowledge and skills. Contradictions and disturbances in the AET programme were also identified, and an intervention was proposed to facilitate a computer programme that includes the curriculum components such as mouse skills, keyboarding, data manipulation and desktop publishing. The study proposes that the AET policy needs to cover Own-Time learners, even though their learning is not obligatory, in the same way as it covers the Full-Time learners. A modified activity theory, contextualised for a situation such as that in the mine in question, was contributed in the study. Furthermore, some limitations and recommendations for future research were identified from the findings. The proposed AT for adult learners suggests a new way of exploring the practices of mind and technology interaction for adult learners, incorporating cultural backgrounds, age, language, values and beliefs as well as reflective decision making. / Science and Technology Education / D. Ed. (Comparative Education)

The purpose of this interpretive qualitative study was to determine the extent of conceptual and procedural difficulties that NCV Level 4 students encountered when factorising and solving problems involving factorisation. This study is based on Kilpatrick, Swafford and Findel’s (2001) ideas on mathematical proficiency, focusing on conceptual knowledge, procedural knowledge and the flexibility of integrating both appropriately to solve algebra problems involving factorisation. This study also explored reasons why NCV Level 4 students demonstrated such difficulties and suggested possible ways that could assist them to understand and flexibly use factorisation to solve problems. A purposive sample consisting of 30 NCV Level 4 students and 5 Subject Matter Experts participated in this study, which adopted a phenomenological case study research design. Triangulation of method was adopted for consistent gathering of information. Data was collected through a written assessment on factorisation under controlled test conditions, and semi-structured interviews. The researcher reduced and analysed data by utilising an integration of constant comparison analysis and classical content analysis. The findings and relevant recommendations concluded this research. / Mathematics Education / M. Ed. (Mathematical Education)

Large underfunded classrooms are indicative of the situation found in many developing countries. These limitations often lead to the ineffective teacher-centred approach dominating classroom instruction in these countries. The aim of the study was to examine active learning strategies used by teachers in large unfunded Physics classrooms, in Kerala, India. This study used a qualitative approach utilising a case study method. The data collection process involved mainly unstructured interviews, lesson observations and the analysis of documents. The results of the study indicate that despite the difficulties faced in India, similar to those faced by other developing countries, the Indian state of Kerala implements and supports active learning rather than the more common teacher-centred approach. The research defines the common forms of active learning in the Physics lessons and critically examines the core elements of the learner-centred teaching approaches. Successes have been found in the implementation of active, collaborative, cooperative and problem-based learning in the large underfunded Physics classes. The results of the research suggest that teachers need to be highly trained, resourceful, creative, hardworking and sometimes go above and beyond the required duties to make active learning in large underfunded Physics classroom a success. / Science and Technology Education / M. Ed. (Natural Science Education)

As theories on the process of reading have advanced so definitional changes have resulted. This in turn has impacted on the teaching of reading. Comprehension is not the result of successful word recognition, rather, meaning is constructed by the reader using various sources as a frame of reference. Reading requires an interaction between the reader, the text read and the context in vvhich it is read. Reader factors involve language competencies, prior knowledge, vocabulary, the use of strategies and attitudes and motivation. The text may be narrative or expository and encompasses instructional materials. The tvvo broad categories in the context are the classroom setting and the instructional context. An 'interactive' or 'organisational' paradigm underlying inclusion recognises individual differences as being a probable cause of failure but postulates that the school and all that it encompasses, can be a barrier to learning and development. As such it is not deficit driven, attributing failure to learners alone. In the case of reading, this means not attributing reading failure to the reader alone but acknowledging the role of the text and the context. This point of departure is confirmed by an interactive model of disability, which, while still explaining reading deficits, advances that alternative areas also be investigated. Focusing on abilities is conducive to proactivity in the prevention of barriers to learning and development. To be inclusive therefore, mainstream schools generally and classrooms specifically, will need to be reformed and restructured to be more responsive to learners experiencing barriers to learning and development. This will require enhanced teaching methods and flexible support systems. Accommodating diversity presupposes the acknowledgement of each learner's uniqueness in order to meet individual needs. This will be facilitated when in the compilation of a reading programme to meet individual needs, the reader, the text and context are matched through assessment and instruction . A reading programme to enhance the teaching of reading and thereby meet individual needs has been compiled and implemented in a mainstream, intermediate phase class. From this practical experience and the literature studied, guidelines for the teaching of reading have been formulated for teachers in the intermediate phase to use within the context of inclusion. / Educational Studies / D. Ed. (Orthopedagogics)

The aim of this study was to investigate the comparative effects of Constructivist-Based Teaching Method (CBTM) and the Traditional Teaching Method (TTM) on Grade 11 Mathematics learners’ errors in algebra. The constructivist learning theory (CLT) was used to frame this study. Mainly, CLT was used to influence the design of CBTI to hone participants’ errors in algebra that militate against their performance in Mathematics. The study was conducted in the Mpumalanga Province of South Africa with a four-week intervention programme in each of the two participating secondary schools. Participants consisted of n=78 Grade 11 Mathematics learners and one Grade 11 Mathematics teacher. A non-equivalent control group design consisting of a pre-test and post-test measure was employed. The Grade 11 teacher in the control school employed the TTM while the researcher implemented CBTM in the experimental school. The main aspects of CBTM entailed participants’ construction of their own knowledge from the base of prior knowledge and through group learning approach and exploratory talk in which discussions included argumentation, verbalising explanations, justifications and reflections. Participants in experimental school became familiar with the basic principles of CBTI such as group work, which enhanced the construction of conceptual understanding of algebraic concepts. This reduced most of the errors they commit in algebra and elevated their performance in Mathematics. The principal instruments for data collection consisted of a standardised Algebra Concept Achievement Test and lesson observations. The pre-test was used to determine participants’ initial errors in algebra before the intervention. A post-test was given at the end of intervention to ascertain change in participants’ errors in algebra over a four-week intervention period. Using descriptive and inferential statistical techniques, the study found that participants in experimental school significantly reduced their errors in algebra than those in control school. The study showed that CBTM was a more effective pedagogy that improved the errors Grade 11 learners commit in algebra than the TTM. / College of Education / M. Ed. (Mathematics Education)

7 Abstract (AJ) Introduction Radiology examination using specialized modern imaging methods, including CT and MRI, is essential in the diagnosis of congenital and acquired diseases of the musculoskeletal system. The first part of the dissertation deals with certain congenital defects of the short femur, known in the literature as proximal femoral focal deficiency (PFFD). This part summarizes our experience with the radiological findings in the preoperative and postoperative period, with the main attention to the vascular supply to the affected area. The second part of the presentation deals with some aspects of autologous chondrocyte transplantation fixed at two different carriers implanted into post-traumatic articular cartilage defects of the distal femur. Radiological findings are evaluated in the relation to the histopathological findings. Objectives The first part of the study after the distribution of patients with PFFD by current commonly used radiographic classification sets the objective in the extent of scans of the hip joints to specify diagnosis PFFD in each patient and to evaluate in detail changes in the area of disability, especially a course of blood vessels. The evaluation of the radiation burden of repeated X-ray measurements was done with respect to the age of the patients. Tissue samples...

[pt] Algoritmos para a resolução de problemas de programação mista 0-1 gerais baseados em cortes derivados dos métodos lift-and-project, tem se mostrado bastante eficientes na prática. Estes cortes são gerados resolvendo um problema que depende de uma certa normalização. Desde um ponto de vista teórico, o bom comportamento destes algoritmos não foi completamente compreendido, especialmente no que diz respeito à normalização. Neste trabalho consideramos normalizações gerais definidas por um conjunto convexo fechado arbitrário, estendendo assim a análise teórica desenvolvida nos anos noventa. Apresentamos um marco teórico que abarca todas as normalizações previamente estudadas e introduzimos novas normalizações, analisando as propriedades dos cortes associados.Introduzimos também uma nova fórmula de atualização do parâmetro proximal para uma variante dos métodos de feixes. Estes métodos são bem conhecidos pela sua eficiência na resolução de problemas de otimização não diferenciável. Por último, propomos uma metodologia para eliminr soluções redundantes de programas inteiros combinatórios. Nossa proposta baseia-se na utilização da informação de simetria do problema, eliminam a simetria sem prejudicar a solução do problema inteiro. / [en] Algorithms for general 0-1 mixed integer programs can be successfully developed by using lift-and-project methods to generate cuts. Cuts are generated by solving a cut- generation-program that depends on a certain normalization. From a theoretical point of view, the good numerical behavior of these cuts is not completely understood yet, specially, concerning to the normalization chosen. We consider a general normalization given by an arbitrary closed convex set, extending the theory developed in the 90's. We present a theoretical framework covering a wide group of already known normalizations. We also introduce new normalizations and analyze the properties of the associated cuts. In this work, we also propose a new updating rule for the prox parameter of a variant of the proximal bundle methods, making use of all the information available at each iteration. Proximal bundle methods are well known for their efficiency in nondifferentiable optimization. Finally, we introduce a way to eliminate redundant solutions ( due to geometrical symmetries ) of combinatorial integer program. This can be done by using the information about the problem symmetry in order to generate inequalities, which added to the formulation of the problem, eliminate this symmetry without affecting solution of the integer problem. / [es] Los algoritmos para la resolución de problemas de programación mixta 0-1 generales que utilizan cortes derivados de los métodos lift-and-project, se han mostrado bastante eficientes en la práctica. Estos cortes se generan resolviendo un problema que depende de una cierta normalización. Desde el punto de vista teórico, el buen comportamiento de estos algoritmos no fue completamente comprendido, especialmente respecto a la normalización. En este trabajo consideramos normalizaciones generales definidas por un conjunto convexo cerrado arbitrario, extendiendo así el análisis teórico desarrollado en los años noventa. Presentamos un marco teórico que abarca todas las normalizaciones previamente estudiadas e introducimos nuevas normalizaciones, analizando las propiedades de los cortes asociados. Introducimos una nueva fórmula de actualización del parámetro de. Estoss métodos son bien conocidos por su eficiencia en la resolución de problemas de optimización no diferenciable. Por último, proponemos una metodología para eliminar soluciones redundantes de programas enteros combinatorios. Nuestra propuesta se basa en la utilización de la información de simetría del problema, eliminan la simetría sin perjudicar la solución del problema entero.