Thesis (MSc (Mathematical Sciences))--Stellenbosch University, 2008. / A novel implementation exploiting modern hardware is explored and found to be a significant improvement over current methods used. A 50 times performance increase in the computation time of DRRs is achieved over the conventional ray casting approach and image registration is performed in under a minute.

Thesis (PhD)--Stellenbosch University, 2012. / Includes bibliography / ENGLISH ABSTRACT: The rationale for this study emerged from a realization that conventional instructional design approaches for introducing Calculus concepts, based on the logical sequencing and structuring of the concepts, did not adequately attend to or address students’ ways of thinking. This was particularly important in a distance education environment where learners depend on instructional texts to make sense of what is being presented, often without support from tutors. The instructional design theory of Realistic Mathematics Education (RME) offered a promising approach for designing learning sequences based on actual investigations of the ways in which students think. This study’s focus was on trialling the process of RME theory-based design using the Fundamental Theorem of Calculus as an example. Curve sketching was prominent in this exercise. Applying RME required developing a hypothetical learning trajectory (HLT) while attempting to adhere to methodological guidelines of design research. In this project, the instructional designer’s conceptualization and interpretation of the derivative-integral construct has had the most immediate implications for the study. The line of inquiry has been largely didactic, in that it was framed by a need to establish ways of introducing the teaching of a mathematical concept following instructional design principles. Throughout the project, the instructional design space has been contested, broken down, rebuilt and, ultimately, enriched by the contributions of the expert teachers and the engagement of participating students. The series of design experiments have revealed knowledge about student reasoning in this learning domain in relation to four main areas of quantifying change, curve sketching, general mathematical reasoning and symbol use. The primary contribution of this research has been a deeper understanding of the extent to which RME can be used as an instruction design theory for planning and introducing a distance teaching Calculus unit. From the study, it is clear that successful adoption of the RME theory is influenced and facilitated by a number of factors, including: careful selection of the concepts and mathematical structures to be presented; a team of experts (mathematicians and mathematics subject didacticians) to research, test and develop the learning activities; opportunities for student interactions; and time and resources for effective RME adoption. More involved research is required to get to the stage of the evolution of a local instructional theory around introducing the derivative-integral relationship as expressed in the Fundamental Theorem of Calculus. / AFRIKAANSE OPSOMMING: Die rasionaal van hierdie studie het uit die besef ontstaan dat konvensionele onderrigontwerpbenaderings vir die bekendstelling van Calculus konsepte, gebaseer op die logiese ordening en strukturering van die konsepte, nie voldoende beantwoord aan die eise van hoe studente dink nie. Dit was van spesifieke belang in die geval van afstandonderwys waar hierdie studente sin moet maak van wat aangebied word, dikwels sonder die ondersteuning van tutors. Die onderrigontwerpteorie van Realistiese Wiskundeonderwys (RWO) bied belowende moontlikhede om leertrajekte te ontwerp wat gebaseer is op werklike ondersoeke van hoe studente dink. Hierdie studie se fokus was om die RWO-gebaseerde teoretiese ontwerp se proses wat die Fundamentele Stelling van Calculus as voorbeeld gebruik, uit te toets. Krommesketsing was prominent in hierdie oefening. Die toepassing van RWO het vereis dat 'n leertrajek ontwikkel moet word terwyl aan die metodologiese vereistes van die ontwikkelingsondersoekbenadering getrou gebly word. In hierdie projek het die onderrigontwerper se konseptualisering en interpretasie van die afgeleide-integraalkonstruk onmiddellike implikasies gehad vir die studie. Die lyn van ondersoek was grootliks didakties van aard. Desnieteenstaande was die instruksionele ontwerpruimte voortdurend beding, afgebreek, herbou en uiteindelik verryk deur die bydraes van die bedrewe onderwysers en die betrokkenheid van die deelnemende studente. Die reeks ontwerpeksperimente het kennis blootgelê van hoe studente in hierdie veld redeneer met betrekking tot die volgende vier hoof areas: kwantifisering van verandering, krommesketsing, algemene wiskundige beredenering en die gebruik van simbole. Die primêre bydrae van hierdie navorsing is die dieper verstaan van die mate waarin RWO gebruik kan word as 'n instruksionele ontwerpteorie vir die beplanning en bekendstelling van 'n Calculus eenheid in afstandsonderrig.Dit is duidelik vanuit die studie dat suksesvolle aanneming van die RWO teorie afhanklik is van 'n aantal faktore: 'n noukeurige seleksie van die konsepte en wiskundige strukture wat aangebied moet word; 'n span van bedrewe wiskundiges en wiskunde vakdidaktici om die leeraktiwiteite na te vors, uit te toets en te ontwikkel; geleenthede vir studente-interaksies, en tyd en bronne vir effektiewe RWO aanpassing. Verdere toegespitsde navorsing hierop is nodig om die fase te bereik van die ontluiking van 'n lokale onderrigteorie oor die bekendstelling van die afgeleide-integraal verwantskap soos uitgedruk in terme van die Fundamentele Stelling van Calculus.

Thesis (MEd)--Stellenbosch University, 2012 / Includes bibliography / ENGLISH ABSTRACT: The use of information and communications technologies (ICTs) in the form of spreadsheet algebra programmes (SAPs) is important in the professional development of high school mathematics teachers. This is in line with The Namibian government‟s Vision 2030 in which ICT skills and competencies are regarded as core elements of living and participating in the 21st century. ICTs are also considered to be fundamental to the development of a dynamic knowledge-based economy (KBE) through the Education and Training Sector Improvement Program (ETSIP). ETSIP‟s aim is to embed ICT at all levels of the education system. It also aims to integrate the use of ICTs as tools in the delivery of curriculum and learning and in so doing, lead to a marked improvement in the quality of the learning and teaching process across all levels. Education has a key role in achieving Vision 2030. The aim of this research was to investigate mathematics teachers‟ perceptions of SAPs as curriculum materials in selected Namibian secondary (high) schools. This research adopted a qualitative methodology, which in this instance was a case study. The sample population consisted of five teachers from Okamu (pseudonym) secondary school in the Ohangwena Region of Namibia. Four of them had been teaching mathematics at different levels in the mentioned school for a period of four years, and one of them was teaching physical science. Three methods of data collection were used. The first two were semi-structured interviews and focus group interviews based on teachers‟ experiences using SAPs. The third method was an audio taped observation of a lesson taught by one of the teachers,. This research provides evidence about teachers‟ perceptions regarding time concerns and constraints with regards to the SAPs and the use of the SAPs. The teachers showed willingness and enthusiasm to use SAPs on linear and quadratic functions in their teaching. Some of the teachers became more aware of the epistemic dimensions associated with mathematical and algebraic symbols. Interview data reveal that the teachers had not considered these dimensions when teaching with the usual paper-and-pen format. The research also provides evidence of a teacher‟s early vision about the use of spreadsheets as an instrument to teach linear functions. This teacher did not consider any epistemic value for the instrumented spreadsheets techniques, or that they might contribute to a deeper understanding of the linear functions. His concern was focused more on getting the learners to acquire computer skills, such as learning how to use spreadsheets. It is recommended that in-service professional development about ICT integration into mathematics teaching be offered. This might help teachers to learn how their knowledge and skills could be used in the classroom more effectively in order to save time. It is also suggested that professional development programmes be designed to stimulate and promote teachers‟ willingness to develop an understanding of the characteristics of ICTs such as SAPs and their uses. Lastly, it is recommended that new SAPs be designed in order to deepen the understanding of algebra at the secondary level. / AFRIKAANSE OPSOMMING: Die gebruik van inligting- en kommunikasietegnologieë (IKT's) in die vorm van sigblad-algebra-programme (SAP's) is belangrik vir die professionele ontwikkeling van hoërskoolwiskunde-onderwysers. Dit is in pas met die Namibiese regering se visie vir 2030, Vision 2030, waarin IKT-vaardighede en -bevoegdhede beskou word as kernelemente van die lewe in en deelname aan die 21ste eeu. IKT's word ook beskou as grondliggend aan die ontwikkeling van ‟n dinamiese kennisekonomie (KE) deur middel van die Verbeteringsprogram vir die Onderwys- en Opleidingsektore (ETSIP). ETSIP het as oogmerk om IKT op alle vlakke van die onderwysstelsel vas te lê. Dit het ook ten doel om die gebruik van IKT's as hulpmiddele te integreer in die lewering van kurrikulum en leer en sodoende ‟n duidelike verbetering in die gehalte van die onderrig-en-leerproses oor alle vlakke heen tot gevolg te hê. Onderwys het ‟n sleutelrol te speel by die bereiking van Vision 2030. Die doel van hierdie navorsing was om wiskundeonderwysers se persepsies van SAP's as kurrikulummateriaal in geselekteerde Namibiese sekondêre (hoër-) skole te ondersoek. Hierdie navorsing het ‟n kwalitatiewe metode gevolg, in hierdie geval ‟n gevallestudie. Die proefgroep het bestaan uit vyf onderwysers van die sekondêre skool Okamu (skuilnaam) in die Ohangwena-streek van Namibië. Vier van hulle het reeds vier jaar lank wiskunde op verskillende vlakke in die betrokke skool gegee en een van hulle het fisiese wetenskap gegee. Drie metodes is ingespan om data in te samel. Die eerste twee was semigestruktureerde onderhoude en fokusgroeponderhoude gebaseer op onderwysers se ervaringe ten opsigte van die gebruik van SAP's. Die derde metode was ‟n klankopname van ‟n waarnemingsessie van ‟n les wat deur een van die onderwysers gegee is. Hierdie navorsing getuig van onderwysers se persepsies ten opsigte van tydskwessies en -beperkinge met betrekking tot die SAP's en die gebruik van die SAP's. Die onderwysers het ‟n gewilligheid en geesdrif geopenbaar om by lineêre en kwadratiese funksies SAP's in hul onderrig te gebruik. Party onderwysers het meer bewus geraak van die epistemiese dimensies in verband met wiskundige en algebraïese simbole. Onderhouddata onthul dat die onderwysers nie hierdie dimensies in ag geneem het toe hulle met die gewone papier-en-pen-formaat klasgegee het nie. Die navorsing bevestig ook ‟n onderwyser se aanvanklike insig oor die gebruik van sigblaaie as 'n instrument om lineêre funksies te onderrig. Hierdie onderwyser het nie die epistemiese waarde vir die geïnstrumenteerde sigbladtegnieke in ag geneem nie, of dat hulle sou kon bydra tot ‟n grondiger begrip van die lineêre funksies nie. Sy belangstelling was eerder daarop gefokus om die leerders sover te kry om rekenaarvaardighede, soos om sigblaaie te kan gebruik, te verwerf. Daar word aanbeveel dat indiens- professionele ontwikkeling oor IKT-integrasie in wiskundeonderrig aangebied word. Dit sal onderwysers moontlik help om te leer hoe hulle kennis en vaardighede meer doeltreffend in die klaskamer gebruik kan word om tyd te bespaar. Daar word ook voorgestel dat professionele ontwikkelingsprogramme ontwerp word, ter stimulering en bevordering van onderwysers se bereidwilligheid om ‟n begrip te ontwikkel van die kenmerke en gebruike van IKT's soos SAP's. Laastens word daar aanbeveel dat nuwe SAP's ontwerp word om die verstaan van algebra op sekondêre vlak te verdiep.

Thesis (MEd)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Many learners, even at high school level, have difficulty with fractions and computations involving fractions. A report from the Department of Basic Education (DBE, 2012c: 15) has highlighted that the lack in basic fraction sense was one of the areas of concern that contributed to the low achievement in matriculation mathematics examinations in 2012. Fractions play an important role in our ever-advancing technological society. Many occupations today rely heavily on the ability to compute accurately, proficiently, and insightfully with fractions. High school learners’ understanding or the lack thereof is carried over to their tertiary studies and workplaces. It is for that reason that in this dissertation, the learning and understanding of fractions and their role in the high school curriculum are studied through a critical literature review. Fractions are compound constructs and can therefore be interpreted in many different ways, depending on the area of study within mathematics. The concept of fractions consists of five sub-constructs, namely, part-whole, ratio, operator, quotient, and measure (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). This thesis starts with discussion of the background of the study and its importance. Thereafter the elements that assist in the understanding of the fraction concept is discussed. Then, the five different sub-constructs are elaborated on, and how these different sub-constructs are used in the high school curriculum is demonstrated. The conclusion offers some implications for classroom teaching and mathematics teachers’ professional development. / AFRIKAANSE OPSOMMING: Talle leerders, tot op hoërskool vlak, ervaar probleme met breuke en berekeninge met breuke nie. ‘n Verslag van die Departement van Basiese Onderwys (DBE, 2012c: 15) het beklemtoon dat die gebrek aan basiese breuk vaardighede een van die oorsake was wat daartoe gely het dat die prestasie in die 2012 matriek wiskunde eksamen so laag was. Breuke speel ‘n belangrike rol in ons voortdurende tegnologiese voor uitgaande samelewing. Talle beroepe vandag is grootliks afhanklik van die akkurate, bekwame en insiggewende berekeninge van breuke. Hoërskool leerders se begrip, of die gebrek daaraan word oorgedra na hul tersiêre studies en werksplekke. Dit is vir dié rede dat hierdie tesis die leer en begrip van breuke en hul rol in die hoërskool kurrikulum bestudeer deur middel van ‘n kritiese literatuur studie. Breuke is ‘n saamgestelde konsep en kan vir hierdie rede op verskillende wyses geïnterpreteer word, afhangende van die area van studie in wiskunde. Die konsep van ‘n breuk bestaan uit vyf sub-konstrukte, naamlik deel-van-‘n-geheel, ‘n verhouding, operateur, kwosiënt en meting (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). Hierdie tesis begin met ‘n bespreking oor die agtergrond van hierdie studie en die belangrikheid daarvan. Daarna word die faktore wat bydra tot die verstaan van die breuk konsep. Dit word gevolg deur ‘n uitbreiding op die vyf verskillende sub-konstrukte en waar hierdie verskillende sub-konstrukte in die hoërskool kurrikulum voorkom. Die bevinding bied ‘n paar implikasies vir onderrig. Hierdie studie fokus nie op die ontwerp van enige take of ander leermateriaal vir ‘n intervensie program nie, maar konsentreer op die belangrike kwessies rondom breuke. My hoop is dat die bevindinge van hierdie studie implikasies inhou vir wiskunde onderwysers se professionele ontwikkeling deur hul te motiveer om nuwe leerondersteuningsmateriaal te ontwikkel en die aanbieding van breuke in klaskamers aan te pas sodat die begrip van breuke by leerders ten volle ontwikkel kan word.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Nurses, midwives and clinical officers referred to as Mid-level Practioners (MLPs) play an important role in the health care system especially in rural Africa. With particular reference to rural Uganda, due to the large shortage of doctors, MLPs handle most of the duties usually meant for doctors, at health centre IV(s). From 2009 to 2011, two training interventions of MLPs were performed at 36 sites in Uganda by the Integrated Infectious Disease Capacity Building Evaluation (IDCAP). The two interventions were: Integrated Management of Infectious Diseases (IMID) and On-site Support Services (OSS) which aimed at improving MLPs’ case management for four diseases: HIV, TB, pneumonia and malaria. In this thesis, we have developed three mathematical models to investigate the effect of the two training interventions on these infectious diseases. All the models are formulated using systems of ordinary differential equations which are structured in three age groups: [0, 5), [5, 14) and [14, 50). We explored the effect of the two training interventions in the context of malaria-pneumonia, HIV-TB co-infections and the four diseases together. Our analysis shows that: i) For malaria-pneumonia, both IMID and the combination of IMID and OSS reduce the number of cases, deaths and prevalence of disease but have no effect on the incident episodes of disease. ii) Results from the HIVTB model propose that HIV and TB testing are important steps in quality of health care and are capable of offsetting slightly negative effects of reduction in ART enrollment and provision of treatment. iii) The HIV-TB-malaria-pneumonia (HTMP) model concurs with the results of the first two models and its results demonstrate that high coverage levels of the training interventions increase the positive effects that the interventions have on mortality and morbidity. Overall, our results suggest that training of MLPs is much more effective for the short term duration diseases such as malaria and pneumonia, where the baseline values for most of the performance indicators are ≥ 0.6, but not so much for long term duration diseases such as HIV and TB, whose baseline values for most of the performance indicators are < 0.6. The results further highlight that problems such as case detection and drug stock-outs need to be addressed in order for training to have substantial impact, especially in instances where the performance indicator proportions are low. / AFRIKAANSE OPSOMMING: Verpleegsters, vroedvroue en kliniese beamptes wat gesamentlik na verwys word as midvlak praktisyns (MVPs) , speel n belangrike rol in die gesondheidsorg sisteem, veral in landelike dele van Afrika. Met spesifieke verwysing na gesondheid sentrums in Uganda, waar daar te min dokters is, hanteer MVPs die meeste van die pligte wat eintlik deur dokters verrig moet word. Vanaf 2009 tot 2011 is twee opleidingsprogramme vir MVPs by 36 fasiliteite in Uganda deur die Integrated Infectious Disease Capacity Building Evaluation (IDCAP) organisasie aangebied. Die twee programme staan bekend as: Integrated Management of Infectious Diseases (IMID) and On-site Support Services (OSS). Beide die programme stel ten doel om die MVPs se pasint bestuur vir die siektes MIV, tuberkulose (TB), longontsteking en malaria te verbeter. Drie wiskundige modelle word in hierdie tesis ontwikkel om die effek van die opleidingsprogramme op hierdie oordraagbare siektes te ondersoek. Al die modelle word geformuleer deur gebruik te maak van stelsels van gewone differensiaal vergelykings wat gestruktureer is in drie ouderdomsgroepe: [0, 5), [5, 14) en [14, 50). Die effek van die opleidings programme word in die konteks van longontstekingmalaria mede-infeksie, MIV- TB mede-infeksie en al vier siektes gelyk, ondersoek. Die analise wys dat: i) Vir longontsteking-malaria mede-infeksie het beide IMID en die kombinasie van IMID en OSS die aantal siekte-gevalle, sterftes en die prevalensie van die siektes verminder, maar het geen effek op die insidensie van siekte-gevalle nie. ii) Resultate van die MIV-TB model dui aan dat MIV en TB toetsing n belangrike aspek van die gehalte van sorg is en dat dit die effense negatiewe effek van die afname in ART inskrywing en voorsiening van behandeling, teenstaan. iii) Die MIV-TB-longontsteking-malaria model (HTMP) stem ooreen met die resultate van die bogenoemde twee modelle en demonstreer dat ho dekking van die opleidingsprogramme die positiewe effek van die programme op mortaliteit en morbiditeit verhoog. In geheel stel die resultate van hierdie studie voor dat die opleiding van MVPs baie meer effektief is vir die korttermyn siektes soos malaria en longontsteking waarvoor die meeste van die beginwaardes van die prestasie-aanwysers ≥ 0.6 is, maar nie soveel vir lang-termyn siektes soos MIV en TB waarvoor die meeste van die beginwaarde van die prestasie-aanwysers < 0.6 is. Die resultate dui verder aan dat opleiding nie voldoende is wanneer die prestasie-aanwysers < 0.6 is nie en dat probleme soos die opsporing van siekte-gevalle en n gebrek aan medisyne by die klinieke aangespreek moet word vir opleiding om aansienlike impak te hê.

The pass rate of first-year university mathematics students at the University of KwaZulu-Natal (Pietermaritzburg Campus) has been low for many years. One cause may be weak algebra skills. At the time of this study, revision of high school algebra was not part of the major first year mathematics course. This study set out to investigate if it would be worthwhile to spend tutorial time on basic algebra when there is already an overcrowded calculus syllabus, or if students refresh their algebra skills sufficiently as they study first year mathematics. Since it was expected that remediation of algebra skills would be found to be worthwhile, two other questions were also investigated: Which remediation strategy is best? Which errors are the hardest to remediate? Five tutorial groups for Math 130 were randomly assigned one of four remediation strategies, or no remediation. Three variations of using cognitive conflict to change students’ misconceptions were used, as well as the strategy of practice. Pre- and post-tests in the form of multiple choice questionnaires with spaces for free responses were analysed. Comparisons between the remediated and non-remediated groups were made based on pre- and post-test results and Math 130 results. The most persistent errors were determined using an 8-category error classification developed for this purpose. The best improvement from pre- to post-test was 12.1% for the group remediated with cognitive conflict over 5 weeks with explanations from the tutor. Drill and practice gave the next-best improvement of 8.1%, followed by self-guided cognitive conflict over 5 weeks (7.8% improvement). A once-off intervention using cognitive conflict gave a 5.9% improvement. The group with no remediation improved by 2.3%. The results showed that the use of tutorintensive interventions more than doubled the improvement between pre-and post-tests but even after remediation, the highest group average was 80%, an unsatisfactory level for basic skills. The three most persistent errors were those involving technical or careless errors, errors from over-generalising and errors from applying a distorted algorithm, definition or theorem. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2009.