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This research focuses on the development of the concept images of the derivative concept of students enrolled in the in-service programme 'Advanced Certificate in Education' at University of KwaZulu-Natal, Pietermaritzburg campus. In addition, two qualified teachers not enrolled in the programme were included. A theoretical framework which describes the derivative as having three layers - the ratio, limit and function layers - that can be represented by a variety of representations - graphical, rate, physical and symbolic - is used to analyse the development of the students' concept images. This framework was adopted from previous research, but expanded to allow for situations where a student's concept image did not fall into any of the layers or representations. In those cases, the concept image was classified into the non-layer section or the instrumental understanding section. The findings of this research show that of the five ACE students who were interviewed, only one had a profound concept image in all the three layers of the derivative, with multiple representations as. well as connections among representations within the layers. This one student also passed the calculus module with a distinction. The other four students had the ratio layer and graphical representation profound in their concept images, while the other layers and representations were pseudostructural with very few connections. Two of these students passed the calculus module while the other two failed. All the students showed progression in their concept images, which can only be credited to the ACE calculus module. However, it is clear that even upon completion of this module, many practicing teachers have concept images of the derivative which are not encompassing all the layers and more than one or two representations. With the function layer absent, it can be difficult to make sense of maximization and minimization tasks. With the limit layer absent or pseudo-structural , the concept Itself and the essence of calculus escapes the teachers - and therefore also will be out of reach of their learners. / Thesis (M.Ed.)-University of KwaZulu-Natal, Pietermaritzburg, 2006.

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One can give various rigorous definitions to the notion of "functional calculus", but a functional calculus is ultimately just a mathematically meaningful way of talking about an operator f(T), where, T is an operator and f is a function. This thesis is concerned with this concept and with one of its applications, the finding of bounds for powers of operators. It is actually this very application that has prompted the entire investigation presented here. This application is relevant to various fields, such as the numerical analysis of PDE and Markov chains. Chapter I presents various abstract approaches to the notion of "functional calculus" that are given content by three major examples: the Riesz-Dunford functional calculus, the Weyl functional calculus and the functional calculus for sectorial operators. Chapter II investigates various conditions that ensure power boundedness for operators, putting the Tadmor-Ritt condition at its center. The Riesz-Dunford calculus is instrumental for the proofs in this chapter. Chapter III investigates Pascale Vitse's use of Cauchy-Stieltjes integrals and their multipliers for obtaining bounds on powers of operators; the chapter closes with an investigation of partially power bounded operators.

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The purpose of this study was to determine the effectiveness of early diagnosis and a summer program to strengthen precalculus skills before students enrolled in Engineering Calculus I. A meta-synthesis of interventions to increase success in college calculus was conducted, with a meta-analysis of studies that contained sufficient quantitative data to calculate Hedge's g effect sizes. Content validity for a mathematics placement exam was confirmed by an expert panel, and internal consistency of scores from 2008-2011 was verified using Cronbach's alpha. Effectiveness of a summer program to strengthen precalculus skills was measured by Hedge's g effect size. Results of content analysis of surveys given to tutors and students in the summer program were presented. ANOVA was used to compare mean GPA's of participants and nonparticipants of the summer program. The meta-synthesis revealed that numerous strategies, some in precalculus and some in calculus, were successful for increasing success in college calculus. For the studies in the meta-analysis, the highest effect sizes were found in studies that used a more comprehensive approach (e.g., collaborative groups and projects) rather than a single strategy (e.g., computer skills practice). An expert panel determined that the exam was a good measure of requisite knowledge for calculus. One question was considered unnecessary for calculus and was not of a type addressed in precalculus and was eliminated from further analysis. Cronbach's alpha was consistently above .8 for each year's scores 2008-2011 and for each subset of scores by gender, ethnicity, and selected majors for 2008-2011. The 122 students who participated in the summer program increased the average score by 6.45 points (total of 33), with 81% of the students raising their scores above the cut score to take Engineering Calculus I. Results of ANOVA to compare mean GPA's for students in the summer program and students who did not participate, both with placement exam scores in the range 16 to 21, inclusive, showed no significant difference. The summer program was successful in allowing some students the opportunity to strengthen their precalculus skills and take Engineering Calculus I a semester earlier than the control group.

iii, 114 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1976