<p>  </p> <p>Radian angle measure is central to learning trigonometry, with researchers providing evidence that a coherent understanding of radian contributes to a coherent understanding of trigonometric and inverse trigonometric functions. However, there are few opportunities for students to engage with curricular situations that involve radian angle measure. The purpose of this dissertation is to explore and provide insights into preservice mathematics teachers’ (PMTs’) conceptions of radian angle measure using three curricular situations. The first chapter reviews the relevant literature, which reported that PMTs’ conceptions of radian angle measure involve angles measured in terms of π, in relation to degrees, and in relation to the unit circle. In chapter two, I explored PMTs’ conceptions of radian angle measure using textbook representations. Seven PMTs participated in a think-aloud semi-structured interviews, where they defined radian angle measure from six textbook diagrams of radian, including a diagram of the unit circle. In chapter three, building on literature that reported that PMTs’ conceptions of radian angle measure involve relating radian to degrees, I explored how PMTs conceptualize this relationship. Five PMTs participated in semi-structured interviews, where they described radian angle measure given the angle measure in degrees. In chapter four, I explored the PMTs’ conceptions of radian angle measure given a novel context. Four PMTs participated in semi-structured virtual interviews, where they engaged with a digital activity that involves radian angle measure in the context of light reflection. Some of the dissertation’s findings align with previous research, where PMTs’ conceptualized radian angle measure in relation to the unit circle. However, this dissertation provides empirical evidence of why the PMTs refer to the unit circle. The PMTs acknowledged knowing the unit circle from memorization, but also explained that the purpose for using the unit circle is efficiency. At the same time, the PMTs acknowledged limitations in the unit circle and in their conceptions of it. Overall findings from the dissertation demonstrate the complexity of PMTs’ conceptions of radian angle measure. The PMTs’ conceptions were reported as concept definitions, ways of thinking, and spatial ways of thinking. The PMTs demonstrated flexibility with reasoning about radian angle measure using foundational conceptions in learning higher mathematics topics (e.g., proportional reasoning concepts, spatial ways of thinking). By positioning the PMTs as knowers and thinkers with valuable insights to provide, I was able to uncover and report a collection of conceptions that were demonstrated by PMTs when a curricular situation involved radian angle measure. The findings from this dissertation extend existing research that explored conceptions of angle measure and radian angle measure by reporting PMTs’ conceptions of radian angle measure given three different curricular situations. While there is still much that needs to be investigated about complexities in PMTs’ conceptions of radian angle measure, this dissertation represents one step toward providing insights about those complexities. </p>

To investigate quantum confinement effects on silicon (Si) light source electroluminescence (EL) properties like quantum efficiency, external power efficiency and spectral emission, thin Si finger junctions with nanometre-scale dimensions were designed and manufactured in a fully customized silicon-on-insulator (SOI) semiconductor production technology. Since commonly available photolithography is unusable to consistently define and align nanometre-scale line-widths accurately and electron-beam lithography (EBL) by itself is too time-expensive to expose complete wafers, the wafer manufacturing process employed a selective combination of photolithography and EBL. The SOI wafers were manufactured in the clean-rooms of both the Carl and Emily Fuchs Institute for Microelectronics (CEFIM) at the University of Pretoria (UP) and the Georgia Institute of Technology’s Microelectronic Research Centre (MiRC), which made a JEOL JBX-9300FS electron-beam pattern generator (EPG) available. As far as is known this was the first project in South Africa (and possibly at the MiRC) that employed EBL to define functional nanometre-scale semiconductor devices. Since no standard process recipe could be employed, the complete design and manufacturing process was based on self-obtained equipment characterization data and material properties. The manufacturing process was unprecedented in both the CEFIM and MiRC clean-rooms. The manufacture of nanometre-scale Si finger junctions not only approached the manufacturing limits of the employed processing machinery, but also had to overcome undesirable physical effects that in larger-scale semiconductor manufacture usually are negligible. The device design, mask layout and manufacturing process therefore had to incorporate various material, equipment limitation and physical phenomena like impurity redistribution occurring during the physical manufacturing process. Although the complicated manufacturing process allowed many unexpected problems to occur, it was expected that at least the simple junction breakdown devices be functional and capable of delivering data regarding quantum confinement effects. Although due to design and processing oversights only 29 out of 505 measured SOI light sources were useful light emitters, the design and manufacture of the SOI light sources was successful in the sense that enough SOI light sources were available to conduct useful optical characterization measurements. In spite of the fact that the functional light sources did not achieve the desired horizontal (width) confinement, measured optical spectra of certain devices indicate that vertical (thickness) confinement had been achieved. All spectrometer-measured thickness-confined SOI light sources displayed a pronounced optical power for 600 nm < λ < 1 μm. The SOI light source with the highest optical power output emitted about 8 times more optical power around λ = 850 nm than a 0.35 μm bulk-CMOS avalanche light-source operating at the same current. Possible explanations for this effect are given. It was shown that the buried oxide (BOX) layer in a SOI process could be used to reflect about 25 % of the light that would usually be lost to downward radiation back up, thereby increasing the external power efficiency of SOI light sources. This document elaborates on the technical objectives, approach, chip and process design, physical wafer manufacture, production process control and measurement of the nanometre-scale SOI light sources. Copyright / Dissertation (MEng)--University of Pretoria, 2010. / Electrical, Electronic and Computer Engineering / unrestricted