Magister Educationis - MEd / Euclidean geometry was recently re-introduced as a compulsory topic in the Mathematics Curriculum for learners in the Further Education and Training (FET) band in 2012. The diagnostic analysis reports on the National Senior Certificate (NSC) Mathematics Paper 2 examinations since 2014 has repeatedly expressed concern of the poor performance of leaners in proof and reasoning items linked to circle geometry. Various efforts have been made to examine the composition of the curriculum to find ways of motivating learners in the study of circle geometry and enhancing their performance but not much has been realized. The use of technology or cooperative learning approaches for the teaching of geometry is beneficial for pedagogical purposes, particularly for improving learners’ performance in geometry. Hence, this study investigated the impact of using technology through cooperative learning on learners’ performance on grade circle 11 geometry. It was thus an attempt to focus on blending these two teaching methods with an emphasis on the use of technology. The research took place at a Khayelitsha school and the scope of technology was limited to using a mathematical computer programme called Heymath. This research was grounded on the cognitive level framework that is used by the Department of Basic Education (DBE) in the setting of National Senior examination mathematics papers, as well as the set of social constructivist views of mathematics teaching and learning. In the case of the latter, both social constructivism and cognitive constructivism views were considered and applied for the purposes of this study. Using a positivist paradigm, this convergent parallel mixed methods study employed a quasi-empirical design, where the control group consisted of a group 26 grade 11 learners who were comparable to the group of 27 grade learners that made up the experimental group.

The development of easily fabricated, highly stable DNA-based microarray and continuous flow concentrating devices is vital for several biomedical and environmental applications. Nucleic acid biosensors can be used for genetic analysis, disease diagnosis, drug discovery, food and water quality control and more, however methods of fabrication are tedious, and the longevity of sensors is compromised by the fragility of the sensing component. In this report, the fabrication and characterization of two biosensing modalities – microarrays and microgels – composed of Rolling Circle Amplification (RCA) product in poly(oligoethylene glycol methacrylate) (POEGMA) hydrogels are investigated. RCA product microarrays were developed by the sequential printing of aldehyde and hydrazide functionalized POEGMA precursors on nitrocellulose paper, exploiting rapid gelling via hydrazone crosslinking to generate thin film hydrogel sensing arrays. POEGMA/RCA product microgels for affinity column applications were synthesized using an inverse emulsion polymerization technique. Inkjet printing evenly deposited RCA product in all wells, with POEGMA effectively stabilizing DNA on the cellulose substrate. Hybridization of complementary probe to the encapsulated RCA product was optimized, yielding a signal to noise ratio of ~4 for a large range of probe concentrations. Microgels were successfully synthesized in the size range of 10-60 μm diameter, and a linear model that can accurately predict size based on initiator and emulsifier concentration was developed. The encapsulation efficiency of RCA product in different sized microgels was explored, with larger microgels entrapping more product and the highest encapsulation efficiency calculated at 56%. These results demonstrate that POEGMA hydrogels can be utilized to encapsulate and stabilize RCA product in two distinct structures, providing a basis for the development of easily fabricated biosensors for more specific applications. / Thesis / Master of Applied Science (MASc)

This is an exploration of the circle through the lens of the African vernacular. The geometric circle is inherently rule based and requires only objective, geometric manipulation. The archetypal circle is subjective, an interpretation based on given characteristics of roundness. The archetypal circle is where abstraction and exploration occur. The plan of traditional African vernacular architecture is a circle, and as such, was the starting point. Rational manipulations of the geometric circle made the building. The mandala studies were subjective manipulations of the archetypal circle made to explore and develop the architecture. This became a Market in Nairobi. / Master of Architecture

This thesis explores architecture's role as host and its relationship with guest through the research and design of a new youth hostel for Washington, D.C. The etymological duality of host is confronted in the project's structure, as well as its liminal spaces. This new hostel offers comfort, protection, affordable accommodation, learning opportunities, and moments of camaraderie to young guests of the nation's capital. / Master of Architecture

To Bach, the Circle of Fifths was the language of the universe. Similar to the constellations used to understand the sky, the Circle of Fifths is a visualization device to understand the fundamental concepts of key signatures, which are the foundation for music. The Circle of Fifths is a guide for writing music because its structure helps compose and harmonize melodies, build chords, and move to different keys within a composition. The Octave is the most significant key signature because it completes the circle of fifths. The Octave, if in perfect tune will create an overtone, which is a tune unable to be created on its own. The movement through the Circle of Fifths led to a contemplation described by Pythagoras as "Music of the Spheres" or meeting between heaven and earth, between spiritual and material realms. The Well Tempered Building uses the Circle of Fifths as the underlying geometry for the foundation of the conservatory. Proportions from the Circle of Fifths, including the Octave, shaped the conservatory making the musicians, audience, sound, light, water and air tuned to each other. / Master of Architecture

This paper will be concerned with the convergence of complex power series.

<p>The purpose of this thesis is to gain insight into educators’ views and work with circle time in preschool. To find out, I have referred to the following questions: <em>How is your circle time performed in preschool? What objectives have teachers with the circle time? What impact has your circle time for the children in preschool?</em></p><p>I have chosen to carry out qualitative interviews with three preschool teachers. After the interviews, I compared the educators’ response to previous research.</p><p>The survey shows that the circle time is one of the day´s highlights for preschool children. It also shows that the most important thing with circle time is to create community, security in the group, and a fun time together. The educators also believe that circle time is an important part in the children's learning process, especially when it comes to social learning. In my study, I can see that the content in the circle time has not changed much during the preschool development; the circle time consists mostly of roll call, rhyme, rhymes and songs.</p><p>Keywords: preschool, circle time, educator, children, circle</p>

The purpose of this thesis is to gain insight into educators’ views and work with circle time in preschool. To find out, I have referred to the following questions: How is your circle time performed in preschool? What objectives have teachers with the circle time? What impact has your circle time for the children in preschool? I have chosen to carry out qualitative interviews with three preschool teachers. After the interviews, I compared the educators’ response to previous research. The survey shows that the circle time is one of the day´s highlights for preschool children. It also shows that the most important thing with circle time is to create community, security in the group, and a fun time together. The educators also believe that circle time is an important part in the children's learning process, especially when it comes to social learning. In my study, I can see that the content in the circle time has not changed much during the preschool development; the circle time consists mostly of roll call, rhyme, rhymes and songs. Keywords: preschool, circle time, educator, children, circle

In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided.

AN ABSTRACT OF THE DISSERTATION OFDinush Lanka Panditharathna Jayasooriya Arachchilage, forthe Doctor of Philosophy degree in MATHEMATICS, presented on June 22, 2020 at SouthernIllinois University Carbondale.TITLE: SPIN NETWORK EVALUATION AND THE ASYMPTOTIC BEHAVIORMAJOR PROFESSOR: Dr. Jerzy KocikGraphically, a spin network is a trivalent graph with weights on each edge. At anyof the vertices, the sum of all three weights is even and the sum of any two weights isgreater than or equal to the remaining weight. If the spin network has no free ends, thenwe can evaluate the spin network. Here, we propose a method to evaluate some basic spinnetworks using the idea of Stirling triangle.Tangent circles with integer curvatures are a natural source to make a spin network.In particular, there are spin networks corresponding the Apollonian circle packing and theFord circle packing. We obtain the recurrence relations using the Descartes circle theoremand we evaluate the Apollonian spin network and the Ford circle spin network. We alsodiscuss the asymptotic behavior of the Ford circle spin network.