Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms. / PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>

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Estudos mostram que motivar professores e alunos para o ensino e aprendizagem de números complexos no Ensino Médio pode ser uma tarefa difícil. Esse trabalho investiga as causas dessa dificuldade e propõe uma abordagem de ensino dos números complexos baseada em história, aplicações e fractais. Além disso, apresenta alguns recursos digitais para explorar lições e atividades mais interativas dos conceitos matemáticos envolvidos. / Literature shows that motivating teachers and students for studying complex numbers in high school can be a challenging task. This work investigates such issue and proposes an approach for teaching complex numbers based on their history, applications and fractals. In addition it provides some digital resources to explore interactive lessons and activities of this content.

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<p dir="ltr">Origami has emerged as a design paradigm to realize morphing structures with rich kinematic and mechanical properties. Biological examples augment the potential folding design space by suggesting intriguing routes for achieving and expanding crease patterns which traditional origami laws are unable to capture. Specifically, spring origami theory exploits the material system architecture and energy storage mechanism of the earwig wing featuring one of the highest folding ratios in the animal kingdom (1:18), minimal energy required for deployment and collapse of the wing, and bistability locking the wing in closed, and open configurations for crawling through tunnels, and flight, respectively. The central mechanism responsible for bistability in the wing features a non-developable crease pattern with a non-zero Gaussian curvature. Reconfiguring, or even flattening a structure with such an intrinsic property requires stretching or tearing; soft, rubbery material found in the creases of the central mechanism allows for stretching enabling shape transformations between open and closed states without tearing. In the first part of this thesis, such characteristics are transferred to a synthetic bistable soft robotic gripper leveraging the shape adaptability and conformability exhibited by the biological organism to minimize actuation energy. This is achieved by integrating soft, flexible material in the bioinspired gripper that allows kinematically driven geometries to grasp and manipulate objects without continuous actuation. Secondly, the stiffening effect from spring origami is utilized in a bioinspired wing for an aerial--aquatic robot. Transitions between air and sea in multimodal robots is challenging, however, a structurally efficient and multifunctional membrane is developed to increase locomotive capabilities and longer flights. This is motivated by the flying fish's locomotive modules and origami design principles for deployment and folding. Additionally, to keep the wing in a stiff state while gliding, spring origami bistable units are integrated into the membrane inducing self-stiffening and a global curvature reducing energy expenditure while generating lift. While the previous examples present solutions to adaptive manipulation and membrane multifunctionality, once programmed, their shapes are fixed. In the third application, a class of multistable self-folding origami architectures that are reprogrammable post fabrication are presented. This is achieved by encoding prestrain in bilayer creases with anisotropic shrinkage that change shape and induce a local curvature in the creases in response to external stimuli. The topology of the energy landscapes can thus be tuned as a function of the stimulation time and adaptable post fabrication. The proposed method and model allows for converting flat sheets with arranged facets and prestrained mountain-valley creases into self-folding multistable structures. Lasty, encoding crease prestrain is leveraged to manufacture a biomimetic earwig wing featuring the complex crease pattern, structural stability, and rapid closure of the biological counterpart. The presented method provides a route for encoding prestrain in self-folding origami, the multistability of which is adaptable after fabrication.</p>