Dissertations / Theses on the topic 'Pre-geometry'

Virginia architecture of the seventeenth and eighteenth centuries prior to the American Revolution has distinctive geometries which determine proportion. The square, root-two rectangle and equilateral triangle are the figures which establish most proportions. Plans and elevations underwent a development based on a rational method of incorporating the figures into a coherent building. This investigation establishes the use of geometry as a starting point for the culmination of many elements which together composed a building.
Master of Architecture

Includes bibliography.
This dissertation examines the impact of a geometry course on pre-service teachers' levels of understanding of geometry. It is located within the Van Hiele model of geometric development which provides a conceptual framework to assess and analyse the progress of students. The study was conducted at a College of Education which prepares teachers for primary school teaching. It involved 26 second year, pre-service teachers over a 9 week period in a geometry course. The students were assessed for their levels of understanding before and after the course using a diagnostic instrument developed by Mayberry (1981) to assess the Van Hiele levels of pre-service teachers. An in-depth investigation of 8 students provided further insight into students' levels of understanding through course work and assignments. The overall findings of the study revealed that the majority of students had low levels of understanding of geometry before and after the course. However, those who had taken high school mathematics performed better than those without, although few managed to reach the higher levels. The results highlight the need for teachers to develop higher levels of understanding before being able to teach and design activities that are appropriate for learners. Little improvement in performance of learners on national and international competitions will occur while teachers continue to register low levels of understanding. It is crucial that time, resources and training are provided to all teachers if effective change is to occur in the mathematics classroom.

Thesis (Ph. D.)--University of North Carolina at Chapel Hill, 2009.
Title from electronic title page (viewed Jun. 26, 2009). "... in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Education." Discipline: Education; Department/School: Education.

Pre-existing structures are ubiquitous throughout the lithosphere, having been imparted during multiple, repeated phases of rifting and orogenesis. These pre-existing structures exist across all scales, from the microstructural to transcontinental, and are present at various structural levels throughout the lithosphere. These pre-existing structures may strongly influence the geometry and evolution of rift systems. Understanding the influence of these pre-existing structures on rifting requires a detailed understanding of their 3D geometry and lithological properties. Seismic reflection data offers detailed imaging of the subsurface over large areas, allowing both pre-existing structures and the overlying rift system to be mapped and analysed. However, in seismic reflection data, it is often difficult to image and constrain the geometry of pre-existing structures which are situated within crystalline basement, at great depths within the lithosphere, or typically display a vertical attitude. In this thesis, I use borehole-constrained 2D and 3D seismic reflection data, covering a 450 x 150 km area offshore southern Norway, to analyse the influence of pre-existing structure on the geometry and evolution of the overlying rift system. In the west of the area, I constrain the offshore continuations and 3D geometry of the Caledonian thrust belt and associated orogenic collapse-related Devonian shear zones within crystalline basement. I document multiple phases of reactivation along these shear zones as well as a range of interactions with rift-related faults. To the east, I identify the upper crustal component of the Sorgenfrei-Tornquist Zone (STZ), a sharp change in lithospheric thickness at sub-crustal depths, represented by a crustal-scale fault system defining the Farsund Basin. Multiple phases of reactivation, including sinistral strike-slip and dextral transtension, are documented along these faults, representing the reactivation of the STZ to accommodate variably oriented regional stress fields. Further east, I identify a Carboniferous-Permian aged dyke swarm, linked to a regional Large Igneous Province (LIP), which represents the first dyke swarm to be identified on seismic reflection data, following passive post-emplacement rotation to shallower dips. Dykes within the swarm form a mechanical anisotropy and thus dictate the geometry of kinematically-linked overlying faults. Geomorphological analyses within the Farsund Basin document a markedly different facies distribution to that in the west, implying a difference in tectono-stratigraphic evolution. I note a major change in the tectonic regime and style of pre-existing structure control between the east of the area, dominated by Proterozoic basement, the dyke swarm and the Sorgenfrei-Tornquist Zone, and the west of the area, dominated by Caledonian orogeny and Devonian orogenic collapse-related upper crustal basement structures. This thesis documents how the presence of different pre-existing structures, and their potential reactivation, can exert a fundamental influence over the evolution of rift systems, displaying a range of relationships with the rift-related faults.

The purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals. The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar quadrilaterals with fixed side lengths) has been well-studied [5], [8], [10]. The symplectic geometry of these spaces has been studied by Kapovich and Millson [6], but the Riemannian geometry of these spaces has not been thoroughly examined. We study paths in the moduli space and the pre-moduli space. We compare intraplanar paths between points in the moduli space to extraplanar paths between those same points. We give conditions on side lengths to guarantee that intraplanar motion is shorter between some points. Direct applications of this result could be applied to motion-planning of a robot arm. We show that horizontal lifts to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine exactly which collections of side lengths allow holonomy.
Ph. D.

This study aimed to investigate the level of understanding of Euclidian geometry, in terms of theoretical knowledge as well as its problem-solving application, in pre-service mathematics education (PME) students at the University of Pretoria. In order to do so, a one group pre-test/ post-test procedure was conducted around an intensive geometry module, and a representational group of students was interviewed before and after the module to discuss their high school experiences of learning geometry and to analyse their attitudes towards the subject. The van Hiele Theory of Levels of Thought in Geometry was used as the theoretical framework for this study. The PME students in this study, prior to their completion of the geometry module, lacked the content knowledge, skills and insight in Euclidian geometry that is expected at matric level (Level 3). The pre-test results revealed that half the group could only be classified as being on Level 0. By the time the post-test was written, 60% of the group had moved onto Level 1 as their maximum competence level. This implies that these students were all brought to greater insight by the teaching they received during the geometry module. However, the overall improvement in the group as revealed in the post-test results, consisted of an upward movement of only one level. Therefore, the geometry module offered did not bring about sufficient improvement for these students to be able to teach geometry adequately (Level 3 is required). The students who were interviewed for this study uniformly expressed their dislike or fear of Euclidian geometry in general, but described the positive change in their attitude during the course of the module because of the way it was presented. Training of students for a career as mathematics educators which includes an in-depth van Hiele-based geometry module would facilitate the acquisition of insight and relational understanding.
Dissertation (MEd)--University of Pretoria, 2008.
Curriculum Studies
MEd
unrestricted

The purpose of this study was to investigate plane geometry problem solving strategies of pre-service elementary mathematics teachers in technology and paper-and-pencil environments after receiving an instruction with GeoGebra. Qualitative research strategies were used to investigate teacher candidates&lsquo
solution strategies. The data was collected and analyzed by means of a multiple case study design. The study was carried out with 7 pre-service elementary mathematics teachers. The main data sources were classroom observations and interviews. After receiving a three-week instructional period, the participants experienced data collection sessions during a week. The data was analyzed by using records of the interviews, answers to the instrument, and transcribing and examining observation records. Results revealed that the participants developed three solution strategies: algebraic, geometric and harmonic. They used mostly algebraic solutions in paper-and-pencil environment and v geometric ones in technology environment. It means that different environments contribute separately pre-service teachers&lsquo
mathematical problem solving abilities. Different from traditional environments, technology contributed students&lsquo
mathematical understanding by means of dynamic features. In addition, pre-service teachers saved time, developed alternative strategies, constructed the figures precisely, visualized them easily, and measured accurately and quickly. The participants faced some technical difficulties in using the software at the beginning of the study but they overcome most of them at the end of instructional period. The results of this study has useful implications for mathematics teachers to use technology during their problem solving activities as educational community encourages to use technology in teaching and learning of mathematics.

In this thesis a method is presented to provide a 3D visualization of the human kidney and surrounding tissue during kidney surgery. The method takes advantage of the high detail of 3D X-Ray Computed Tomography (CT) and the high time resolution of Ultrasonography (US). By extracting the geometry from a single preoperative CT scan and animating the kidney by tracking its position in real time US images, a 3D visualization of the surgical volume can be created. The first part of the project consisted of building an imaging phantom as a simplified model of the human body around the kidney. It consists of three parts: the shell part representing surrounding tissue, the kidney part representing the kidney soft tissue and a kidney stone part embedded in the kidney part. The shell and soft tissue kidney parts was cast with a mixture of the synthetic polymer Polyvinyl Alchohol (PVA) and water. The kidney stone part was cast with epoxy glue. All three parts where designed to look like human tissue in CT and US images. The method is a pipeline of stages that starts with acquiring the CT image as a 3D matrix of intensity values. This matrix is then segmented, resulting in separate polygonal 3D models for the three phantom parts. A scan of the model is then performed using US, producing a sequence of US images. A computer program extracts easily recognizable image feature points from the images in the sequence. Knowing the spatial position and orientation of a new US image in which these features can be found again allows the position of the kidney to be calculated. The presented method is realized as a proof of concept implementation of the pipeline. The implementation displays an interactive visualization where the kidney is positioned according to a user-selected US image scanned for image features. Using the proof of concept implementation as a guide, the accuracy of the proposed method is estimated to be bounded by the acquired image data. For high resolution CT and US images, the accuracy can be in the order of a few millimeters.

The 9/11 Act mandates the inspection of 100 % of cargo shipments entering the U.S by 2012 and 100 % inspection of air cargo by March 2010. So far, only 5% of inbound shipping containers are inspected thoroughly while air cargo inspections have fared better at 50 %. Government officials have admitted that these milestones cannot be met since the appropriate technology does not exist. This research presents a novel planar solid phase microextraction (PSPME) device with enhanced surface area and capacity for collection of the volatile chemical signatures in air that are emitted from illicit compounds for direct introduction into ion mobility spectrometers (IMS) for detection. These IMS detectors are widely used to detect particles of illicit substances and do not have to be adapted specifically to this technology. For static extractions, PDMS and sol-gel PDMS PSPME devices provide significant increases in sensitivity over conventional fiber SPME. Results show a 50-400 times increase in mass detected of piperonal and a 2- 4 times increase for TNT. In a blind study of 6 cases suspected to contain varying amounts of MDMA, PSPME-IMS correctly detected 5 positive cases with no false positives or negatives. One of these cases had minimal amounts of MDMA resulting in a false negative response for fiber SPME-IMS. A La (dihed) phase chemistry has shown an increase in the extraction efficiency of TNT and 2,4-DNT and enhanced retention over time. An alternative PSPME device was also developed for the rapid (seconds) dynamic sampling and preconcentration of large volumes of air for direct thermal desorption into an IMS. This device affords high extraction efficiencies due to strong retention properties under ambient conditions resulting in ppt detection limits when 3.5 L of air are sampled over the course of 10 seconds. Dynamic PSPME was used to sample the headspace over the following: MDMA tablets (12-40 ng detected of piperonal), high explosives (Pentolite) (0.6 ng detected of TNT), and several smokeless powders (26-35 ng of 2,4- DNT and 11-74 ng DPA detected). PSPME-IMS technology is flexible to end-user needs, is low-cost, rapid, sensitive, easy to use, easy to implement, and effective.

For all the successes of the two edifices of modern physics, quantum theory and Einstein's relativity, a fundamental description of the Universe as a whole -- a theory that informs as to the true nature of reality -- has continued to elude science. This thesis describes the development and evolution of a new paradigm called Process Physics, a radical information-theoretic modelling of reality. It is argued that the failure of the extant approaches in physics is the direct consequence of limitations stemming from the mathematization, language and methodology of theoretical physics: the limitations of the postulated background spatial concepts and geometric modelling of time, the limitations of quantum theory in its failure to account for the measurement process and classicality; and the limitations of formal systems. In contrast, Process Physics utilizes the limitations of logic first identified by Godel and asserts the priority of process and relational endophysics, realized via a stochastic, autopoietic bootstrap system whose properties emerge a posteriori rather than being assumed a priori. The work is arranged in two parts. Part I discusses the historical, philosophical, and metaphysical foundations of physics to consider how the prevailing views in modern physics arose and what this revealed and contributed to the development of Process Physics. Part II describes the fundamentals of the new theory and its implementation, and demonstrates the viability of looking outside the current paradigms by showing that Process Physics yields unified emergent phenomena that permit an understanding of fundamental processes and penultimately motivate both quantum theory and relativity as relevant higher-level descriptors within their respective domains.

L’estudi del coneixement matemàtic i les creences de l’eficàcia de l’ensenyament de les matemàtiques en la formació inicial dels futurs mestres és fonamental, ja que influencia el rendiment acadèmic dels seus estudiants. L’objectiu d’aquesta tesi és estudiar tant el coneixement matemàtic inicial dels futurs mestres com també les seves creences sobre l’eficàcia matemàtica i la seva actitud envers les matemàtiques. Per a complir amb l’objectiu, es realitzen vàries investigacions. Primer, s’estudien els coneixements inicials de nombres i geometria dels estudiants del primer curs del Grau d’Educació Primària a la Universitat Rovira i Virgili (URV). En segon lloc, s’estudien les creences de l’eficàcia de l’ensenyament de les matemàtiques dels futurs mestres durant el grau. En tercer lloc, en aquesta Tesi es compara l’autoeficàcia i l’expectativa de resultats de l’ensenyament de les matemàtiques de futurs mestres, mestres novells i mestres experimentats. En quart lloc, s’estudia la relació entre les creences de l’ensenyament de les matemàtiques, l’actitud envers les matemàtiques i el rendiment acadèmic dels futurs mestres. En cinquè lloc, s’estudia la influència dels factors experiència docent, nivell d’educació i nivell d’ensenyament sobre les creences de l’eficàcia de l’ensenyament de les matemàtiques en mestres en actiu. Finalment, es compara l’autoeficàcia de l’ensenyament de les matemàtiques entre els estudiants del quart any del grau de mestres a la Universitat del Azuay i a la URV. Els resultats d’aquesta Tesi ofereixen informació potencialment important sobre el coneixement matemàtic, les creences, l’autoeficàcia de l’ensenyament de les matemàtiques i l’actitud envers les matemàtiques dels futurs mestres i dels mestres en actiu. Aquests resultats poden ajudar a desenvolupar polítiques adients a l’hora de dissenyar plans d’estudis i també assessorar als professors dels graus de mestre en les institucions d’educació superior.
El estudio del conocimiento matemático y las creencias de la eficacia de la enseñanza de las matemáticas en la formación inicial de los futuros maestros es fundamental, ya que influye en el rendimiento académico de los estudiantes. El objetivo de esta tesis es estudiar tanto el conocimiento matemático inicial de los futuros maestros como sus creencias sobre la eficacia matemática y su actitud hacia las matemáticas. Para cumplir con el objetivo se realiza varias investigaciones. Primero, se estudia los conocimientos iniciales de números y geometría de los estudiantes de primer año del Grado de Educación Primaria en la Universidad Rovira y Virgili (URV). En segundo lugar, se estudia las creencias de la eficacia de la enseñanza de las matemáticas de los futuros maestros a lo largo del grado. Tercero, esta Tesis compara la autoeficacia y la expectativa de resultados de la enseñanza de las matemáticas de futuros maestros, maestros novatos y maestros experimentados. Cuarto, se estudia la relación entre las creencias de la enseñanza de las matemáticas, la actitud hacia las matemáticas y su rendimiento académico. Quinto, se estudia la influencia de los factores experiencia docente, nivel de educación y nivel de enseñanza, sobre las creencias de la eficacia de la enseñanza de las matemáticas en maestros en servicio. Finalmente, se compara la autoeficacia de la enseñanza de las matemáticas entre los estudiantes de cuarto año del grado de maestro en la Universidad del Azuay y en la URV. Los resultados de esta Tesis ofrecen información potencialmente importante sobre el conocimiento matemático, las creencias, la autoeficacia de la enseñanza de las matemáticas y la actitud hacia las matemáticas de los futuros maestros y maestros en servicio. Estos resultados pueden ayudar a desarrollar políticas adecuadas para diseñar planes de estudios y también asesorar a los profesores de los grados de maestro en las instituciones de educación superior.
The study of mathematical content knowledge and teachers’ mathematics teaching beliefs of the pre-service teachers is fundamental, since it influences the academic performance of students. The objective of this Thesis is to study the initial mathematical knowledge of pre-service teachers and also their teachers’ mathematics teaching beliefs and their attitude towards mathematics. To meet the objective, various investigations are carried out. First, the initial knowledge of numbers and geometry of first-year students of the primary education degree at the Rovira and Virgili University (URV) is studied. Second, pre-service teachers’ mathematics teaching beliefs are studied throughout the grade. Third, this Thesis compares the self-efficacy and the expectation of results of the teaching of mathematics of pre-service teachers, novice in-service teachers and experienced in-service teachers. Fourth, the relationship between the teachers’ mathematics teaching beliefs, the attitude towards mathematics and their academic performance is studied. Fifth, the influence of the factors teaching level factor and level of training on the teachers’ mathematics teaching beliefs of in-service teachers is studied. Finally, the self-efficacy of mathematics teaching of fourth-year students at the Azuay University and at the URV is compared. The results of this Thesis offer potentially important information on the mathematical knowledge, beliefs, self-efficacy of mathematics teaching and the attitude towards mathematics of pre-service teachers and in-service teachers. These results can help develop policies for curriculum developers and teaching professors at institutes of higher education.

The client-server architecture of STAR-CCM+ allows multiple users to collaborate on a simulation set-up. The effectiveness of collaboration with this architecture is tested and evaluated on five models. The testing of these models is a start to finish set-up of an entire simulation excluding computational time for generating mesh and solving the solution. The different models have distinct differences which test every operation that would be used in a general CFD simulation. These tests focus on reducing the time spent preparing the geometry to be meshed, including setting up for a conformal mesh between multiple regions in conjugate heat transfer models. Results from these five tests show a maximum speed up of 36%.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
De acordo com documentos oficiais brasileiros, a Ãrea da MatemÃtica escolar à dividida em quatro blocos ou eixos: NÃmeros e OperaÃÃes, EspaÃo e Forma (Geometria), Grandezas e Medidas, e Tratamento da InformaÃÃo (EstatÃstica). O ensino da MatemÃtica, entretanto, ainda à muito focado no primeiro, em detrimento dos outros blocos. Hà duas dÃcadas, vÃrios pesquisadores (PAVANELLO, 1993; ARAÃJO, 1994; LORENZATO, 1995; FAINGUELERNT, 1995) denunciaram o abandono da Geometria, a qual contempla conteÃdos referentes a forma e espaÃo, na EducaÃÃo BÃsica e defenderam a sua valorizaÃÃo. Na dÃcada passada, dispositivos legais determinaram a inclusÃo da PrÃ-Escola na EducaÃÃo BÃsica obrigatÃria. No que se refere ao trabalho pedagÃgico com a Geometria nesta fase da EducaÃÃo Infantil, muitas vezes o educador aborda apenas o (re)conhecimento de figuras geomÃtricas planas â cÃrculo, triÃngulo, retÃngulo, quadrado â em atividades de pintura e nomeaÃÃo. O ensino e a aprendizagem de Geometria, no entanto, precisa oportunizar, tal como propÃem vÃrios pesquisadores â (GRANDE, 1994), (CERQUETTI-ABERKNE; BERDONNEAU, 1997), (DUHALDE; CUBERES, 1998), (SMOLE; DINIZ; CÃNDIDO, 2003), (LORENZATO, 2006) â o desenvolvimento de conceitos referentes a espaÃo e forma, oferecendo Ãs crianÃas oportunidades de perceberem e conhecerem os espaÃos em que vivem, se locomovem, nos quais elas aprendem a explorar, conquistar, ordenar e representar. Este estudo teve como objetivo identificar os saberes docentes de pedagogos que lecionam na PrÃ-Escola sobre orientaÃÃo espacial. A pesquisa de natureza qualitativa, do tipo estudo de caso, foi realizada numa instituiÃÃo de EducaÃÃo Infantil e Ensino Fundamental do sistema municipal de Fortaleza. Participaram do estudo duas professoras, sendo uma do Infantil IV e uma do Infantil V, e uma formadora da EducaÃÃo Infantil. Ao longo de nove encontros, foram realizadas visitas periÃdicas para a realizaÃÃo das observaÃÃes em campo das aulas ministradas pelas professoras participantes da pesquisa e entrevistas (iniciais e reflexivas), que foram gravadas, e, posteriormente, transcritas, gerando textos e reflexÃes. A partir dos resultados, constatou-se que, apesar de possuÃrem algum conhecimento de Geometria, os saberes docentes das professoras e formadora referentes à orientaÃÃo espacial, um conteÃdo importante no desenvolvimento e na aprendizagem das crianÃas da PrÃ-Escola, sÃo fragmentados, sendo necessÃrio proporcionar, com urgÃncia, oportunidades de formaÃÃo que ampliem e articulem tais saberes.
According to Brazilian official documents, the area of school mathematics is divided into four blocks or axes: Numbers and Operations, Space and Shape (Geometry), Quantities and Measurements, and Treatment Information (Statistics). The teaching of mathematics, however, is still very focused on the first, at the expense of other blocks. Two decades ago, several researchers (PAVANELLO, 1993; ARAÃJO, 1994; LORENZATO, 1995; FAINGUELERNT, 1995) denounced the abandonment of geometry, which includes content related to form and space, in Basic Education and defended their appreciation. In the past decade, legal provisions determined the inclusion of Pre-School in compulsory basic education. With regard to the pedagogical work with the geometry at this stage of early childhood education, often the educator addresses only the (re) knowledge of plane geometric figures - circle, triangle, rectangle, square - in painting and appointment activities. Teaching and Geometry learning, however, need to create opportunities, as proposed by several researchers - (GRANDE, 1994) (CERQUETTI-ABERKNE; BERDONNEAU, 1997) (DUHALDE; Cuberes, 1998) (Smole; DINIZ; CANDID , 2003), (Lorenzato, 2006) - the development of concepts for space and form, giving children opportunities to realize and know the areas in which they live, they move, where they learn to explore, conquer, order and represent. This study aimed to identify the teachersÂs knowledge of teachers who teach the spatial orientation on Pre-School. The qualitative research, case study type, was carried out in an institution of Early Childhood Education and Elementary Education municipal fortress system. Study participants were two teachers, one of child IV and of child V, and a trainer of early childhood education. Over nine meetings were held periodic visits to the realization of field observations of classes taught by teachers participating research and interviews (initial and reflective), which were recorded and later transcribed, generating texts and reflections. From the results, it was found that, despite having some geometry knowledge, the teacherÂs knowledge of teachers and trainer regarding the spatial orientation, an important content in the development and learning of children from pre-school, are fragmented, requiring provide urgently training opportunities that enhance and articulate such knowledge.

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

[EN] In the current work, the characterization of the combustion process inside a stratified pre-chamber spark ignition (PCSI) system is performed. An extensive bibliographical review about the pre-chamber systems developed from the second half of the 20th century until modern times is presented. The review shows that the latest generation systems have the potential to accomplish the emissions limits while providing high performance and low fuel consumption. Nevertheless, many efforts of the scientific community are still needed to allow the large-scale application of the technology. Indeed, based on the outstanding challenges observed, the investigation plan is developed including both experimental and numerical parts. All experiments were performed by means of the rapid compressionexpansion machine (RCEM) in the CMT-Motores Térmicos laboratory. The original cylinder head layout was modified to allow the housing of the prechamber itself, fuel injectors, spark plug, pressure transducers in both chamber, and a thermocouple. The test methodology involved the acquisition of the pressure evolution in both main chamber and pre-chamber, the piston position (used to compute the instantaneous cylinder volume), the duration of the auxiliary injection, and the spark ignition point. These are used as input for the zero-dimensional thermodynamic model which simulates the fundamental parameters aims to characterize the PCSI system working cycle. Therefore, a deeper knowledge of the mass interchanged process, induced turbulence field, heat release rate, combustion speed, and flame regime is generated. Subsequently, to calibrate the zero-dimensional model coefficients under motoring conditions, several 3D CFD simulations were carried out by means of Converge software. Hence, the results of the simulations in terms of interchanged mass and pre-chamber turbulent kinetic energy have been used to calibrate the nozzle discharge coefficient and the turbulence sub-model coefficients for all the pre-chamber geometries. Furthermore, the 3D CFD simulations outputs are analysed to fully understand the flow field structure and the local effect induced by the different nozzles at the spark activation time. The turbulent kinetic energy in terms of intensity and orientation is investigated over several relevant pre-chamber sections. The results reveal a clear relationship between the turbulence developed within the pre-chamber and the orifices structure. Straight orifices or perpendicular jets impact, promote more intense local turbulence due to direct collision while tilted orifices guarantee more homogeneity due to the swirling motion. Additionally, increase the orifice numbers shows benefits on the fluid dynamic homogeneity. Thus, preceding the experimental campaign several fundamental aspects of the system are evaluated. The cycle-to-cycle dispersion is explored by means of the statistical assessment showing low pressure peak deviation. The auxiliary injection pressure and timing are optimized for avoiding wall wetting phenomena while ensuring proper air/fuel mixing. Finally, the spark activation point is chosen as a function of the theoretically maximum turbulent flame speed. Thereby, the experimental campaign is carried out according to tests matrix, in order to evaluate the effect of the equivalence ratio of both chambers, and how the orifices diameter, number, and distribution affect the combustion process. Moreover, chemiluminescence visualization tests, performed by means of the available optical access of the RCEM, are combined with zerodimensional and 3D CFD results to shed light on the work cycle. Conclusions suggest a slightly rich mixture inside the pre-chamber combined with the highest number of tilted orifices as the better configuration for improving combustion efficiency under lean and ultra-lean main chamber mixture conditions. Nevertheless, axial orifices should be considered for further investigations. Finally, the author proposes a series of developments considered interesting in both the experimental and numerical fields.
[ES] En el presente trabajo se realiza la caracterización del proceso de combustión dentro de un sistema de encendido por pre-cámara bajo carga estratificada. Por lo tanto, se presenta una extensa revisión bibliográfica sobre los sistemas de pre-cámara desarrollados desde la segunda mitad del siglo XX hasta los tiempos modernos. El resumen muestra que los sistemas de última generación tienen el potencial de cumplir con los límites de las emisiones, al tiempo que proporcionan un alto rendimiento y un bajo consumo de combustible. No obstante, todavía se necesitan muchos esfuerzos de la comunidad científica para permitir la difusión a gran escala de la tecnología. De hecho, sobre la base de los desafíos abiertos observados, se desarrolla el plan de investigación incluyendo tanto una parte experimental como numérica. Todos los experimentos se realizan mediante la máquina de compresión-expansión rápida (RCEM) de que dispone el laboratorio CMT-Motores Térmicos . La disposición original de la culata se modificó para permitir el alojamiento de la propia pre-cámara, los inyectores , la bujía, los sensores de presión y un termopar. La metodología de ensayo implica la adquisición de la evolución de la presión tanto en cámara principal como en pre-cámara, el volumen del cilindro, la duración de la inyección auxiliar y el punto de ignición de la bujía. Estos se utilizan como parámetros de entrada para el modelo termodinámico cero-dimensional que devuelve los parámetros fundamentales que caracterizan ciclo de trabajo del sistema PCSI. Por lo tanto, se genera un conocimiento más profundo del proceso de intercambio de masas, del campo de turbulencias inducidas, de la tasa de liberación de calor, de la velocidad de combustión y del régimen de la llama. Posteriormente, para calibrar los coeficientes del modelo cero-dimensional bajo condiciones de arrastre, se llevaron a cabo varias simulaciones CFD en 3D mediante el software Converge. Por lo tanto, los resultados de las simulaciones en términos de masa intercambiada y energía cinética turbulenta de la precámara se han utilizado para calibrar el coeficiente de descarga de la tobera y los coeficientes del sub-modelo de turbulencia para todas las geometrías de la pre-cámara. Además, se analizan los resultados de las simulaciones CFD para comprender plenamente la estructura del campo de flujo y el efecto local inducido por las diferentes geometriás en el tiempo de activación de la chispa. La energía cinética turbulenta en términos de intensidad y orientación se investiga en varias secciones relevantes de la pre-cámara. Los resultados revelan una clara relación entre la turbulencia desarrollada dentro de la pre-cámara y la estructura de los orificios. Los orificios rectos o los chorros perpendiculares, promueven una turbulencia local más intensa debido a la colisión directa mientras que los orificios inclinados del campo fluido y del dosado. Precedentemente al desarrollo de la campaña experimental se evalúan varios aspectos fundamentales del sistema. La dispersión ciclo a ciclo se explora por medio de la evaluación estadística que muestra una baja desviación de los picos de presión. La presión y el punto de inyección auxiliar se optimizan para evitar los fenómenos de mojado de las paredes, asegurando al mismo tiempo una mezcla adecuada de aire/combustible. Finalmente, el punto de activación de la chispa se elige en función de la velocidad máxima teórica de la llama turbulenta. De este modo, la campaña experimental se lleva a cabo de acuerdo con la matriz de pruebas, con el fin de evaluar el efecto del dosado equivalente de ambas cámaras, y cómo el diámetro, el número y la distribución de los orificios afectan al proceso de combustión. Además, las pruebas de visualización de quimioluminiscencia, realizadas mediante el acceso óptico disponible de la RCEM, se combinan con resultados de CFD y resultados del modelo cerodimen para arrojar luz sobre el ciclo de trabajo. Las conclusiones sugieren que una mezcla ligeramente rica dentro de la pre-cámaracombinadaconelmayornúmerodeorificiosdesfasadoseslamejor configuración para garantizar un elevada eficiencia de la combustión en condiciones de mezcla pobre y ultra-pobre de la cámara principal. No obstante, los orificios axiales deben ser considerados para investigaciones futuras. Por último, el autor propone una serie de desarrollos considerados interesantes tanto en el campo experimental como en el numérico.
[CA] En el present treball es realitza la caracterització del procés de combustió dins d'un sistema d'encesa de pre-cambra soto càrrega estratifi-cada. Per tant, es presenta una extensa revisió bibliogràfica sobre els sistemes de precambra desenvolupats des de la segona meitat del segle XX fins als temps moderns. El resum mostra que els sistemes d'última generació tenen el potencial de complir amb els límits de les emissions, al mateix temps que proporcionen un alt rendiment i un baix consum de combustible. No obstant això, encara es necessiten molts esforços de la comunitat científica per a permetre la difusió a gran escala de la tecnologia. De fet, sobre la base dels desafiaments oberts observats, es desenvolupa el pla d'investigació incloent tant una part experimental com numèrica. Tots els experiments es realitzen mitjançant la màquina de compressió-expansió ràpida (RCEM) de què disposa el laboratori CMT-Motors Tèrmics. La disposició original de la culata es va modificar per a permetre l'allotjament de la pròpia pre-cambra, els injectors , la bugia, els sensors de pressió i un termoparell. La metodologia d'assaig implica l'adquisició de l'evolució de la pressió tant en cambra principal com en pre-cambra, el volum del cilindre, la duració de la injecció auxiliar i el punt d'ignició de l'espurna. Aquests s'utilitzen com a paràmetres d'entrada per al model termodinàic zero-dimensional que retorna els paràmetres fonamen-tals que caracteritzen cicle de treball del sistema PCSI. Per tant, es genera un coneixement més profund del procés d'intercanvi de masses, del camp de turbulències induïdes, de la taxa d'alliberament de calor, de la velocitat de combustió i del règim de la flama. Posteriorment, per a calibrar els coefi-cients del model zerodimensional sota condicions d'arrossegament, es van dur a terme diverses simulacions CFD en 3D mitjançant el programari Converge. Per tant, els resultats de les simulacions en termes de massa intercanviada i energia cinètica turbulenta de la pre-cambra s'han utilitzat per a calibrar el coeficient de descàrrega de la tovera i els coeficients del sub-model de turbulència per a totes les geometries de la pre-cambra. A més, s'analitzen els resultats de les simulacions CFD per a comprendre plenament l'estructura del camp de flux i l'efecte local induït per les diferents geometries en el temps d'activació de l'espurna. L'energia cinètica turbulenta en termes d'intensitat i orientació s'investiga en diverses seccions rellevants de la pre-cambra. Els resultats revelen una clara relació entre la turbulència desenvolupada dins de la pre-cambra i l'estructura dels orificis. Els orificis rectes o els dolls perpendiculars, promouen una turbulència local més intensa a causa de la col·lisió directa mentre que els orificis inclinats garanteixen una major homogeneïtat a causa de la generació d'un macro-remolì. A més, l'augment del nombre d'orificis mostra beneficis en l'homogeneïtat fluid-dinàmica. Llavors, abans de la campanya experimental s'avaluen diversos aspectes fonamentals del sistema. La dispersió cicle a cicle s'explora per mitjà de l'avaluació estadística que mostra una baixa desviació dels pics de pressió. La pressió i el punt d'injecció auxiliar s'optimitzen per a evitar els fenòmens de mullat de les parets, assegurant al mateix temps una mescla adequada d'aire/combustible. Finalment, el punt d'activació de l'espurna es tria en funció de la velocitat màxima teòrica de la flama turbulenta. D'aquesta manera, la campanya experimental es duu a terme d'acord amb la matriu de proves, amb la finalitat d'avaluar l'efecte del dosatge equivalent de totes dues cambres, i com el diàmetre, el numero i la distribució dels orificis afecten el procés de combustió. A més, les proves de visualització de quimioluminescència, realitzades mitjançant l’accés òptic disponible de la RCEM, es combinen amb resultats de CFD i resultats del model zero-dimensional per a llançar llum sobre el cicle de treball. Les conclusions suggereixen que una mescla lleugerament rica dins de la pre-cambra combinada amb el major nombre d’orificis desfasats és la millor configuració per a garantir un elevada eficiència de la combustió en condicions de mescla pobra i ultra-pobre de la cambra principal. No obstant això, els orificis axials han de ser considerats per a investigacions futures. Finalment, l’autor proposa una sèrie de desenvolupaments considerats interessants tant en el camp experimental com en el numèric.
Pagano, V. (2020). Analysis of a stratified pre-chamber spark ignition system under lean mixture conditions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/152486
TESIS

Les artistes traditionnels peignent directement sur une toile et créent des apparences plausibles de scènes qui ressemblent au monde réel. A l’opposé, les artistes en informatique graphique définissent des objets dans une scène virtuelle (maillages 3D, matériaux et sources de lumière), et utilisent des algorithmes complexes (rendu) pour reproduire leur apparence. D’un côté, les techniques de peinture permettent de librement définir l’apparence. D’un autre côté, les techniques de rendu permettent de modifier séparément et dynamiquement les différents éléments qui définissent l’apparence. Dans cette thèse, nous présentons une approche intermédiaire pour manipuler l’apparence, qui permettent certaines manipulations en 3D en travaillant dans l’espace 2D. Mous étudions d’abord l’impact sur l’ombrage des matériaux, tenant en compte des matériaux comme des filtres passe-bande d’éclairage. Nous présentons ensuite un petit ensemble de relations statistiques locales entre les matériaux / l’éclairage et l’ombrage. Ces relations sont utilisées pour imiter les modifications sur le matériaux ou l’éclairage d’une image d’une sphère créée par un artiste. Les techniques connues sons le nom de LitSpheres / MatCaps utilisent ce genre d’images pour transférer leur apparence `a des objets de forme quelconque. Notre technique prouve la possibilité d’imiter les modifications 3D de la lumière et de matériaux à partir d’une image en 2D. Nous présentons une technique différente pour modifier le troisième élément impliqué dans l’aspect visuel d’un objet, sa géométrie. Dans ce cas, on utilise des rendus comme images d’entrée avec des images auxiliaires qui contiennent des informations 3D de la scène. Nous récupérons un ombrage indépendant de la géométrie pour chaque surface, ce qui nous demande de supposer qu’il n’y a pas de variations spatiales d’éclairage pour chaque surface. L’ombrage récupéré peut être utilisé pour modifier arbitrairement la forme locale de l’objet de manière interactive sans la nécessité de rendre `a nouveau la scène
Traditional artists paint directly on a canvas and create plausible appearances of real-world scenes. In contrast, Computer Graphics artists define objects on a virtual scene (3D meshes, materials and light sources), and use complex algorithms (rendering) to reproduce their appearance. On the one hand, painting techniques permit to freely define appearance. On the other hand, rendering techniques permit to modify separately and dynamically the different elements that compose the scene. In this thesis we present a middle-ground approach to manipulate appearance. We offer 3D-like manipulation abilities while working on the 2D space. We first study the impact on shading of materials as band-pass filters of lighting. We present a small set of local statistical relationships between material/lighting and shading. These relationships are used to mimic modifications on material or lighting from an artist-created image of a sphere. Techniques known as LitSpheres/MatCaps use these kinds of images to transfer their appearance to arbitrary-shaped objects. Our technique proves the possibility to mimic 3D-like modifications of light and material from an input artwork in 2D. We present a different technique to modify the third element involved on the visual appearance of an object: its geometry. In this case we use as input rendered images alongside with 3D information of the scene output in so-called auxiliary buffers. We are able to recover geometry-independent shading for each object surface, assuming no spatial variations for each recovered surface. The recovered shading can be used to modify arbitrarily the local shape of the object interactively without the need to re-render the scene

In this thesis we review the status of an approach to Quantum Gravity through lattice toy models, Quantum Graphity. In particular, we describe the two toy models introduced in the literature and describe with a certain level of details the results obtained so far. We emphasize the connection between Quantum Graphity and emergent gravity, and the relation with Variable Speed of Light theories.

Over the past years geometry has posed a challenge to most learners in South African schools. The Government, in particular the Department of Basic Education (DBE), have tried and are still trying to implement new innovations and strategies for teaching mathematics more effectively. South Africa has experienced many changes in mathematics curriculum with an aim of placing the country on an equal footing with countries globally. This study was conducted while there was the implementation of the new Curriculum and Assessment Policy Statement (CAPS), which reinstated the geometry section within the curriculum. Geometry was relegated to an optional paper in mathematics in 2006, 2007 and 2008 in Grades 10, 11 and 12 respectively. This study is framed within the theoretical framework lens of social constructivism and situated learning, and is located within the qualitative research paradigm. It takes the form of survey research in one of the universities in KwaZulu-Natal, South Africa. This university is referred to as the University of Hope (UOH) in this study to protect its identity. The main aim of this study was to explore the pre-service teachers' (PSTs) knowledge of proof in geometry. The study used qualitative analysis of data generated through a survey questionnaire, task-based worksheets and semi-structured interviews for both the focus group and individual interviews. In total 180 PSTs completed task-based worksheets. Within this group of 180 students, 47 were 4th year students, 93 were 3rd year and 40 were 2nd year students. After the analysis of a task-based worksheet, a total of 20 participants from the 3rd and 4th year were invited to participate in focus group interviews. The findings of the study exhibit that the PSTs have very little knowledge of proof in geometry. The study revealed that this lack of the knowledge stems from the knowledge proof in geometry the PSTs are exposed to at school level.
Thesis (M. Ed.)-University of KwaZulu-Natal, Durban, 2012.

博士
大同大學
機械工程研究所
88
The objective of this thesis is to develop an algorithm of pre-processing the data points, measured from existing objects, for feature-curve fitting and geometry reconstruction in reverse engineering. The problem in reverse engineering is that the huge amounts of data points are hard to be handled in most CAD systems. Not only is file size a concern, but also the original measured points are with irregular format, unequal distribution and even incompleteness to be difficultly fitted into the B-spline form directly. Additionally, in product design, the design features, which are useful for further manipulation, must be extracted while rebuilding the geometric model in reverse engineering. Therefore, the technology of restoring and segmenting data points for feature curve regeneration is very important to reconstruct the geometric model of an existing object. By the proposed approach, the original measured data points can be regrouped to extract the design features for regenerating individual curves, with the format in which the data points achieve the requirements of fitting into the B-spline form with good shape and high quality. This algorithm consists of four parts: (1) segmentation of data points; (2) regressing data points into an explicit non-parametric equation; (3) regeneration of data points; and (4) determining knot points. The proposed approach has been implemented by using several examples for practical application in reverse engineering. The result of the implementation proves the viability of integrating the approach with commercial CAD systems.

Neste relatório final apresentamos a descrição e a reflexão da Prática de Ensino Supervisionada desenvolvida numa sala de creche e numa sala de jardim de infância, da cidade de Portalegre. Nele permitimo-nos realçar a importância do domínio da Matemática na Educação Pré-Escolar, especificamente no que diz respeito a conceitos ligados à Geometria e a processos matemáticos, no início da aprendizagem matemática das crianças. Ao longo deste relatório são apresentadas e discutidas tarefas exploradas com as crianças, analisadas as suas contribuições orais e os seus registos escritos. Optámos pela metodologia de investigação-ação, por possibilitar uma observação e reflexão aprofundada da prática e, consequentemente, a sua melhoria, e por contribuir para a nossa formação profissional.
In this final report we present the description and reflection of the Supervised Teaching Practice developed in a nursery school and a kindergarten in the city of Portalegre. It enables us to highlight the importance of the domain of mathematics in Preschool Education, specifically regarding to concepts related to geometry and mathematical processes, at the beginning of the children mathematical learning. Throughout this report are presented and discussed the tasks explored with the children, examining their oral contributions and their written records. We have chosen the action research methodology, by enabling an observation and reflection of the practice and thus its improvement, and for contributing to our professional training.

There are numerous problems associated with the teaching of Euclidean geometry at secondary schools today. Students do not see the necessity of proving results which have been obtained intuitively. They do not comprehend that the validity of a deduction is independent of the 'truth' of the initial assumptions. They do not realise that they cannot reason from diagrams, because these may be misleading or inaccurate. Most importantly, they do not understand that Euclidean geometry is a particular interpretation of physical space and that there are alternative, equally valid interpretations. A possible means of addressing the above problems is tbe introduction of nonEuclidean geometry at school level. It is imperative to identify those students who have the pre-requisite knowledge and skills. A number of interesting teaching strategies, such as debates, discussions, investigations, and oral and written presentations, can be used to introduce and develop the content matter.
Mathematics Education
M. Sc. (Mathematics)

A subject didactic analysis of the development of the spatial knowledge of young children through a problem-centred approach to mathematics teaching and learning Researchers and educators are in agreement that it is very important that the spatial knowledge of the young child should be developed from the first years of school. In order to develop these skills the appropriate materials and activities need to be designed. This can only be realised through proper research methods that not only acknowledge the cognitive abilities of the young child, but also the social and cultural backgrounds of the children. This implies that due attention should be given to language, beliefs, cognitive skills, socio-economic background, schooling and teaching systems. The immediate worlds of the children should be used in developing the spatial skills of these children. The work that is described in this document is an effort to describe the complexity of such a research endeavour. The development of the spatial skills of young children were investigated through three different instructional/executional media namely language, drawing/writing and physical constructions. The three major variables that were described as influencing the spatial development in the different media were the task that was given to the children, the objects that the children worked with and the dimension and viewpoints of the objects and situations. It was clear from the research that in the development of the spatial skills of children, attention should. be given to: the real-world of the children, hands-on-experiences of the children, the cultural background, the language of instruction, the socio-economic background the classroom culture, the media of instruction and the cognitive skills of the children.
Proefskrif (PhD (Didaktiese Opvoedkunde))--PU vir CHO, 1997

The subject of teacher competencies or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competencies necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer's Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competencies were identified: mathematical/geometrical competencies. pedagogical competencies. computer and software competences, language and assessment competencies.
Mathematical Sciences
M. Ed. (Mathematical Education)

The subject of teacher competences or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competences necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer’s Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competences were identified: mathematical/geometrical competences, pedagogical competences, computer and software competences, language and assessment competencies.
Mathematics Education
M. Ed. (Mathematics Education)

This study was carried out to investigate the effect of using animated computer 3-D figures illustration (ACTDFI) in the learning of polyhedron in geometry. By random sampling, intact group of four grade 9 classes in four different schools from a cluster of four educational district schools of Limpopo province in South Africa were selected. The study involved quasi-experimental and inquiry research approaches, the quasi-experimental approach involved pre and posttest design while the inquiry research approach involve classroom observation. There were three experimental groups and a control group with a total of 174 study participants. ACTDFI was used as an intervention for two weeks in the three experimental groups while in the control group, chalk-talk traditional teaching approach was used. Pre-test and post-test was used to collect quantitative data while classroom observation was used to collect qualitative data. This study was carried out to investigate the effect of using animated computer 3-D figures illustration (ACTDFI) in the learning of polyhedron in geometry. By random sampling, intact group of four grade 9 classes in four different schools from a cluster of four educational district schools of Limpopo province in South Africa were selected. The study involved quasi-experimental and inquiry research approaches, the quasi-experimental approach involved pre and posttest design while the inquiry research approach involve classroom observation. There were three experimental groups and a control group with a total of 174 study participants. ACTDFI was used as an intervention for two weeks in the three experimental groups while in the control group, chalk-talk traditional teaching approach was used. Pre-test and post-test was used to collect quantitative data while classroom observation was used to collect qualitative data. The findings from the quantitative Classroom observations were carried out to collect relevant data on how the study participants were taught stationary points in differential calculus, especially with the use of the constructivist pedagogical approach. A suitable observation checklist was developed for this purpose (Appendix 6 refers). Classroom observation checklist is a list of factors to be considered while observing a class. It gives a structure and framework for the observation. suggested that the use of ACTDFI might have improved academic achievement in learning of polyhedron during the intervention, while the qualitative data analysis indicated that the use of ACTDFI in the experimental groups might have facilitated the learning of the concepts of polyhedron. It is therefore recommended that further research is necessary on the application of ACTDFI in the teaching of 3-dimensional shapes at the primary schools
Mathematics Education
M. Sc. (Mathematics Education)

This thesis studies the pre-quantization of quasi-Hamiltonian group actions from a cohomological viewpoint. The compatibility of pre-quantization with symplectic reduction and the fusion product are established, and are used to understand the necessary and sufficient conditions for the pre-quantization of M(G,S), the moduli space of at flat G-bundles over a closed surface S. For a simply connected, compact, simple Lie group G, M(G,S) is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this thesis determines the obstruction, namely a certain 3-dimensional cohomology class, that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G. Partial results are obtained for the case of a surface S with marked points. Also, it is shown that via the bijective correspondence between quasi-Hamiltonian group actions and Hamiltonian loop group actions, the corresponding notions of prequantization coincide.

The high school pass rate, for mathematics, in South Africa is very low. This is particularly so in trigonometry functions. One of the possible factors leading to this is the traditional method of teaching and learning. This study was undertaken to determine whether the use of Information and Communication Technology (ICT) would influence students’ learning of trigonometry functions. In order to answer this question the teaching and learning instructions developed were based on activity theory (AT) and action, process, object, and schema (APOS) theory. The study followed a non-equivalent control group, quasi-experimental design with a pre- post-test approach. Since it was not possible to randomly select participants for the study, intact groups were used. There were two groups: a control and an experimental one. Both groups wrote a standardized achievement pre-test to establish their comparability at the beginning of the study. While the control group was taught in the traditional way (grade 10-12 syllabus), the experimental group used the software Geogebra. The computer software (Geogebra) and the South African grade 10-12 syllabus for trigonometry functions were used during the lessons of the experimental group. At the end of the study, a similar post-test was administered on both groups to measure the comparative effects of either of the teaching methods on the performance of students. A t-test independent sample statistical analysis was performed on the findings using a statistics package, SPSS. The results of this investigation indicated that the use of the computer software, Geogebra, in the teaching and learning of trigonometry functions improved the performance of the Grade 12 students.
Mathematics Education
M. Sc. (Mathematics Education)

Researchers conceptually recommend inquiry-based learning as a necessary means to alleviate the problems of learning but this study has embarked on practical implementation of inquiry-based facilitation and learning in Euclidean Geometry. Inquiry-based learning is student-centred. Therefore, the teaching or monitoring of inquiry-based learning in this study is referred to as inquiry-based facilitation. The null hypothesis discarded in this study explains that there is no difference between inquiry-based facilitation and traditional axiomatic approach in teaching Euclidean Geometry, that is, H0: μinquiry-based facilitation = μtraditional axiomatic approach. This study emphasises a pragmatist view that constructivism is fundamental to realism, that is, inductive inquiry supplements deductive inquiry in teaching and learning. Participants in this study comprise schools in Tshwane North district that served as experimental group and Tshwane West district schools classified as comparison group. The two districts are in the Gauteng Province of South Africa. The total number of students who participated is 166, that is, 97 students in the experimental group and 69 students in the comparison group. Convenient sampling applied and three experimental and three comparison group schools were sampled. Embedded mixed-method methodology was employed. Quantitative and qualitative methodologies are integrated in collecting data; analysis and interpretation of data. Inquiry-based-facilitation occurred in experimental group when the facilitator probed asking students to research, weigh evidence, explore, share discoveries, allow students to display authentic knowledge and skills and guiding students to apply knowledge and skills to solve problems for the classroom and for the world out of the classroom. In response to inquiry-based facilitation, students engaged in cooperative learning, exploration, self-centred and self-regulated learning in order to acquire knowledge and skills. In the comparison group, teaching progressed as usual. Quantitative data revealed that on average, participant that received intervention through inquiry-based facilitation acquired inquiry-based learning skills and improved (M= -7.773, SE= 0.7146) than those who did not receive intervention (M= -0.221, SE = 0.4429). This difference (-7.547), 95% CI (-8.08, 5.69), was significant at t (10.88), p = 0.0001, p<0.05 and represented a large effect size of 0.55. The large effect size emphasises that inquiry-based facilitation contributed significantly towards improvement in inquiry-based learning and that the framework contributed by this study can be considered as a framework of inquiry-based facilitation in Euclidean Geometry. This study has shown that the traditional axiomatic approach promotes rote learning; passive, deductive and algorithmic learning that obstructs application of knowledge in problem-solving. Therefore, this study asserts that the application of Inquiry-based facilitation to implement inquiry-based learning promotes deeper, authentic, non-algorithmic, self-regulated learning that enhances problem-solving skills in Euclidean Geometry.
Mathematics Education
Ph. D. (Mathematics, Science and Technology Education)