Dissertations / Theses: 'Mathematical visualisation'

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Dongwi, Beata Lididimikeni. "Examining mathematical reasoning through enacted visualisation." Thesis, Rhodes University, 2019. http://hdl.handle.net/10962/68192.

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This study sets out to analyse the co-emergence of visualisation and reasoning processes when selected learners engaged in solving word problems. The study argues that visualisation processes and mathematical reasoning processes are closely interlinked in the process of engaging in any mathematical activity. This qualitative research project adopted a case study methodology embedded within a broader interpretative orientation. The research participants were a cohort of 17 mixedgender and mixed-ability Grade 11 learners from a private school in southern Namibia. Data was collected in three phases and comprised of one-on-one task-based interviews in the first phase, focus group task-based interviews in the second, and semi-structured reflective interviews in the third. The analytical framework was informed by elements of enactivism and consisted of a hybrid of observable visualisation and mathematical reasoning indicators. The study was framed by an enactivist perspective that served as a linking mediator to bring visualisation and reasoning processes together, and as a lens through which the coemergence of these processes was observed and analysed. The key enactivist concepts of structural coupling and co-emergence were the two mediating ideas that enabled me to discuss the links between visualisation and reasoning that emerged whilst my participants solved the set word problems. The study argues that the visualisation processes enacted by the participants when solving these problems are inseparable from the reasoning processes that the participants brought to bear; that is, they co-emerged.

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Tolmie, Julie, and julie tolmie@techbc ca. "Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1." The Australian National University. School of Mathematical Sciences, 2000. http://thesis.anu.edu.au./public/adt-ANU20020313.101505.

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There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.

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Melander, Emma. "Applied VMS to handle mathematical misconception in algebra : Metacognition through interactive visualisation prototype." Thesis, Linköpings universitet, Institutionen för datavetenskap, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-132316.

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Swedish education needs to change in order to reverse the negative trend in PISA and TIMSS. At the same time digital artifacts are increasingly present in the classroom and commercial applications more available. It is difficult to determine which technical tool that provide real results and it is not yet clear what role technology should have in the classroom. This study examines how visualisation and technology can be used to promote deeper understanding of algebra by addressing the misconception that frequently occurs among students. Observation and interviews were held with students from two schools, one in Norrköping and one in Östersund. Theory regarding algebraic misconceptions have been studied and compiled into two categories. In addition, three prototypes were developed with and later evaluated by a focus group. This work has resulted in suggestions on what technical and interactive activities could be used in algebraic education. The study shows that more research is needed to determine how technology best can be implemented and used effectively in teaching. The work also shows that such research and development needs to be implemented with a user-centered approach with the involvement of relevant target groups.
Svensk undervisning är i behov att förändras för att vända den negativa trenden i PISA och TIMSS. Samtidigt är digitala artefakter alltmer närvarande i klassrummet och kommersiella applikationer alltmer tillgängliga. Det är svårt att avgöra vilka tekniska verktyg som ger faktiska resultat och det är ännu inte klarlagt vilken roll tekniken bör spela i en klassrumssituation. I detta arbete undersöks hur visualiseringar och teknik kan användas för att främja djupare förståelse inom algebra genom att adressera missuppfattningar som ofta förekommer hos elever. Observation och intervjuer genomfördes med elever på två skolor, en i Norrköping och en i Östersund. Teori om algebraiska missuppfattningar har studerats och sammanställts i två kategorier. Dessutom har tre prototyper utvecklats tillsammans med, och utvärderats av, en fokusgrupp. Arbetet har resulterat i förslag på vad tekniska och interaktiva aktiviteter skulle kunna innebära i algebraisk matematikundervisning. Det visar att mer forskning behövs för att avgöra hur teknik på bästa sätt kan implementeras och utnyttjas effektivt inom undervisning. Arbetet visar också att sådan forskning och utveckling behöver genomföras med en användarcentrerad ansats med delaktighet från berörda målgrupper.

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Tolmie, Julie. "Visualisation, navigation and mathematical perception : a visual notation for rational numbers mod 1." View thesis entry in Australian Digital Theses Program, 2000. http://thesis.anu.edu.au/public/adt-ANU20020313.101505/index.html.

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Vittadello, Sean T. "Mathematical models for cell migration and proliferation informed by visualisation of the cell cycle." Thesis, Queensland University of Technology, 2020. https://eprints.qut.edu.au/204074/1/Sean_Vittadello_Thesis.pdf.

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Cell migration and proliferation are essential for normal physiological processes, however their misregulation contributes to pathologies including cancer. In this thesis we develop and analyse new mathematical models of cell migration and proliferation, based on new experimental studies that provide visualisation of cell cycle progression, to improve understanding of the migration and proliferation of cells. In particular, we investigate cell migration as a function of cell cycle dynamics, normally-hidden cell synchronisation in cellular assays, whether cell migration and proliferation are mutually exclusive processes, and cellular mechanisms that contribute to heterogeneous cell proliferation.

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Quan, Chaoyu. "Mathematical methods for implicit solvation models in quantum chemistry." Thesis, Paris 6, 2017. http://www.theses.fr/2017PA066587/document.

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Cette thèse est consacrée à étudier et à améliorer les modèles mathématiques et les méthodes utilisées pour les modèles de solvatation implicite en chimie quantique. Ce manuscrit est composée de deux parties. Dans la première partie où nous analysons l'interface soluté-solvant, nous donnons, pour la première fois, une caractérisation complète de la surface moléculaire lisse, c'est-à-dire la surface exclue du solvant (SES). À partie de cette caractérisation, nous développons un algorithme de maillage par morceaux pour les surfaces moléculaires différentes, en particulier pour la SES, en utilisant la triangulation à front avançant. De plus, la cavité de la SES (la région entourée par la SES) est une description plus précise de la cavité de soluté. Dans la deuxième partie, nous construisons donc un modèle de continuum polarisable basé (PCM) sur la SES, dans lequel le paramètre de permittivité diélectrique est continu. Le problème électrostatique de ce modèle consiste à résoudre une équation de Poisson définie sur R3. Nous développons ensuite une méthode de Schwarz particulière, où seules les équations locales restreintes à des boules doivent être résolues. Enfin, nous étudions le modèle de solvatation de Poisson-Boltzmann, un autre modèle de solvatation implicite, qui tient compte à la fois de la permittivité diélectrique et de la force ionique du solvant. Une méthode de Schwarz similaire est proposée pour résoudre l'équation de Poisson-Boltzmann associée en résolvant des équations locales restreintes aux boules comme pour le PCM basé sur la SES
This thesis is devoted to study and improve the mathematical models and methods used in implicit solvation models in quantum chemistry. The manuscript is composed of two parts. In the first part where we analyze the solute-solvent interface, we give, for the first time, a complete characterization of the so-called “smooth” molecular surface, i.e., the solvent excluded surface (SES). Based on this characterization, we develop a piecewise meshing algorithm for different molecular surfaces, especially the SES, using the advancing-front triangulation. Further, it has been pointed out in the literature that the SES-cavity (the region enclosed by the SES) is a more accurate description of the solute cavity. In the second part, we therefore construct an SES-based polarizable continuum model (PCM), in which the dielectric permittivity parameter is continuous. The electrostatic problem of this model involves solving a Poisson equation defined in R3. We then develop a particular Schwarz domain decomposition method where only local equations restricted to balls need to be solved. Finally, the Poisson-Boltzmann solvation model, another implicit solvation model, is also investigated, which takes into account both the dielectric permittivity and the ionic strength of the solvent. A similar Schwarz domain decomposition method is proposed to solve the associated Poisson-Boltzmann equation by solving local equations restricted to balls as it is for the SES-based PCM

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Hurdal, Monica Kimberly. "Mathematical and computer modelling of the human brain with reference to cortical magnification and dipole source localisation in the visual cortx." Thesis, Queensland University of Technology, 1998.

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Ivana, Milanović. "Oбрада функције дате помоћу одређеног интеграла у процесу математичког моделирања." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2015. https://www.cris.uns.ac.rs/record.jsf?recordId=90360&source=NDLTD&language=en.

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У докторској дисертацији презентовано је педагошко истраживање које се односи на увођењематематичког моделирања као савременог методичког приступа учењу математике, његову примену унаставној пракси, као и на могућност да се кроз математичко моделирање проблема, појава и процеса којисе обрађују у неком другом наставном предмету оствари модернији, интердисциплинарни приступсредњошколској настави. Све то у циљу афирмације иновативног, креативног и напредног математичкогмишљења, квалитетног, структуираног и функционалногзнања, које се у овом истраживању односи наодређене наставне садржаје математичке анализе. У наставни процес интегрисан је осмишљен и израђенпројекат чија је реализација омогућила успостављањемеђупредметне корелације, изучавање функција ињихових примена у научним проблемима кроз процесе математичког моделирања. Нарочита пажњапосвећена је реализацији когнитивних активности ученика у свакој етапи процеса математичкогмоделирања и остваривању њиховог вертикално-кумулативног поретка. Као посебна активност усмерена напотребе и циљеве одабраних процеса математичког моделирања и усклађена са наставним садржајимакоји су изучавани, уведен је и примењен нови методички приступ у обради функција датих помоћуодређеног интеграла, са посебним акцентом на анализу логаритамске функције дате помоћу одређеногинтеграла, а затим и анализу експоненцијалне функције, као инверзне логаритамској. Један деоистраживања фокусиран је на увођење и реализацију назначене нове методичке концепције у обрадифункција датих помоћу одређеног интеграла и на високошколском нивоу учења математичке анализе.Испитивање ефикасности примењених нових, савременихметодичких приступа у раду са ученицима,односно студентима, уз планску и систематску употребу рачунара са одговарајућом софтверскомподршком, обрађено је компаративном анализом резултата педагошких експеримената. На основурезултата истраживања утврђен је позитиван утицај предложених методичких приступа на квалитетматематичког знања ученика, односно студената, и оствареност оптималних резултата у учењу и изучавањунаставних садржаја из области функције.
U doktorskoj disertaciji prezentovano je pedagoško istraživanje koje se odnosi na uvođenjematematičkog modeliranja kao savremenog metodičkog pristupa učenju matematike, njegovu primenu unastavnoj praksi, kao i na mogućnost da se kroz matematičko modeliranje problema, pojava i procesa kojise obrađuju u nekom drugom nastavnom predmetu ostvari moderniji, interdisciplinarni pristupsrednjoškolskoj nastavi. Sve to u cilju afirmacije inovativnog, kreativnog i naprednog matematičkogmišljenja, kvalitetnog, struktuiranog i funkcionalnogznanja, koje se u ovom istraživanju odnosi naodređene nastavne sadržaje matematičke analize. U nastavni proces integrisan je osmišljen i izrađenprojekat čija je realizacija omogućila uspostavljanjemeđupredmetne korelacije, izučavanje funkcija injihovih primena u naučnim problemima kroz procese matematičkog modeliranja. Naročita pažnjaposvećena je realizaciji kognitivnih aktivnosti učenika u svakoj etapi procesa matematičkogmodeliranja i ostvarivanju njihovog vertikalno-kumulativnog poretka. Kao posebna aktivnost usmerena napotrebe i ciljeve odabranih procesa matematičkog modeliranja i usklađena sa nastavnim sadržajimakoji su izučavani, uveden je i primenjen novi metodički pristup u obradi funkcija datih pomoćuodređenog integrala, sa posebnim akcentom na analizu logaritamske funkcije date pomoću određenogintegrala, a zatim i analizu eksponencijalne funkcije, kao inverzne logaritamskoj. Jedan deoistraživanja fokusiran je na uvođenje i realizaciju naznačene nove metodičke koncepcije u obradifunkcija datih pomoću određenog integrala i na visokoškolskom nivou učenja matematičke analize.Ispitivanje efikasnosti primenjenih novih, savremenihmetodičkih pristupa u radu sa učenicima,odnosno studentima, uz plansku i sistematsku upotrebu računara sa odgovarajućom softverskompodrškom, obrađeno je komparativnom analizom rezultata pedagoških eksperimenata. Na osnovurezultata istraživanja utvrđen je pozitivan uticaj predloženih metodičkih pristupa na kvalitetmatematičkog znanja učenika, odnosno studenata, i ostvarenost optimalnih rezultata u učenju i izučavanjunastavnih sadržaja iz oblasti funkcije.
In this PhD dissertation pedagogical research related to introduction of mathematical modeling as amodern methodical approach to learning mathematics is presented, its application in teaching practice,as wellas possibilities to achieve modern, interdisciplinary approach to high school teaching, using mathematicalmodeling of problems, phenomenons and processes which are being looked at in another ciruculum subject. Theintention is to affirm innovative, creative and advanced mathematical thinking, high quality, structured andfunctional knowledge, which in this study refers tospecific areas of mathematical analysis. A well thoughtproject was designed and constructed in such a way to integrate within the teaching process, and itsimplementation has enabled the establishment of interdisciplinary correlation and the study of functions andtheir use in scientific problems through the process of mathematical modelling. Special attention is paid to therealization of cognitive activities of pupils in each stage of the process of mathematical modelling, and acheivingtheir vertically-cumulative order. As a separate activity focused on the goals and needs of selected processes ofmathematical modeling, harmonized with teaching content being studied, a new methodical approach wasintroduced and applied to treat functions given by a definite integral, with special emphasis on treatment oflogarithmic function given by a definite integral, and followed by treatment of the exponential function as theinverse logarithmic. A part of the research is focused on the introduction and implementation of new methodicalconcept in treatment of functions given by a definite integral and on university level of studying mathematicalanalysis. Efficiency of these new and modern methodical approaches was tested on pupils and students, withplanned and systematic usage of computers and appropriate software, and processed by comparative analysis ofresults obtained by pedagogical experiments. Results have shown positive impact of proposed methodicalapproaches to the quality of pupils’ and students’ mathematical knowledge, and the achievement of optimalresults in learning and studying of teaching content on mathematical functions.

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Crawford, Carl Jonathan. "Dynamic multilevel graph layout and visualisation." Thesis, University of Greenwich, 2016. http://gala.gre.ac.uk/18104/.

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This thesis addresses the issue of efficient dynamic graph drawing for large scale connected graphs with around 10,000 vertices. It contains three main contributions. Firstly, an efficient method for approximating the n-body calculations used in Force Directed Placement (FDP) is described, exploiting use of a multilevel scheme to approximate distance between groups of vertices much like the Barnes Hut Octree. The method suggests better representation of the graphs underlying relationships. In experiments this algorithm, referred to as Multilevel Global Force (MGF), reduces running time by an average of 40% compared to the popular Barnes Hut Octree approximation method. Secondly, optimisation methods used in static graph drawing (such as multilevel and approximation schemes) are adapted for use in dynamic graph drawing, simultaneously improving the quality of layouts produced and reducing the complexity such that large graphs with thousands of vertices can be drawn in interactive time. Thirdly, several techniques are introduced for incorporating graph changes into the dynamic graph drawing to differing extents, allowing the viewer to decide whether the ongoing layout should preserve the original layout or prioritise the graph changes. The works are combined to form an efficient multilevel dynamic graph drawing algorithm.

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Sun, Yi. "Non-linear hierarchical visualisation." Thesis, Aston University, 2002. http://publications.aston.ac.uk/13263/.

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This thesis applies a hierarchical latent trait model system to a large quantity of data. The motivation for it was lack of viable approaches to analyse High Throughput Screening datasets which maybe include thousands of data points with high dimensions. We believe that a latent variable model (LTM) with a non-linear mapping from the latent space to the data space is a preferred choice for visualising a complex high-dimensional data set. As a type of latent variable model, the latent trait model can deal with either continuous data or discrete data, which makes it particularly useful in this domain. In addition, with the aid of differential geometry, we can imagine that distribution of data from magnification factor and curvature plots. Rather than obtaining the useful information just from a single plot, a hierarchical LTM arranges a set of LTMs and their corresponding plots in a tree structure. We model the whole data set with a LTM at the top level, which is broken down into clusters at deeper levels of the hierarchy. In this manner, the refined visualisation plots can be displayed in deeper levels and sub-clusters may be found. Hierarchy of LTMs is trained using expectation-maximisation (EM) algorithm to maximise its likelihood with respect to the data sample. Training proceeds interactively in a recursive fashion (top-down). The user subjectively identifies interesting regions on the visualisation plot that they would like to model in a greater detail. At each stage of hierarchical LTM construction, the EM algorithm alternates between the E - and M - step. Another problem that can occur when visualising a large data set is that there may be significant overlaps of data clusters. It is very difficult for the user to judge where centres of regions of interest should be put. We address this problem by employing the minimum message length technique, which can help the user to decide the optimal structure of the model.

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Mårtensson, Jonathan. "Visualisation of Risk Level in Portfolio Management." Thesis, Uppsala University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121375.

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Aguilera, cabanas Jorge antonio. "Robustesse et visualisation de production de mélanges." Phd thesis, Université de Grenoble, 2011. http://tel.archives-ouvertes.fr/tel-00647297.

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Le procédé de fabrication de mélanges (PM) consiste à déterminer les proportions optimales à mélanger d'un ensemble de composants de façon que le produit obtenu satisfasse un ensemble de spécifications sur leurs propriétés. Deux caractéristiques importantes du problème de mélange sont les bornes dures sur les propriétés du mélange et l'incertitude répandue dans le procédé. Dans ce travail, on propose une méthode pour la production de mélanges robustes en temps réel qui minimise le coût de la recette et la sur-qualité du mélange. La méthode est basée sur les techniques de l'Optimisation Robuste et sur l'hypothèse que les lois des mélange sont linéaires. On exploite les polytopes sous-jacents pour mesurer, visualiser et caractériser l'infaisabilité du PM et on analyse la modification des bornes sur les composants pour guider le procédé vers le ''meilleur'' mélange robuste. On propose un ensemble d'indicateurs et de visualisations en vue d'offrir une aide à la décision.

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Tshithigona, Gaus. "An investigation into teaching mathematics using a visualisation approach to recontextualise indigenous knowledge." Thesis, Rhodes University, 2018. http://hdl.handle.net/10962/62461.

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It can be argued that the Namibian curriculum is largely influenced by a Western epistemology. However, many studies suggest that learners make powerful meanings of mathematical concepts they are learning if they visualise these and experience them in relation to what they already know from their own cultural backgrounds. According to the Namibian National Curriculum for Basic Education (NNCBE, 2010), it is not only important for learners to acquire mathematical knowledge and skills, but also to develop and grow their identities, cultures and values as individuals. The aim of this study was to explore and investigate how selected mathematics teachers employ visualisation as a teaching approach to re-contextualise indigenous knowledge (IK). The study set out to explore how conceptual understanding is enhanced by participation in an intervention programme. The use of visualisation is considered an important mediating and pedagogical tool in the mathematics classroom to enhance the mathematics learning of learners. The research is informed by a socio-cultural theory of learning and is located within an interpretive paradigm. The study was conducted at four schools in the Oshana region of Namibia and involved four mathematics teachers who were purposefully selected due to their willingness to use visualisation-IK approaches in their teaching, based on their responses to the survey. The methodologies used are qualitative and quantitative case study. To gather data, document analysis, a survey, lesson observations and focus group interviews were used. The study found that most teachers in the Oshana region have an understanding of the effective use of visualisation-IK approaches. However, the study revealed that visualisation- IK approaches are mostly used in grades 4 to 7 mathematics classrooms. It also discovered that most of the visualisation approaches that teachers employed aligned well with the curriculum and promoted conceptual understanding in the teaching of mathematics. The results of the study showed that teachers have experience of the enabling and constraining factors in using a visualisation-IK approach to teaching mathematics.

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Malabar, Ian. "The use of computer technology and constructivism to enhance visualisation skills in mathematics education." Thesis, Liverpool John Moores University, 2003. http://researchonline.ljmu.ac.uk/5627/.

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Harris, Richard Alan. "An analysis of the genetic algorithm and abstract search space visualisation." Thesis, University of Plymouth, 1996. http://hdl.handle.net/10026.1/2087.

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The Genetic Algorithm (Holland, 1975) is a powerful search technique based upon the principles of Darwinian evolution. In its simplest form the GA consists of three main operators - crossover, mutation and selection. The principal theoretical treatment of the Genetic Algorithm (GA) is provided by the Schema Theorem and building block hypothesis (Holland, 1975). The building block hypothesis describes the GA search process as the combination, sampling and recombination of fragments of solutions known as building blocks. The crossover operator is responsible for the combination of building blocks, whilst the selection operator allocates increasing numbers of samples to good building blocks. Thus the GA constructs the optimal (or near-optimal) solution from those fragments of solutions which are, in some sense, optimal. The first part of this thesis documents the development of a technique for the isolation of building blocks from the populations of the GA. This technique is shown to extract exactly those building blocks of interest - those which are sampled most regularly by the GA. These building blocks are used to empirically investigate the validity of the building block hypothesis. It is shown that good building blocks do not combine to form significantly better solution fragments than those resulting from the addition of randomly generated building blocks to good building blocks. This results casts some doubt onto the value of the building block hypothesis as an account of the GA search process (at least for the functions used during these experiments). The second part of this thesis describes an alternative account of the action of crossover. This account is an approximation of the geometric effect of crossover upon the population of samples maintained by the GA. It is shown that, for a simple function, this description of the crossover operator is sufficiently accurate to warrant further investigation. A pair of performance models for the GA upon this function are derived and shown to be accurate for a wide range of crossover schemes. Finally, the GA search process is described in terms of this account of the crossover operator and parallels are drawn with the search process of the simulated annealing algorithm (Kirkpatrick et al, 1983). The third and final part of this thesis describes a technique for the visualisation of high dimensional surfaces, such as are defined by functions of many parameters. This technique is compared to the statistical technique of projection pursuit regression (Friedman & Tukey, 1974) and is shown to compare favourably both in terms of computational expense and quantitative accuracy upon a wide range of test functions. A fundamental flaw of this technique is that it may produce poor visualisations when applied to functions with a small high frequency (or order) components.

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Samson, Duncan Alistair. "The heuristic significance of enacted visualisation." Thesis, Rhodes University, 2012. http://hdl.handle.net/10962/d1003434.

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This study is centred on an analysis of pupils' lived experience while engaged in the generalisation of linear sequences/progressions presented in a pictorial context. The study is oriented within the conceptual framework of qualitative research, and is anchored within an interpretive paradigm. A case study methodological strategy was adopted, the research participants being the members of a mixed gender, high ability Grade 9 class of 23 pupils at an independent school in South Africa. The analytical framework is structured around a combination of complementary multiple perspectives provided by three theoretical ideas, enactivism, figural apprehension, and knowledge objectification. An important aspect of this analytical framework is the sensitivity it shows to the visual, phenomenological and semiotic aspects of figural pattern generalisation. It is the central thesis of this study that the combined complementary multiple perspectives of enactivism, figural apprehension and knowledge objectification provide a powerful depth of analysis to the exploration of the inter-relationship between the embodied processes of pattern generalisation and the visualisation of pictorial cues. The richly textured tapestry of activity captured through a multi-systemic semiotic analysis of participants' generalisation activity stands testament to this central thesis. Insights gleaned from this study are presented as practical strategies which support and encourage a multiple representational approach to pattern generalisation in the pedagogical context of the classroom.

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Gerussi, Alexandre. "Analyse multirésolution non emboîtée : applications à la visualisation scientifique." Phd thesis, Université Joseph Fourier (Grenoble), 2000. http://tel.archives-ouvertes.fr/tel-00006745.

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Cette thèse présente une construction générale d'ondelettes de seconde génération dont l'originalité est de distinguer deux points vues complémentaires : le point de vue de subdivision, qui souligne le lien bien connu entre les schémas de subdivision et l'analyse multirésolution, et d'autre part le point de vue non emboîté, dans lequel les espaces d'approximation, qui remplacent les espaces d'échelle traditionnels, ne sont plus nécessairement imbriqués. Dans la première partie de la thèse, le cadre multirésolution est présenté puis divers aspects théoriques, essentiellement relatifs au point de vue non emboîté, sont étudiés. En particulier, plusieurs techniques de constructions des opérateurs d'analyse ou de synthèse sont présentées. La deuxième partie de la thèse est consacrée aux applications. Le point de vue non emboîté est utilisé pour développer un cadre multirésolution pour fonctions constantes ou linéaires par morceaux définies sur des triangulations irrégulières d'un domaine planaire ou sphérique, permettant notamment la visualisation progressive de grands volumes de données. Les algorithmes de décomposition et de reconstruction des données sont discutés en détails notamment du point de vue de leur implémentation effective, plus délicate que dans le cas des ondelettes classiques. Des applications traditionnelles telles que la compression ou l'édition à différents niveaux de détails sont également généralisées à ces fonctions. D'autre part, est également discutée l'utilisation du cadre non emboîté pour l'approximation et la reconstruction de fonctions définies sur des maillages surfaciques construits via des modèles multirésolution basés sur les techniques de décimation de maillages. Enfin, on montre à diverses reprises que le point de vue non emboîté permet un abord unifié des algorithmes basés sur les ondelettes et des techniques décimatoires, traditionnellement opposées.

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Cena, Bernard Maria. "Reconstruction for visualisation of discrete data fields using wavelet signal processing." University of Western Australia. Dept. of Computer Science, 2000. http://theses.library.uwa.edu.au/adt-WU2003.0014.

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The reconstruction of a function and its derivative from a set of measured samples is a fundamental operation in visualisation. Multiresolution techniques, such as wavelet signal processing, are instrumental in improving the performance and algorithm design for data analysis, filtering and processing. This dissertation explores the possibilities of combining traditional multiresolution analysis and processing features of wavelets with the design of appropriate filters for reconstruction of sampled data. On the one hand, a multiresolution system allows data feature detection, analysis and filtering. Wavelets have already been proven successful in these tasks. On the other hand, a choice of discrete filter which converges to a continuous basis function under iteration permits efficient and accurate function representation by providing a “bridge” from the discrete to the continuous. A function representation method capable of both multiresolution analysis and accurate reconstruction of the underlying measured function would make a valuable tool for scientific visualisation. The aim of this dissertation is not to try to outperform existing filters designed specifically for reconstruction of sampled functions. The goal is to design a wavelet filter family which, while retaining properties necessary to preform multiresolution analysis, possesses features to enable the wavelets to be used as efficient and accurate “building blocks” for function representation. The application to visualisation is used as a means of practical demonstration of the results. Wavelet and visualisation filter design is analysed in the first part of this dissertation and a list of wavelet filter design criteria for visualisation is collated. Candidate wavelet filters are constructed based on a parameter space search of the BC-spline family and direct solution of equations describing filter properties. Further, a biorthogonal wavelet filter family is constructed based on point and average interpolating subdivision and using the lifting scheme. The main feature of these filters is their ability to reconstruct arbitrary degree piecewise polynomial functions and their derivatives using measured samples as direct input into a wavelet transform. The lifting scheme provides an intuitive, interval-adapted, time-domain filter and transform construction method. A generalised factorisation for arbitrary primal and dual order point and average interpolating filters is a result of the lifting construction. The proposed visualisation filter family is analysed quantitatively and qualitatively in the final part of the dissertation. Results from wavelet theory are used in the analysis which allow comparisons among wavelet filter families and between wavelets and filters designed specifically for reconstruction for visualisation. Lastly, the performance of the constructed wavelet filters is demonstrated in the visualisation context. One-dimensional signals are used to illustrate reconstruction performance of the wavelet filter family from noiseless and noisy samples in comparison to other wavelet filters and dedicated visualisation filters. The proposed wavelet filters converge to basis functions capable of reproducing functions that can be represented locally by arbitrary order piecewise polynomials. They are interpolating, smooth and provide asymptotically optimal reconstruction in the case when samples are used directly as wavelet coefficients. The reconstruction performance of the proposed wavelet filter family approaches that of continuous spatial domain filters designed specifically for reconstruction for visualisation. This is achieved in addition to retaining multiresolution analysis and processing properties of wavelets.

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Pendlington, Sandra. "Teaching mathmatics to low achievers in the primary school : strategies and visualisations to raise self-esteem and improve mathematical competence." Thesis, University of Bristol, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.404150.

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Kroliczek, Filip. "Matematický model parní turbíny." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2014. http://www.nusl.cz/ntk/nusl-221118.

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The goal of this thesis was to create a mathematical model of a steam turbine based on the data acquired by measurement, and to verify its behaviour. The first part contains research, which is supposed to introduce basic principles of the steam turbine and description of important construction parts and the possibilities of control. The second part describes the experimental identification method of least squares, used for the calculation of an ARX model of the steam turbine. Finally, the last part focuses on the program environment used for creation of the mathematical model and explanation of measured data analysis process. Furthermore this segment describes the created simulation program as well as a visualisation of the dynamic processes in the steam turbine, including the design of control.

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Nghifimule, Selma Ndilipomwene. "An analysis of the nature of visualisation objects in three Namibian grade 9 mathematics textbooks: a case study in Namibia." Thesis, Rhodes University, 2017. http://hdl.handle.net/10962/6334.

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Textbooks are a universal and central element of teaching and learning mathematics (Namibia. Ministry of Education [MoE], 2008). Steenpaß and Steinbring (2014) state that diagrams in mathematics textbooks are often used as Visualisation Objects (VOs) to enhance learning of mathematical concepts. VOs in textbooks are thus important teaching and learning tools (Fotakopoulou and Spiliotopoulou, 2008). This Namibian interpretive case study analysed the nature of VOs used in the three approved grade 9 Namibian mathematics textbooks namely: y=mx+c to success, Maths for Life 9 and Discover Mathematics 9. The VOs were analysed by using an analytical framework adapted from Fotakopoulou and Spiliotopoulou (2008). This analytic tool was specifically used to interrogate the following categories: the type of VOs, the roles of VOs, the relation of VOs to mathematical content, the relation of VOs to reality, and their properties. The 266 VOs under study were collected from the Algebra and Geometry chapters of each book. This study also included survey questionnaires with the 50 selected mathematics teachers, which sought their views and perceptions on the use of the identified VOs. In addition, the authors’ rationale in selecting the identified VOs used in their textbooks was sought through interviews. This research study is part of the “Visualisation in Namibia and Zambia” (VISNAMZA) project which seeks to research the effective use of visualisation processes in the mathematics classroom in Namibia and Zambia (Schäfer, 2015). It is hoped that this study contributes towards improving the quality of textbook evaluations, and design of suitable and more comprehensive assessment procedures in Namibia. It is also hoped that it creates a critical awareness of the roles of VOs in textbooks amongst teachers, inspiring them to help their learners interpret VOs effectively. It should also inspire potential authors to use suitable and appropriate VOs that enhance conceptual teaching and learning of mathematics. The study discovered that most of the VOs used in the selected textbooks align well with the mathematical content. The VOs can help make abstract ideas concrete, stimulate learning, simplify and clarify written texts. In addition, VOs can also be used as a tool for reasoning and an instrument for problem solving. The findings however also indicate that some of the VOs used are not self-explanatory; they are vague, unfamiliar and confusing, leading to misinterpretations by some learners. Another interesting finding was that some of the learners found it difficult to interpret VOs on their own without the help of the teacher.

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Bergh, Johann Rath. "Ontology comprehension." Thesis, Stellenbosch : University of Stellenbosch, 2011. http://hdl.handle.net/10019.1/6517.

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Thesis (MSc (Mathematical Sciences. Computer Science))--University of Stellenbosch, 2011.
ENGLISH ABSTRACT: Ontologies are conceptual models of a domain of discourse and are used in a number of applications to model a field of knowledge. For example, SNOMED, an ontology of medical terminology, is widely used among medical professionals. Commercial ontologies, such as SNOMED, can have hundreds of thousands of concepts. People who want to use these ontologies need an understanding thereof, but the sheer magnitude of these ontologies hampers comprehension. It was within this context that the need arose for software tools that facilitate the understanding of ontologies. Given this background, our aim is to investigate a new area within the field of ontologies, namely, ontology comprehension. We make a contribution to it by developing an ontology comprehension framework and writing a software tool of our own. This software tool, PathViz, helps users to understand how different concepts in an ontology are related to each other and what effect entailments have on the way concepts in an ontology relate to each other. The ontology comprehension framework, PathViz and the reasoning measurement instruments were found useful for ontology comprehension by participants at an ontology workshop.
AFRIKAANSE OPSOMMING: Ontologieë is konseptuele modelle van ’n domein en word in verskeie toepassings gebruik om ’n kennisveld te modelleer. SNOMED is ’n voorbeeld van ’n ontologie van mediese terme wat baie gebruik word deur die mediese beroepslui. Kommersiële ontologieë, soos SNOMED, kan bestaan uit duisende konsepte. Dit is belangrik om hierdie ontologieë wat gebruik word te verstaan, maar die enorme omvang van hierdie ontologieë belemmer die verstaanproses. In hierdie konteks het die behoefte ontstaan vir programmatuur wat die verstaanproses van ontologieë vergemaklik. Met hierdie agtergrond inaggenome, is dit ons doel om ’n nuwe area in die veld van ontologieë te ondersoek, naamlik, Ontologie-begrip. Ons maak ’n bydra tot hierdie veld deur ’n raamwerk vir ontologie-begrip te ontwikkel en programmatuur van ons eie te skryf. Hierdie programmatuur, PathViz, help gebruikers om te verstaan hoe verskillende konsepte in ’n ontologie aan mekaar verwant is. Verder help dit gebruikers om te verstaan watter invloed afleidings uit die ontologieë het op konsepverwantskappe. Deelnemers aan ’n ontologie-werkswinkel het gevind dat die raamwerk vir ontologie-begrip, PathViz en die instrumente wat die invloed van die ontologie-redeneerder meet, ontologie-begrip bevorder.

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Boutin, François. "Filtrage, partitionnement et visualisation multi-échelles de graphes d'interactions à partir d'un focus." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2005. http://tel.archives-ouvertes.fr/tel-00113852.

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Cette thèse étudie les caractéristiques de réseaux interactions réels, notamment les propriétés « petit monde » et « sans échelle ». Elle présente diverses techniques de filtrage, partitionnement et visualisation de ces réseaux.
La propriété « sans échelle » provient du fait que ces réseaux d'interactions sont le plus souvent dans un processus de croissance : les nouveaux noeuds établissent, de préférence, des liens avec des noeuds existants à fort degré. On parle d'« attachement préférentiel ». La propriété « petit monde » s'explique par l'adage : « les amis de mes amis sont mes amis ».
Les réseaux d'interaction présentent souvent un noyau dense difficilement analysable et visualisable à l'aide de techniques de partitionnement et de dessin classiques.
Cette étude introduit une nouvelle technique de filtrage permettant l'extraction d'une structure dite « arborée » ayant également les propriétés « petit monde » et « sans échelle ». Le réseau, ainsi filtré, est organisé en un arbre de silhouettes emboîtées. Cette structure multi-échelles, facilement visualisable et navigable, présente une organisation contextuelle du réseau autour d'un focus
utilisateur.
La nouvelle technique de partitionnement optimise les critères de qualité recensés et introduits dans cette thèse. Par ailleurs, l'étude du contenu des silhouettes, menée a posteriori, souligne la qualité de l'utilisation conjointe du filtrage et du partitionnement.

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Latapie, Simon. "Visualisation interactive de simulations à grand nombre d'atomes en Physique des Matériaux." Phd thesis, Ecole Centrale Paris, 2009. http://tel.archives-ouvertes.fr/tel-00453299.

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Afin de répondre au besoin croissant de puissance de calcul qui intéresse les applications scientifiques, des architectures de calculateurs de plus en plus parallèles se répandent, de type cluster, MPP, ou autre, avec de plus en plus d'unités de calcul. Les codes de calcul, en s'adaptant à ce parallélisme, produisent de fait des résultats de plus en plus volumineux. Dans le domaine plus spécifique de la dynamique moléculaire appliquée à la physique des matériaux, le nombre de particules d'un système étudié est aujourd'hui de l'ordre de quelques millions à quelques centaines de millions, ce qui pose d'importants problèmes d'exploitation des données produites. Les solutions de visualisation utilisées auparavant ne sont plus capables de traiter une telle quantité d'information de façon interactive. Le but de cette thèse est de concevoir et d'implémenter une solution de manipulation de données composées d'un très grand nombre de particules, formant un ensemble .dense. ( par exemple, une simulation en dynamique moléculaire de quelques dizaines de millions d'atomes ). Cette solution devra être adaptative pour pouvoir fonctionner non seulement sur un poste de bureau de type .desktop., mais aussi dans des environnements collaboratifs et immersifs simples, par exemple de type mur d'image, avec éventuellement des éléments de réalité virtuelle (écrans stéréo, périphériques 3D). De plus, elle devra augmenter l'interactivité, c'est-à-dire rendre le système plus apte à réagir avec l'utilisateur, afin d'effectuer des phases d'exploration des données plus efficaces. Nous proposons une solution répondant à ces besoins de visualisation particulaire dense, en apportant des améliorations : - à l'interaction, par un système permettant une exploration simple et efficace d'une simulation scientifique. A cette fin, nous avons conçu, développé, et testé le .FlowMenu3D., une extension tridimensionnelle du .FlowMenu., un menu octogonal utilisé en environnement 2D. - à l'interactivité, par une architecture optimisée pour le rendu de systèmes particulaires denses. Cette dernière, intégrée au framework de visualisation VTK, est conçue sur le principe d'un rendu parallèle hybride sort-first/sort-last, dans lequel le système sort-last Ice-T est couplé à des extensions de partitionnement spatial des données, d'occlusion statistique par une méthode de Monte-Carlo, mais aussi de programmation GPU à base de shaders pour améliorer la performance et la qualité du rendu des sphères représentant les données.

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Dufour, Alexandre. "Segmentation, suivi et visualisation d'objets biologiques en microscopie 3D par fluorescence : Approches par modèles déformables." Phd thesis, Université René Descartes - Paris V, 2007. http://tel.archives-ouvertes.fr/tel-00271191.

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Nous nous intéressons à la détection et au suivi d'objets biologiques divers (cellules, noyaux, etc.) dans des images et séquences tri-dimensionnelles acquises en microscopie par fluorescence. L'observation de phénomènes biologiques in situ étant de plus en plus cruciale pour les experts, il est nécessaire, en plus de l'analyse quantitative, d'effectuer un rendu volumique 3D de la scène et des objets qui y évoluent. De plus, l'automatisation des techniques d'acquisition d'images requiert un haut niveau de reproductibilité des algorithmes et induit souvent des contraintes de temps de calcul que nous nous efforçons de prendre en compte.

Les modèles déformables, également connus sous le nom de contours actifs, font actuellement partie des méthodes de pointe en analyse d'images pour la segmentation et le suivi d'objets grâce à leur robustesse, leur flexibilité et leur représentation à haut niveau sémantique des entités recherchées. Afin de les adapter à notre problématique, nous devons faire face à diverses difficultés. Tout d'abord, les méthodes existantes se réfèrent souvent aux variations locales d'intensité (ou gradients) de l'image pour détecter le contour des objets recherchés. Cette approche est inefficace en microscopie tridimensionnelle par fluorescence, où les gradients sont très peu prononcés selon l'axe de profondeur de l'image. Ensuite, nous devons gérer le suivi d'objets multiples susceptibles d'entrer en contact en évitant leur confusion. Enfin, nous devons mettre en place un système permettant de visualiser efficacement les contours durant leur déformation sans altérer les temps de calcul.

Dans la première partie de ce travail, nous pallions à ces problèmes en proposant un modèle de segmentation et de suivi multi-objets basé sur le formalisme des lignes de niveaux (ou level sets) et exploitant la fonctionnelle de Mumford et Shah. La méthode obtenue donne des résultats quantitatifs satisfaisants, mais ne se prête pas efficacement au rendu 3D de la scène, pour lequel nous sommes tributaires d'algorithmes dédiés à la reconstruction 3D (e.g. la méthode des "Marching Cubes"), souvent coûteux en mémoire et en temps de calcul. De plus, ces algorithmes peuvent induire des erreurs d'approximation et ainsi entraîner une mauvaise interprétation des résultats.

Dans la seconde partie, nous proposons une variation de la méthode précédente en remplaçant le formalisme des lignes de niveaux par celui des maillages triangulaires, très populaire dans le domaine de la conception assistée par ordinateur (CAO) pour leur rendu 3D rapide et précis. Cette nouvelle approche produit des résultats quantitatifs équivalents, en revanche le formalisme des maillages permet d'une part de réduire considérablement la complexité du problème et autorise d'autre part à effectuer un rendu 3D précis de la scène parallèlement au processus de segmentation, réduisant d'autant plus les temps de calculs.

Les performances des deux méthodes proposées sont d'abord évaluées puis comparées sur un jeu de données simulées reproduisant le mieux possible les caractéristiques des images réelles. Ensuite, nous nous intéressons plus particulièrement à l'évaluation de la méthode par maillages sur des données réelles, en évaluant la robustesse et la stabilité de quelques descripteurs de forme simples sur des expériences d'imagerie haut-débit. Enfin, nous présentons des applications concrètes de la méthode à des problématiques biologiques réelles, réalisées en collaboration avec d'autres équipes de l'Institut Pasteur de Corée.

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Muhembo, Gottfried Mbundu. "An analysis of how visualisation processes can be used by teachers participating in an intervention programme to teach for conceptual understanding of geometry." Thesis, Rhodes University, 2018. http://hdl.handle.net/10962/62439.

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Visualisation in general and visualisation processes in particular have received much attention in the mathematics education research literature. Literature suggests that the appropriate use of visualisation helps learners to develop their conceptual understanding and skills of geometry as it allows them to visually interpret and understand fundamental mathematical and geometrical concepts. It is claimed that visual tools play an important role in communicating mathematical ideas through diagrams, gestures, images, sketches or drawings. Learning mathematics through visualisation can be a powerful tool to explore mathematical problems and give meaning to mathematical concepts and relationships between them. This interpretive case study focused on how selected teachers taught concepts in geometry through visualisation processes for conceptual understanding as a result of an intervention programme. The study was conducted at four high schools by four mathematics teachers in the Kavango East Region in Northern Namibia. The participants were involved in a three-week intervention programme and afterwards taught three lessons each on the topic of geometry. The data collection method of this research was: focus group and stimulus recall interviews, classroom observations and recorded videos. This research is located in constructivism. I used vertical and horizontal analysis strategies to analyse the data. My analytical instrument consisted of an observation schedule which I used in each lesson to identify how each of the visualisation processes was evident in each of the observed lessons. This study revealed that the participant teachers used visualisation processes in most of their lessons and these processes were used accurately in line with the requirements of the grade 8 mathematics syllabi. The visualisation processes were used through designed visual materials, posters and through the use of geometrical objects such as chalkboard ruler, protractor and compass. The results from this study also confirmed that visualisation processes can be a powerful instructional tool for enhancing learners’ conceptual understanding of geometry.

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Baroin, Gilles. "Applications de la théorie des graphes à des objets musicaux : modélisations, visualisations en hyperespace." Phd thesis, Université Toulouse le Mirail - Toulouse II, 2011. http://tel.archives-ouvertes.fr/tel-00943407.

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A la frontière entre musique et mathématiques, cette étude présente un espace musical géométrique original utilisé pour l'analyse et la pédagogie.En utilisant différentes méthodes, les mathématiciens et théoriciens de la musique ont démontré que notre espace des hauteurs tempéré à douze notes peut être considéré comme une combinaison de tierces mineurs et majeures. Nous utilisons le produit cartésien de deux graphes circulaires C3□C4 pour construire le graphe Planet qui répond à ce concept. Comme la décomposition implique deux ensembles et que chaque classe de hauteur est la combinaison unique de ces deux sous-groupes, nous utilisons une coloration en termes de graphes par des nombres complexes et introduisons le concept d'idéogrammes à deux dimensions. Nous effectuons une analyse spectrale du graphe Planet pour déterminer ses espaces propres et obtenir des coordonnées géométriques. Le modèle qui en résulte est appelé Planet-4D, il offre à chaque symbole une position physiquement équivalente. Il comporte plus de symétries que tout modèle discret 3D. A partir de ce modèle, nous construisons une représentation en quatre dimensions où les accords parfaits se trouvent en surface d'une hypersphère. Nous étendons enfin le concept principal pour afficher n'importe quel agrégat de notes sur l'hypersphère dans un cadre atonal. Dans une seconde partie, nous modélisons sous forme de graphes des objets musicaux existants : claviers, réseaux de notes (Tonnetze) ou d'accords ainsi que des schémas de modulation. Nous appliquons des projections spectrales afin de visualiser les symétries inhérentes à ces objets et terminons par des études d'œuvres tonales et atonales, effectuées avec le système de visualisation inventé.

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Kangwa, Lemmy. "An investigation of how Visual Technology for the Autonomous Learning of Mathematics (VITALmaths) video clips on mobile phones can be used by student teachers as a visualisation tool in the teaching of Number Sense: a case study." Thesis, Rhodes University, 2017. http://hdl.handle.net/10962/6930.

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Visualisation is increasingly being recognised as having a significant role in the learning of mathematics especially when students are solving mathematical problems (Thornton, 2001). It is argued that visualisation is a powerful tool for learners to construct mental and physical representations that correctly mirror mathematical relationships and concepts. To gain a thorough understanding of the scope of visualisation, three Visual Technology for Autonomous Learning of Mathematics (VITALmaths) (www.vitalmaths.com) video clips were uploaded on mobile phones of each of the eleven participating student teachers who used them in their teaching practice. This is in cognisance of the educational potential offered by mobile phones and their current pervasiveness in the daily lives of both teachers and learners in Zambia (Zambia. Ministry of Education [M.O.E], 2013]). This study sought to investigate how VITALmaths video clips on mobile phones could be used by student teachers as a visualisation tool in the teaching of Number Sense. The videos of the lessons formed the core of my analysis. The study was conducted at four primary schools by eleven student teachers of a public university in Zambia. The study is framed as a case study and is grounded within the interpretive paradigm. The findings revealed that the student teachers’ use of the video clips in the classrooms for teaching was generally approached from two perspectives: at the beginning of the lesson or at the end. The videos were used at the beginning of the lesson as a means to introduce a topic or an idea, and at the end of the lesson to consolidate what was taught. The videos were also used to enhance the conceptual understanding of Number Sense. The findings also revealed that students encountered both enabling and constraining factors in their use of mobile phones to teach number sense. The overall findings revealed that, if well utilised, mobile phones as visualisation tools had the potential to enhance the teaching of Mathematics in general and Number Sense in particular, and therefore teachers should be encouraged to use them in their teaching.

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Brown, Steven Richard. "A design of experiments approach for engineering carbon metabolism in the yeast Saccharomyces cerevisiae." Thesis, University of Exeter, 2016. http://hdl.handle.net/10871/26158.

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The proven ability to ferment Saccharomyces cerevisiae on a large scale presents an attractive target for producing chemicals and fuels from sustainable sources. Efficient and predominant carbon flux through to ethanol is a significant engineering issue in the development of this yeast as a multi-product cell chassis used in biorefineries. In order to evaluate diversion of carbon flux away from ethanol, combinatorial deletions were investigated in genes encoding the six isozymes of alcohol dehydrogenase (ADH), which catalyse the terminal step in ethanol production. The scarless, dominant and counter- selectable amdSYM gene deletion method was optimised for generation of a combinatorial ADH knockout library in an industrially relevant strain of S. cerevisiae. Current understanding of the individual ADH genes fails to fully evaluate genotype-by-genotype and genotype-by-environment interactions: rather, further research of such a complex biological process requires a multivariate mathematical modelling approach. Application of such an approach using the Design of Experiments (DoE) methodology is appraised here as essential for detailed empirical evaluation of complex systems. DoE provided empirical evidence that in S. cerevisiae: i) the ADH2 gene is not associated with producing ethanol under anaerobic culture conditions in combination with 25 g l-1 glucose substrate concentrations; ii) ADH4 is associated with increased ethanol production when the cell is confronted with a zinc-limited [1 μM] environment; and iii) ADH5 is linked with the production of ethanol, predominantly at pH 4.5. A successful metabolic engineering strategy is detailed which increases the product portfolio of S. cerevisiae, currently used for large-scale production of bioethanol. Heterologous expression of the cytochrome P450 fatty acid peroxygenase from Jeotgalicoccus sp., OleTJE, fused to the RhFRED reductase from Rhodococcus sp. NCIMB 978 converted free fatty acid precursors to C13, C15 and C17 alkenes (3.81 ng μl-1 total alkene concentration).

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Blaise, Jean-Yves. "Le formalisme objet appliqué à l'étude de l'édifice patrimonial : Problèmes de modélisation et d'échanges de données sur le réseau Internet." Phd thesis, Aix-Marseille 3, 2003. http://tel.archives-ouvertes.fr/tel-00268228.

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Dans ce travail nous évaluons la pertinence et la portée d'un ensemble de formalismes informatiques récents sur l'étude du patrimoine architectural, champ d'application dont nous montrons les spécificités et leurs implications. Le problème que pose ce travail peut être présenté ainsi : le raisonnement par classifications que le formalisme objet autorise est-il pertinent pour décrire un modèle de l'édifice patrimonial; les concepts classifiés peuvent-ils être le maillon fédérateur d'une chaîne d'applications relevant de points de vues bien distinct sur l'édifice patrimonial ? Nous définissons une méthodologie d'analyse de l'édifice qui se veut générique, et un ensemble d'outils d'évaluation et d'exploitation du modèle.
Nous montrons qu'une analyse a priori du corpus d'éléments physiques formant l'édifice peut grandement en faciliter l'étude. Mesure, représentations et documentation, trois exemples d'applications abordés, deviennent dans ce cadre des processus visant à renseigner le modèle et ses instances. Nous montrons que la compréhension globale de l'édifice bâti et de son évolution est mieux assurée.

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Mithalal, Joris. "Déconstruction instrumentale et déconstruction dimensionnelle dans le contexte de la géométrie dynamique tridimensionnelle." Phd thesis, Université de Grenoble, 2010. http://tel.archives-ouvertes.fr/tel-00590941.

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Ce travail de thèse porte sur le passage, dans l'enseignement secondaire, d'une géométrie du concret à une géométrie portant sur des objets idéaux. Nous montrons que des environnements de géométrie dynamique tridimensionnelle offrent des conditions favorables à ce passage, que nous détaillons. La réflexion théorique s'appuie sur la Théorie des situations didactiques (Brousseau, 1998) pour proposer des hypothèses quant aux conditions et mécanismes d'apprentissage. En outre, notre questionnement initial est interprété à l'aide de deux cadres principaux. Le point de vue épistémologique des paradigmes géométriques (Houdement et Kuzniak, 2006) permet d'identifier la référence à GII comme un objectif fondamental. L'approche cognitive de Duval (2005, 1994) montre qu'à cette fin, l'élève doit abandonner la visualisation iconique et s'appuyer sur déconstruction dimensionnelle pour l'interprétation et la résolution des problèmes de géométrie. La géométrie dynamique dans l'espace est envisagée comme moteur de cette double perspective, et la déconstruction instrumentale y joue un rôle clef. Ce rôle, ainsi que des hypothèses d'émergences de la déconstruction dimensionnelle, sont précisés par un important travail théorique s'appuyant sur le modèle cKc (Balacheff, 1995; Balacheff et Margolinas, 2005), ainsi que la mise en œuvre d'une ingénierie didactique. Celle-ci apporte une validation expérimentale de plusieurs plusieurs résultats, au nombre desquels : - la pertinence d'analyser l'activité géométrique simultanément en termes de visualisations, déconstructions, et paradigmes géométriques ; - l'intérêt de la géométrie dynamique dans l'espace pour déstabiliser la visualisation iconique ; - l'existence de deux déconstructions instrumentales, et leur rôle fondamental pour l'émergence de la déconstruction dimensionnelle ; - les interactions entre les différentes déconstructions, qui n'étaient pas établies dans les travaux antérieurs ; - la possibilité de produire des situations s'appuyant sur la géométrie dynamique dans l'espace favorisant l'émergence de la déconstruction dimensionnelle ; - l'intérêt du modèle cKc pour la modélisation et l'analyse des phénomènes observés.

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Ahn, Yun-Kang. "L'analyse musicale computationnelle : rapport avec la composition, la segmentation et la représentation à l'aide de graphes." Phd thesis, Université Pierre et Marie Curie - Paris VI, 2009. http://tel.archives-ouvertes.fr/tel-00447778.

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Ce travail est axé autour de l'analyse assistée par ordinateur, dans un cadre musicologique ciblant la musique contemporaine. Une approche computationnelle de l'analyse musicale pose la question de sa modélisation et de sa formalisation, puisque touchant à une activité souvent empirique et dont la pratique n'est pas définie à l'aide de normes communes. Cette question implique la mesure de l'importance de la représentation graphique dans l'interprétation analytique. Face au rôle croissant de la représentation, nous nous intéressons d'une part à la manière dont elle relie l'analyse et la composition en proposant, à partir d'une analyse connue des Structures IA de Pierre Boulez réalisée par le compositeur György Ligeti, des modèles de machinerie informatique en vue de générer des pièces musicales sous deux interfaces, OpenMusic et Rubato. Parmi ces deux environnements de programmation visuelle, le second se base sur un puissant paradigme mathématique reposant sur la théorie des catégories. Ce paradigme constitue également une base pour modéliser les K-réseaux, outil d'analyse reposant sur des graphes décrivant les relations entre les notes. Ces relations sont formalisées dans la Set Theory d'Allen Forte qui propose une méthode de classification des structures musicales à partir des douze hauteurs de la gamme chromatique occidentale. Une application de ces réseaux et plus généralement l'apport des structures de graphe est abordée ensuite dans la modélisation de l'analyse d'une autre pièce, le Klavierstück III de Karlheinz Stockhausen : cette analyse a été faite par le théoricien David Lewin en deux étapes que nous étudierons dans leurs aspects à la fois computationnel et d'interface-homme machine. En premier lieu, la segmentation, non expliquée par l'analyste, fait l'objet d'une reconstitution dans laquelle nous tentons de la retrouver par le biais d'une exploration dans un ensemble de segmentations possibles. Il s'agit ainsi non seulement de reconstituer mais également de voir s'il est possible de la dépasser et l'améliorer selon des critères comme la couverture de la partition. Nous verrons dans ce cadre si les K-réseaux peuvent affiner cette analyse et en constituer un angle d'attaque pertinent. En second lieu, Lewin organise ses segments selon un réseau qui ne suit plus un quelconque agencement chronologique mais une disposition spatiale qui offre un nouveau parcours de la pièce. Cette difficulté s'inscrit dans le contexte plus général de la sélection et de la formation d'un graphe décrivant la structure d'une oeuvre musicale, que nous envisageons à l'intérieur du problème de l'utilisation d'un réseau en analyse computationnelle.

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Lamirel, Jean-Charles. "Vers une approche systémique et multivues pour l'analyse de données et la recherche d'information : un nouveau paradigme." Habilitation à diriger des recherches, Université Nancy II, 2010. http://tel.archives-ouvertes.fr/tel-00552247.

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Le sujet principal de notre travail d'habilitation concerne l'extension de l'approche systémique, initialement implantée dans le Système de Recherche d'Information NOMAD, qui a fait l'objet de notre travail de thèse, pour mettre en place un nouveau paradigme général d'analyse de données basé sur les points de vue multiples, paradigme que nous avons baptisé MVDA (Multi-View Data Analysis). Ce paradigme couvre à la fois le domaine de l'analyse de données et celui de la fouille de données. Selon celui-ci, chaque analyse de données est considérée comme une vue différente sur les données. Le croisement entre les vues s'opère par l'intermédiaire d'un réseau bayésien construit, de manière non supervisée, à partir des données ou des propriétés partagées entre ces dernières. Le paradigme MDVA repose également sur l'exploitation de méthodes spécifiques de visualisation, comme la visualisation topographique ou la visualisation hyperbolique. La mise en place de nouveaux estimateurs de qualité de type Rappel/Précision non supervisés basés sur l'analyse de la distribution des propriétés associées aux classes, et qui à la fois sont indépendants des méthodes de classification et des changements relatifs à leur mode opératoire (initialisation, distances utilisées ...), nous a permis de démontrer objectivement la supériorité de ce paradigme par rapport à l'approche globale, classique en analyse de données. Elle nous a également permis de comparer et d'intégrer dans le paradigme MVDA des méthodes de classification non supervisées (clustering) neuronales qui sont plus particulièrement adaptées à la gestion des données ultra-éparses et fortement multidimensionnelles, à l'image des données documentaires, ainsi que d'optimiser le mode opératoire de telles méthodes. Notre démarche a par ailleurs impliqué de développer la cohabitation entre le raisonnement neuronal et le raisonnement symbolique, ou entre des modèles de nature différente, de manière à couvrir l'ensemble des fonctions de la recherche et de l'analyse de l'information et à éliminer, sinon à réduire, les défauts inhérents à chacun des types d'approche. A travers de nombreuses applications, notamment dans le domaine de l'évaluation des sciences, nous montrons comment l'exploitation d'un tel paradigme peut permettre de résoudre des problèmes complexes d'analyse de données, comme ceux liés l'analyse diachronique à grande échelle des données textuelles polythématiques. Nous montrons également comment l'ensemble des outils développés dans le cadre de ce paradigme nous ont permis mettre en place de nouvelles méthodes très robustes et très performantes pour la classification supervisée et pour le clustering incrémental. Nous montrons finalement comment nous envisageons d'étendre leur application à d'autres domaines très porteurs, comme ceux du traitement automatique des langues ou de la bioinformatique.

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Tolmie, Julie. "Visualisation, navigation and mathematical perception: a visual notation for rational numbers mod1." Phd thesis, 2000. http://hdl.handle.net/1885/6969.

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There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.

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Li, Rongxin. "Towards a perceptually meaningful texture space for scientific visualisation." Phd thesis, 1999. http://hdl.handle.net/1885/148075.

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Muzangwa, Jonatan. "Imagery and visualisation characteristics of undergraduate students' thinking processes in learning selected concepts of mathematical analysis." Thesis, 2018. http://uir.unisa.ac.za/handle/10500/25649.

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The present study investigated imagery and visualisation characteristics of undergraduate students’ thinking processes in learning selected concepts of mathematical analysis. The aim was to discover the nature of images evoked by these undergraduate students and the role of imagery and visualisations when students were solving some selected problems related to mathematical analysis. The study was guided by the theory of registers of semiotic representations. Psychological notions on imagery were also fused to cater for a cognitive approach to the study. A sample of 50 undergraduate mathematics students participated in the study. The researcher employed both quantitative and qualitative methods. Before the main study, a pilot study was conducted to account for the reliability and validity of the research instruments. The data were collected through use of a cognitive test that was composed of 12 tasks with items selected from mathematical analysis. These tasks were specially designed to capture the variables of imagery and visualisations. A structured interview was also conducted as a follow-up to the results of the cognitive test. The study found that visual images were noticeable in the thinking processes of undergraduate students in solving problems related to mathematical analysis. The nature of the visual images evoked by the students varied from person to person. The nature of these images was also determined by the nature of the task. The most common types of imagery were diagrams, prototypes and symbols. On rare occasions the students also evoked metaphoric images. It was also observed that these images were used for illustrative purposes and to spark the idea for a proof. It was also interesting to note that some images were used to discover the limit of a converging series. The results confirmed the need to use visualisation with caution, especially when treating concepts which involve infinity. The study recommends that instructors of mathematics should encourage visual thinking in the learning and teaching of mathematical analysis. Knowledge of the students’ concept images helped the researcher to understand the nature of the learning difficulties of the students. Further research should focus on the strengths and weaknesses of visual-mediated learning and also on the relationship between creativity and visual thinking.
Mathematics Education
D. Phil. (Mathematics, Science and Technology Education (Mathematics Education))

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Ivan, Matúš. "Modelování představ žáků o chemických principech s využitím matematiky." Doctoral thesis, 2018. http://www.nusl.cz/ntk/nusl-388766.

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Key words mathematical modelling, chemical principles, shape and structure of the molecules, visualisation of chemical structures, chemistry and mathematics interdisciplinarity Ivan, Matúš: Modelling students' conceptions on chemical principles using mathematics. (Ph.D. Thesis). Prague: Faculty of Science, Charles University, 2018. Abstract The primary aims of the dissertation project is identification of "contact areas" of subjects of chemistry and mathematics and creation of interventions to solve the problems of chosen areas. At first, it was necessary to analyse available literature by the authors from the field, that have mentioned the problematics of interdisciplinarity between mathematics and chemistry. A part of literature research was dedicated to a search for the expected outcomes in curricular documents in Czech Republic and in Slovak Republic. In Czech Republic these were Framework Educational Programmes (RVP ZV, 2017 a RVP G, 2007). In Slovak Republic thee documents are State Educational programme ISCED 2 - lower secondary education (2015) and State Educational programme ISCED 3A - upper secondary education (2015). After that the analysis of a sample of School educational programmes of chosen schools in Czech Republic was conducted. The sample was chosen through proportional stratified...

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Du, Plooy Maryna C. "Visualisation as a metacognitive strategy in learning multiplicative concepts : a design research intervention." Thesis, 2016. http://hdl.handle.net/2263/51258.

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At the primary school level, the understanding of multiplication and division is critical, and division is a particularly challenging concept for most learners. I argue that the concept of division can be grasped more readily if learners learn to regulate their own mental processes intentionally; and that metacognitive strategies can be cultivated. Learners’ cognitive development and their propensity for visual imagery at age 11 to 12 years provide an opportunity to mediate such a strategy while learning mathematics concepts. Visualisation gives learners access to what I call “the virtual space of the mind”, where they can reconstruct an external life situation as an internal reality, upon which they can act mathematically. Based on a review of the relevant literature, the use of visualisation in problem solving (division in particular) at this developmental age was found to be under-explored, which gave rise to the need to develop a useful and feasible instructional design. Design Research was applied, as its iterative, cyclic nature allows the researcher to work through phases while developing various prototypes of the design. The fourth prototype of this design was tested with sixteen Grade 6 participants in the classroom setting of an English-medium primary school in Gauteng Province, South Africa, where the strategy was mediated for division in money-, area- and rate contexts. The research enabled the identification of specific design principles to underpin a final design for the mediation of visualisation as a metacognitive strategy in learning multiplicative concepts.
Thesis (PhD)--University of Pretoria, 2016.
Science, Mathematics and Technology Education
Unrestricted

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Nixon, Edith Glenda. "An investigation of the influence of visualisation, exploring patterns and generalisation on thinking levels in the formation of the concepts of sequences and series." Diss., 2002. http://hdl.handle.net/10500/977.

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Piaget and Freudenthal advocated thinking levels. In the 1950's the van Hieles developed a five level model of geometric thought. Judith Land adapted the model in 1990, utilising four levels to teach the concept of functions. These four levels have been considered here In the formation of concepts of sequences and series. The origin and relevance of sequences and series have been studied and the importance of visualisation, patterning and generalisation in the instructional process investigated. A series of lessons on these topics was taught to a group of six higher grade matriculation students of mixed ability and gender. Questionnaires related to student progress through the various levels were answered, categorised, graphed and analysed. Despite the small number of students, results seem to indicate that emphasising visualisation, exploring patterns and generalisation and teaching the topics as a reinvention had made a positive contribution towards progress through the various thought levels.

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Martins, Manuel Carlos Machado. "Francy an interactive discrete mathematics framework for GAP." Doctoral thesis, 2019. http://hdl.handle.net/10400.2/8666.

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Em Julho de 2014, uma sessão entitulada A Proposal of scalable web framework for running GAP on the cloud foi apresentada no Workshop International em Algebra Computacional em Lisboa. Até ali, a utilizacão do GAP na Web era uma tarefa difícil, e esta apresentação trouxe algumas ideias novas na tentativa de resolver este problema. Foi então que surgiu um pequeno projeto chamado WebGAP. Em 2015, surge o projeto OpenDreamKit. O projecto OpenDreamKit junta universidades de toda a Europa e visa fortalecer a investigação através da disponibilização de uma plataforma colaborativa virtual. Um dos seus objetivos é enriquecer o ecossistema do Jupyter, fornecendo meios de reprodutibilidade na ciência computacional. Em 2017, a package de software para o GAP Jupyter Kernel é finalmente disponibilizada, tornando possível utilizar o GAP num Web Browser, de forma simples e segura. Foi então identificada a falta de uma package de software para o GAP que permitisse criar interfaces gráficas interactivas, que deu origem a uma package de software para o GAP chamada Francy. Francy é uma package de software para o GAP que permite criar interfaces gráficas interactivas facilitando a representação e exploração de estruturas de dados. Francy é uma package independente de qualquer linguagem de programação ou sistema operativo e permite que virtualmente qualquer outra package de software para o GAP seja transformada numa ferramenta interativa, dando representação gráfica a estruturas matemáticas na forma de grafos ou gráficos topológicos.
A Proposal of scalable web framework for running GAP on the cloud was presented at the international Workshop on Computational Algebra in Lisbon in July of 2014. At that time, running GAP on the web was proven to be a difficult task, and this solution brought some fresh new ideas to solve it. A small project aroused, it was called WebGAP. In 2015, the OpenDreamKit project started. The OpenDreamKit project brings together universities across Europe and aims to strengthen virtual research. One of its goals is to enrich the Jupyter ecosystem by providing means to reproducibility in computational science. In 2017, the GAP Jupyter Kernel package is finally made available making it possible to run GAP on a web browser, in a simple and secure way. By that time, it was obvious that a package for providing a graphical user interface for GAP was missing. Francy is a package for GAP, and aims to provide graphical interfaces for representation and exploitation of data structures, cross platform wise. Francy allows virtually any GAP package to be transformed into an interactive tool, by giving graphical representation to mathematical structures in the form of topological graphs or charts, making it independent of any particular programming language or operating system.

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Masilo, Motshidisi Marleen. "Implementing inquiry-based learning to enhance Grade 11 students' problem-solving skills in Euclidean Geometry." Thesis, 2018. http://hdl.handle.net/10500/24966.

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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)

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Makandidze, Lancelot Sibanengi. "Exploring learners' understanding of trigonometric functions using GeoGebra software : a case of grade 11 Mathematics learners at a school in Tshwane South District." Diss., 2020. http://hdl.handle.net/10500/26949.

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Abstracts in English, Afrikaans and Zulu
The purpose of this study was to explore the learners’ understanding of trigonometric functions using GeoGebra software. A qualitative case study approach was used with six Grade 11 learners at a school in Tshwane South District. The data were collected during a seven-day period using multiple methods: a diagnostic test, worksheets, a smart recorder, a trigonometric functions test, one-on-one interviews and focus-group discussions. The findings showed clearly that the use of GeoGebra enhanced Grade 11 learners’ understanding of trigonometric functions. The findings also showed that the use of GeoGebra helped the learners to understand the basic trigonometric functions graphs. This enabled them to sketch freely without using the point-by-point method. Based on this study, it is recommended that GeoGebra should be made available to all Grade 11 learners. This will encourage them use the software out of regular school hours.
Die doel van hierdie studiegids was om die leerder se kennis van die gebruik van GeoGebra sagteware, ten opsigte van trigonometriese funksies te ondersoek. ‘n Kwalitatiewe benadering is gevolg met ses Graad 11 leerders by ‘n skool in die Tshwane Suid Distrik. Die data is ingewin oor ‘n periode van sewe dae, deur gebruik te maak van verskeie metodes: ‘n diagnostiese toets, werkkaarte, ‘n “smart” opname, ‘n trigonometriese funksies toets, een-tot-een onderhoude en fokusgroepe waar besprekings plaasgevind het. Die data-analise wys duidelik dat die gebruik van GeoGebra, die Graad 11 leerders gehelp het om trigonometriese funksies beter te verstaan. Die uitkomste wys ook duidelik dat GeoGebra die leerders gehelp het met ‘n beter begrip van die basiese trigonometriese funksies. Dit het hulle gehelp om vryhand sketse te doen en nie noodwendig die punt-tot-punt metode nie. Gebaseer op hierdie studie beveel ons aan dat GeoGebra beskikbaar gemaak moet word aan aale Graad 11 leerders. Dit sal leerders motiveer om ook die sagteware op hulle eie buite skoolure te gebruik.
Injongo yocwaningo lolu bekuyikuhlola ukuzwisisa kwabantwana amagrafu e-trigonometry ma bewafunda nge softhiwe ye GeoGebra. Abantwana abayisithupha abakubanga letshumi lanye abaphuma kusigodi se Tshwane South babambiqhaza kulolucwaningo lwendlela ye ‘qualitative case study’. Imininingo iqoqwe kumalanga ayisikhombisa kusetshenziswa indlela ezilandelayo: ukuhlolwa kwe-diagnostic, amaphepha okusebenzela, isingxoxo zamunye ngamunye lezingxoxo leqembu. Iziphumo ezinkulu zikhombe ngokucacile ukuthi ukusetshenziswa kwesofthiwe yeGeoGebra kukhulise ukuzwisisa kwamagrafu e-trigonometry ngabafundi bebanga letshumi lanye.Iziphumo njalo zibonise ukuthi abafundi bazwisisa izinto eziyisiseko ngala magrafu ma bewafunda ngesofthiwe ye GeoGebra. Lokhu kunike abafundi amandla okudwebadweba lamagrafu ngokushesha ngokukhululeka. Kusekelwa ngalezi ziphumo, kunconyiwe ukuthi abafundi bafumane isofthiwe yeGeoGebra ngaso sonke isikhathi. Lokhu kuzabakhuthaza ukuthi basebenzise lesofthiwe noma bengaphandle kwesikolo.
Mathematics Education
M. Ed. (Mathematics Education)

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Dissertations / Theses on the topic 'Mathematical visualisation'

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