Mathematics and Computer Science Education

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A comparative study of collocation methods for the numerical solution of differential equations.
(2008) Kajotoni, Margaret Modupe.; Parumasur, Nabendra.; Singh, Pravin.
The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature.
A simulation modeling approach to aid research into the control of a stalk-borer in the South African Sugar Industry.
(2008) Horton, Petrovious Mitchell.; Sibanda, Precious.; Hearne, John W.; Conlong, Desmond Edward.; Apaloo, Joseph.
The control of the African stalk borer Eldana saccharina Walker (Lepidoptera: Pyralidae) in sugarcane fields of KwaZulu-Natal, South Africa has proved problematical. Researchers at the South African Sugarcane Research Institute (SASRI) have since 1974 been intensively investigating various means of controlling the pest. Among the methods of control currently being investigated are biological control, chemical control, production of more resistant varieties and crop management. These investigations, however, require many years of experimentation before any conclusions can be made. In order to aid the research currently being carried out in the Entomology Department at SASRI (to investigate biological control strategies, insecticide application strategies and the carry-over decision), a simulation model of E. saccharina growth in sugarcane has been formulated. The model is cohort-based and includes the effect of temperature on the physiological development of individuals in each life-stage of the insect. It also takes into account the effect of the condition of sugarcane on the rate of E. saccharina infestation, by making use of output from the sugarcane growth model CANEGRO. Further, a crop damage index is defined that gives an indication of the history of E. saccharina infestation levels during the sugarcane’s growth period. It is linked to the physiological activity of the borer during the period spent feeding on the stalk tissue. The damage index can further be translated into length of stalks bored and hence the percentage of the stalk length bored can be calculated at each point in the simulation using the total length of stalks calculated in the CANEGRO model. Using an industry accepted relationship between percent stalks damaged and reduction in sucrose content of the crop, reductions in losses in the relative value of the crop when the various control measures are implemented can be compared. Relationships between the reduction in percent stalk length bored (and hence gains in the relative value of the crop) and the various control strategies are obtained.
Algebraic graph theoretic applications to cryptography.
(2015) Mafunda, Sonwabile Templeton.; Amery, Gareth.; Mukwembi, Simon.; Swart, Christine Scott.
Abstract available in PDF file.
An algebraic study of residuated ordered monoids and logics without exchange and contraction.
(1998) Van Alten, Clint Johann.; Raftery, James Gordon.
Please refer to the thesis for the abstract.
Amplitude-shape method for the numerical solution of ordinary differential equations.
(1997) Parumasur, Nabendra.; Banasiak, Jacek.; Mika, Janusz R.
In this work, we present an amplitude-shape method for solving evolution problems described by partial differential equations. The method is capable of recognizing the special structure of many evolution problems. In particular, the stiff system of ordinary differential equations resulting from the semi-discretization of partial differential equations is considered. The method involves transforming the system so that only a few equations are stiff and the majority of the equations remain non-stiff. The system is treated with a mixed explicit-implicit scheme with a built-in error control mechanism. This approach proved to be very effective for the solution of stiff systems of equations describing spatially dependent chemical kinetics.
An exploration of preservice teachers’ use of educational technologies as visualization tools when teaching mathematics.
(2022) Zulu, Mzwandile Wiseman.; Mudaly, Vimolan.
This interpretive qualitative study explores the use of educational technologies by preservice teachers as visualization tools during mathematics teaching at secondary schools. Sfard’s commognitive framework and Koehler and Mishra’s technological pedagogical content knowledge theoretical frameworks undergird the study. Data were collected from ten preservice mathematics teachers at a university in the province of KwaZulu-Natal, South Africa. Performance tests, semi-structured interviews, focus group discussions, and observations were employed to collect data, which was analyzed using reflexive thematic analysis. The study found that preservice teachers employed two primary visualization strategies when they engaged in mathematical problem-solving: symbolic mental visualization, which they combined with their understanding of word usage, endorsed narratives and routines to arrive at a solution; and graphic visual mediators, such as diagrams, which they sketched to contextualize the problem statement and verify they solutions and use of mathematical word usage, routines, and endorsed narratives. Participants were found to be unable to solve a mathematics problem if they had not visualized it effectively; using a graphic visual mediator to understand the problem statement did not, however, guarantee success when solving a problem. A relationship was found between the visualization techniques that the participants used in their own attempts to solve mathematical problems and the visualization techniques they used in their lesson planning and teaching of mathematics in the same content area. Participants used innovative strategies to mediate learning, including educational technologies that facilitated visual mediators to enhance learners’ engagement with concepts. Synergies were found between the elements of the commognition and TPACK frameworks as these were used in tandem to analyze data. A model was developed (C+TPACK) that integrates the key elements of these frameworks. Further research is recommended to establish the viability, credibility and generalizability of the model.
Analysis of co-infection of human immunodeficiency virus with human papillomavirus.
(2014) Maregere, Bothwell.; Chirove, Faraimunashe.
We formulate a deterministic mathematical model for the co-infection of HPV with HIV without treatment. Mathematical techniques were used to analyze the stability of the models in terms of basic reproduction numbers for disease-free equilibrium point and fixed point theory used for analysis of the endemic equilibrium point. The model incorporating HIV and HPV co-infection sought to investigate the impact of HIV infection in the natural history of HPV infection, and the impact of HPV infection in the natural history of HIV infection, over a period of time. Numerical simulations were carried out to illustrate the trends of progression of HIV and HPV in the case of co-infection. The results from our study showed that when both HIV and HPV infected individuals are active in the system then co-infection grows faster compared to one infection which is active in the system. Our study also showed that when we started with HPV infection in the community and introduces HIV infection after sometime has more impact in the growth of co-infection population compared to start with HIV infection and introduces HPV infection after sometime in the community.
Anisotropic stars in general relativity.
(2004) Chaisi, Mosa.; Maharaj, Sunil Dutt.
In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile f-L ex r-2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (f-L ex r-2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (f-L ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr - pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models.
An APOS analysis of the teaching and learning of factorisation of quadratic expressions in grade 10 mathematics classrooms.
(2021) Vilakazi, Aubrey Sifiso.; Bansilal, Sarah.
The South African Curriculum and Assessment Policy Statements (CAPS) document, for the Further Education and Training Phase (FET) Mathematics Grades 10-12 (2011) shows that the factorisation of algebraic quadratic expressions or equations pervades the mathematics of the secondary school. As a result, for learners to be successful at mathematics in Grade 12, they need to know a great deal of algebra, particularly the factorisation of quadratics. It is therefore important for us as mathematics educators to identify areas in the factorization of quadratics that teachers and learners are struggling to learn and apply. With this in mind, the study sets to embark on an APOS analysis of the teaching and learning of factorisation of quadratic expressions in Grade 10 mathematics classrooms. Following on from the research questions, this study is located within the principles of the mixed methods case study approach. The combination of methodologies has allowed me to identify broad trends across the groups of learners and those of educators as a whole as well as differences within the participants of the groups themselves. The participants of the study were the groups of Grade 10 learners from the two participating schools, as well as the Grade 10 mathematics teachers from the two circuits of Ilembe District. Five sources of data were used. Firstly, data were generated from 25 teachers from the two circuits who participated in the teachers’ questionnaires. A second data collection instrument was the classroom lessons’ observations of the six teachers. A third data source was the learner group activity and learners’ interviews administered to 12 learners. A fourth data source was the unstructured interviews with six teachers. The final instrument was the analysis of the 205 Grade 10 mathematics 2019 March common paper learners’ scripts. This study was guided by the theory of constructivism and more specifically Action, Process, Object, Schema (APOS) theory which views learning as changes in conception. As an individual engages with a concept, the conception changes from an initial external view towards seeing the concept as a totality upon which other Actions and Processes can act. This study has found that, firstly, teachers and learners tend to rely too much on the use of rules in factoring certain quadratics. In so doing, a prototype of the quadratic expression concept is perceived which consists of isolated and disconnected concepts. As a result, most learners were not able to factor the trinomial quadratic of 𝑎≠1, since they perceived the factoring of 𝑎𝑥2+𝑏𝑥+𝑐 with 𝑎=1 and that of 𝑎≠1 as two different procedures. Secondly, there are also students whose mental constructions (conception) are limited to Action levels in terms of APOS theory. The findings of the study suggest that teachers and learners should be able to consider quadratic expressions as one big idea and follow the fundamental considerations when factoring the quadratic expressions. Furthermore the use of multi-methods in factoring quadratics is encouraged and needed for students to better understand the connections between different methodologies for conceptual development.
An APOS exploration of conceptual understanding of the chain rule in calculus by first year engineering students.
(2011) Jojo, Zingiswa Mybert Monica.; Brijlall, Deonarain.; Maharaj, Aneshkumar.
The main issue in this study is how students conceptualise mathematical learning in the context of calculus with specific reference to the chain rule. The study focuses on how students use the chain rule in finding derivatives of composite functions (including trigonometric ones). The study was based on the APOS (Action-Process-Objects-Schema) approach in exploring conceptual understanding displayed by first year University of Technology students in learning the chain rule in calculus. The study consisted of two phases, both using a qualitative approach. Phase 1 was the pilot study which involved collection of data via questionnaires which were administered to 23 previous semester students of known ability, willing to participate in the study. The questionnaire was then administered to 30 volunteering first year students in Phase 2. A structured way to describe an individual student's understanding of the chain rule was developed and applied to analyzing the evolution of that understanding for each of the 30 first year students. Various methods of data collection were used namely: (1) classroom observations, (2) open-ended questionnaire, (3) semi-structured and unstructured interviews, (4) video-recordings, and (5) written class work, tests and exercises. The research done indicates that it is essential for instructional design to accommodate multiple ways of function representation to enable students to make connections and have a deeper understanding of the concept of the chain rule. Learning activities should include tasks that demand all three techniques, Straight form technique, Link form technique and Leibniz form technique, to cater for the variation in learner preferences. It is believed that the APOS paradigm using selected activities brought the students to the point of being better able to understand the chain rule and informed the teaching strategies for this concept. In this way, it is believed that this conceptualization will enable the formulation of schema of the chain rule which can be applied to a wider range of contexts in calculus. There is a need to establish a conceptual basis that allows construction of a schema of the chain rule. The understanding of the concept with skills can then be augmented by instructional design based on the modified genetic decomposition. This will then subject students to a better understanding of the chain rule and hence more of calculus and its applications.
The application of artifacts in the teaching and learning of grade 9 geometry.
(2005) Jojo, Zingiswa Mybert Monica.; Brijlall, Deonarain.; Maharaj, A.
The main focus of the study was to explore how the experiences that the learners went through in the Technology class during the construction and design of artifacts, could be used to inform the teaching of Geometry in the mainstream Mathematics classes. It was important to find out how the teaching of Geometry would allow the learners to both reflect and utilize the Geometry they know, as a starting point or springboard for further study of Geometry. Data was collected through observations, structured and semi-structured interviews of a sample of twenty grade 9 learners of Mashesha Junior Secondary School of Margate in KwaZulu Natal. It was collected through observation of drawings and completely constructed double-storey artifacts at different intervals of designing. Observations and notes on every activity done by the learners for example, measurements, comparisons, estimations, scaling, drawings use of symmetry and perspective drawing were kept and analyzed. Data for the interviews was collected in the form of drawings, photographs, transcriptions of video and audiotapes. The observations in particular were looking for the Geometry in finished artifacts. Interviews with the learners were directed at how each learner started drawing a house to the finish. When and how scale drawing, projections, angles made and length preservation were used by the learner, was of utmost importance. It is believed that grade 9 learners of Mashesha have Geometric experiences which can be used to inform the teaching of Geometry in mainstream mathematics. It was found that this experience brought by the learners from the Technology construction of artifacts could cause the learners to find mainstream mathematics interesting and challenging. It is also believed that the use of projective Geometry already employed by the learners can be incorporated in mainstream mathematics so as to improve how learners understand Euclidean Geometry. In this way, it is believed, that the teaching of Geometry will allow the learners to utilize and reflect the Geometry already known to them. This Geometry would therefore be used as a starting point for further study of Geometry. Suggestions for further research and recommendations for the improvement of Geometry teaching and learning have also been made.
Application of bivariate spectral quasilinearization method to second grade fluid flow equations.
(2020) Dlongolo, Simphiwe Gloria.; Sibanda, Precious.; Goqo, Sicelo Praisegod.
In this study, the steady flow of a second grade magnetohydrodynamic fluid in a porous channel is investigated. We further investigate the hydromagnetic flow of a second grade fluid over a stretching sheet. The partial differential equations that describe the flows are solved numerically using the bivariate spectral quasilinearization method. The method is extended to a system of non-similar partial differential equations that model the steady two dimensional flow of Falkner-Skan flow of an incompressible second grade nano fluid. The work is also concerned with heat and the mass transfer from the electrically conducting second grade magnetohydrodynamic fluid over a stretching sheet. The sensitivity of the flow characteristics with respect to the second grade fluid parameter, magnetic field parameter, thermal radiation parameter, and the chemical reaction parameter are investigated. The accuracy of the numerical method is determined using the residual error analysis.
The application of Rasch measurement theory to improve the functioning of a mathematics assessment instrument.
(2021) Ngirishi, Harrison.; Bansilal, Sarah.
Assessment is an integral part of the teaching and learning process. Concerns about student performance in assessments often drive the teaching and learning. In South Africa there has been numerous concerns about poor learning outcomes in mathematics and this has led to calls for all stakeholders to work together to try and find solutions. This study focuses on the assessment of mathematics with particular interest in the KZN provincial Grade 12 mathematics trial examination paper 2. The study explored the use of Rasch analysis in improving the functioning of the mathematics assessment instrument. The aim of the study was to use the Rasch analysis to report on the functioning of the test instrument in measuring proficiency in mathematics, checking on the targeting and reliability of the test instrument, explain anomalies where data did not fit the Rasch model, investigate differential item functioning (DIF), response dependency and multidimensionality. The study also sought the teachers’ views about the findings of the Rasch analysis. A sequential explanatory design was used in this study, where the Rasch analysis provided the theoretical framework for the analysis of the quantitative data. The qualitative analysis of the teachers’ responses helped to get more understanding of the results of the quantitative analysis of the leaners’ responses. The study found that the assessment instrument was difficult for this particular cohort, some items displayed DIF for language and response dependency due to some teachers not applying continuous accuracy marking. The study revealed that some teachers were not applying the continuous accuracy marking process. Items which carried more than two accuracy marks, showed misfit to the Rasch model. Teachers cited not applying continuous accuracy marking due to time constraints and large number of learners in classes. Teachers blamed poor performance on learners’ lack of basic understanding, adequate preparation and motivation, societal influences, poor understanding of proof type questions, allocation of many accuracy marks on one item and the language barrier. The recommendations of this study if implemented may help teachers in the teaching process and examiners in producing fair assessment instruments. The recommendations may lead to improvement of mathematics results.
The application of the multigrid algorithm to the solution of stiff ordinary differential equations resulting from partial differential equations.
(1992) Parumasur, Nabendra.; Mika, Janusz R.
We wish to apply the newly developed multigrid method [14] to the solution of ODEs resulting from the semi-discretization of time dependent PDEs by the method of lines. In particular, we consider the general form of two important PDE equations occuring in practice, viz. the nonlinear diffusion equation and the telegraph equation. Furthermore, we briefly examine a practical area, viz. atmospheric physics where we feel this method might be of significance. In order to offer the method to a wider range of PC users we present a computer program, called PDEMGS. The purpose of this program is to relieve the user of much of the expensive and time consuming effort involved in the solution of nonlinear PDEs. A wide variety of examples are given to demonstrate the usefulness of the multigrid method and the versatility of PDEMGS.
Applications of symmetry analysis to physically relevant differential equations.
(2005) Kweyama, Mandelenkosi Christopher.; Govinder, Keshlan Sathasiva.; Maharaj, Sunil Dutt.
We investigate the role of Lie symmetries in generating solutions to differential equations that arise in particular physical systems. We first provide an overview of the Lie analysis and review the relevant symmetry analysis of differential equations in general. The Lie symmetries of some simple ordinary differential equations are found t. illustrate the general method. Then we study the properties of particular ordinary differential equations that arise in astrophysics and cosmology using the Lie analysis of differential equations. Firstly, a system of differential equations arising in the model of a relativistic star is generated and a governing nonlinear equation is obtained for a linear equation of state. A comprehensive symmetry analysis is performed on this equation. Secondly, a second order nonlinear ordinary differential equation arising in the model of the early universe is described and a detailed symmetry analysis of this equation is undertaken. Our objective in each case is to find explicit Lie symmetry generators that may help in analysing the model.
Aspects of distance and domination in graphs.
(1995) Smithdorf, Vivienne.; Swart, Hendrika Cornelia Scott.; Dankelmann, Peter A.
The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph.
Aspects of functional variations of domination in graphs.
(2003) Harris, Laura Marie.; Henning, Michael Anthony.; Hattingh, Johannes H.
Let G = (V, E) be a graph. For any real valued function f : V >R and SCV, let f (s) = z ues f(u). The weight of f is defined as f(V). A signed k-subdominating function (signed kSF) of G is defined as a function f : V > {-I, I} such that f(N[v]) > 1 for at least k vertices of G, where N[v] denotes the closed neighborhood of v. The signed k-subdomination number of a graph G, denoted by yks-11(G), is equal to min{f(V) I f is a signed kSF of G}. If instead of the range {-I, I}, we require the range {-I, 0, I}, then we obtain the concept of a minus k-subdominating function. Its associated parameter, called the minus k-subdomination number of G, is denoted by ytks-101(G). In chapter 2 we survey recent results on signed and minus k-subdomination in graphs. In Chapter 3, we compute the signed and minus k-subdomination numbers for certain complete multipartite graphs and their complements, generalizing results due to Holm [30]. In Chapter 4, we give a lower bound on the total signed k-subdomination number in terms of the minimum degree, maximum degree and the order of the graph. A lower bound in terms of the degree sequence is also given. We then compute the total signed k-subdomination number of a cycle, and present a characterization of graphs G with equal total signed k-subdomination and total signed l-subdomination numbers. Finally, we establish a sharp upper bound on the total signed k-subdomination number of a tree in terms of its order n and k where 1 < k < n, and characterize trees attaining these bounds for certain values of k. For this purpose, we first establish the total signed k-subdomination number of simple structures, including paths and spiders. In Chapter 5, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus domination function of weight at most k may be NP-complete, even when restricted to bipartite or chordal graphs. Also in Chapter 5, linear time algorithms for computing Ytns-11(T) and Ytns-101(T) for an arbitrary tree T are presented, where n = n(T). In Chapter 6, we present cubic time algorithms to compute Ytks-11(T) and Ytks-101l(T) for a tree T. We show that the decision problem corresponding to the computation of Ytks-11(G) is NP-complete, and that the decision problem corresponding to the computation of Ytks-101 (T) is NP-complete, even for bipartite graphs. In addition, we present cubic time algorithms to computeYks-11(T) and Yks-101(T) for a tree T, solving problems appearing in [25].
Aspects of spherically symmetric cosmological models.
(1998) Moodley, Kavilan.; Maharaj, Sunil Dutt.; Govinder, Keshlan Sathasiva.
In this thesis we consider spherically symmetric cosmological models when the shear is nonzero and also cases when the shear is vanishing. We investigate the role of the Emden-Fowler equation which governs the behaviour of the gravitational field. The Einstein field equations are derived in comoving coordinates for a spherically symmetric line element and a perfect fluid source for charged and uncharged matter. It is possible to reduce the system of field equations under different assumptions to the solution of a particular Emden-Fowler equation. The situations in which the Emden-Fowler equation arises are identified and studied. We analyse the Emden-Fowler equation via the method of Lie point symmetries. The conditions under which this equation is reduced to quadratures are obtained. The Lie analysis is applied to the particular models of Herlt (1996), Govender (1996) and Maharaj et al (1996) and the role of the Emden-Fowler equation is highlighted. We establish the uniqueness of the solutions of Maharaj et al (1996). Some physical features of the Einstein-Maxwell system are noted which distinguishes charged solutions. A charged analogue of the Maharaj et al (1993) spherically symmetric solution is obtained. The Gutman-Bespal'ko (1967) solution is recovered as a special case within this class of solutions by fixing the parameters and setting the charge to zero. It is also demonstrated that, under the assumptions of vanishing acceleration and proper charge density, the Emden-Fowler equation arises as a governing equation in charged spherically symmetric models.
A case study: the use of GeoGebra to alleviate learner difficulty in learning the similarity of triangles in a South African grade 9 classroom.
(2022) Mpanza, Nompumelelo.; Shongwe, Michael Bafana Mthembiseni.
The mixed methods study investigated the use of GeoGebra as a dynamic geometry software (DGS) to alleviate the learning similarity of triangles. A mixed methods philosophical framework in the form of an exploratory case study was used to conveniently and purposively select a sample of 60 Grade 9 learners enrolled at Sondelani Full-Service School (pseudonym), a township school in the Pinetown district in KwaZulu–Natal province, South Africa. During this research, GeoGebra and the concept of similarity of triangles were introduced to the participants. Then, participants answered several (Euclidean geometry) Similarity Achievement Test (SAT) questions prescribed by the National Mathematics pacesetter for Grade 9 and 10. A 10-item Likert scale questionnaire intended to elicit participants’ attitudes about GeoGebra and its impact on Euclidean geometry and mathematics was administered to these participants. The questionnaire also included four open-ended items, asking participants to reflect on the application of GeoGebra. The analysis of SAT data revealed that performance was higher after GeoGebra instruction (𝑀 = 22.50) than during traditional instruction, which did not feature GeoGebra (𝑀 = 11.65). Thus, it was found that the use of GeoGebra is an appropriate tool to increase achievement in learning geometry concepts; to promote accuracy, visualization; learner participation; and to create enjoyment and learner interest towards learning mathematics. It is recommended that mathematics teachers need to use GeoGebra for effective teaching and learning of similarity of triangles.
Character tables of the general linear group and some of its subgroups
(2008) Basheer, Ayoub Basheer Mohammed.; Moori, Jamshid.
The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned.

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