3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item A study of the constitutive criteria for algebraic explanations and acts of explaining from a professional development course and in teachers’ practices(2019) Luxomo, Nontsikelelo NtsikiPedagogically, it is important to know what an explanation in itself can be understood to be. It is also useful for the mathematics teacher to have explicit guiding principles that are constitutive criteria of an explanation and acts of explaining. I set out on this study to first find theoretically the constitutive criteria of an explanation, and acts of explaining and secondly to empirically apply the constitutive criteria using data from the course, Wits Maths Connect Secondary Professional Development (WMCS PD), and teachers’ practices. A conceptual separation between explanation and acts of explaining was adopted from the philosophical literature, where I used Ruben’s (1992) interpretation of Aristotle’s four criteria of explanation. I then mobilised the PD and mathematics education literature in order to particularise and re-describe the criteria for an algebraic expressions mathematics education focus. The separation between explanation and acts of explaining served as an organisational structure through which I then read and engaged with the literature. In this methodology, the four criteria of explanation were then operationalised by translating them from four criteria into 8 codes for explanation. These were matter (M1 and M2), form (F1 and F2), process (P1 and P2) and goal (G1 and G2). I found that the course distinguished between what and how explanations with a possibility of why and when attachments for both types of explanations. Criteria transmitted by the course for acts of explaining were examples and their representations, as well as language which I coded as Rx and Rn respectively. I found from the classroom data that there were communicative techniques which I classified as acts of explaining such as re-voicing Rv, finishing teachers’ sentence Rf, gestures Rt, chorusing Rc, and evaluating Re. There was regulative talk in the classroom which I coded as Rg. The dominant and most privileged criteria of explanation in teachers’ practices were the reading of constitutive elements (M1) and process (P1). For acts of explaining the dominant criteria were regulation (Rg) and re-voicing (Re). I concluded that there is merit in having the analytic separation between criteria for explanation, and acts of explaining. The implication the findings had for PD was that more attention ought to be focused on criteria for explanation and acts of explaining as well as effective ways of communicating and transmitting these to teachers. The interpretation I made about the findings was that poor learner performances are linked to the most dominant criteria of explanation transmitted by teachers.Item A study of toeplitz decorrelation techniques for direction of arrival estimation of coherent sources(2019) Shafuda, Frans ShiwovanhuThe Direction of Arrival (DOA) is one of the features of propagating waves that is of interest in sensor array signal processing. Applications such as direction finding and source location make use of DOA estimation. A number of DOA estimation methods have been developed over time with special focus on achieving high resolution performance. Subspace-based, also known as eigenstructure-based; methods such as Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) are amongst the excellent and widely applied methods for DOA estimation. However, the performance of these methods is highly degraded in the presence of coherent or highly correlated incident sources caused by multipath propagation and electronic jamming. Conventional techniques such as spatial smoothing have been introduced to better the performance of these methods by removing correlation between coherent sources. However, this is attained at the expense of reduced array aperture (degree of freedom) and increased computational complexity. Of late, a variation of decorrelation techniques based on Toeplitz matrix theory gained much interest in overcoming the drawbacks of conventional decorrelation techniques. Based on a narrowband signal propagation model and a Uniform Linear Array (ULA) in the presence of white additive Gaussian noise, this study carried out an algebraic analysis of two Toeplitz decorrelation techniques. These are, the correlation Toeplitz (CTOP) and the average Toeplitz (AVTOP) decorrelation techniques. The techniques were studied for DOA estimation of coherent sources in conjunction with the MUSIC algorithm. The goal of the study is to provide an understanding of how and why these techniques work. Through the algebraic analysis the study found that, DOA information is perfectly preserved during decorrelation when retained as sums of individual sources of information (i.e. in a superposition form). This explains why the CTOP technique perfectly decorrelates coherent sources unlike the AVTOP technique. This is because decorrelation based on the CTOP technique retains superimposed sources’ DOA information. Based on the assumption that the exact signal plus noise array covariance matrix is known, the MUSIC algorithm was applied algebraically in order to validate the findings from the analysis. A maximum of four array elements and three coherent narrowband sources were considered. The algebra becomes intractable when the ULA elements are more than four. The algebraic results have further shown that when the exact array covariance matrix is known, the noise variance has no influence on DOA information and DOA information can be accurately obtained using the MUSIC algorithm. Numerical simulations were also conducted in order to confirm the superiority of the CTOP decorrelation technique. Through numerical experiments, the performance of these techniques was evaluated in terms of Root Mean Square Error, standard deviation and probability of success in DOA estimation. The performance of the classic MUSIC algorithm was also evaluated to serve as a baseline for comparison. Regardless of the coherence among the sources, the CTOP MUSIC (CTOP technique applied in conjunction with the MUSIC algorithm) returns more accurate estimates even at low levels of SNR and minimum number of array sensors. The study has been able to provide an understanding of the working principles behind the two Toeplitz decorrelation techniques.Item Resourcing learner errors and misconceptions on grade 10 fractional equations at a mathematics clinic(2016) Khanyile, Duduzile WinnieThe purpose of this study, conducted at a mathematics clinic, was to investigate the misconceptions that learners display through errors they make when solving algebraic equations involving fractions. A teaching intervention to address those errors and misconceptions was done at a mathematics clinic. A mathematics clinic is a remedial facility where low-attaining students attend sessions, by choice or by referrals. In this study teaching intervention was used to address learners’ errors and misconceptions. The assumption of the study was that learners are knowledge constructors that use previously-learned knowledge as the basis of new knowledge. Since their previous knowledge contains errors and misconceptions, the construction of new knowledge results in errors. This research was mainly qualitative. Data were collected, using a sample of 17 grade 10 learners, though the work of only 13 of them was analysed. Two participants wrote the pre-test, but did not participate in the subsequent data collection, and the other two did not solve some of the equations in the pre- and post-tests. There were three stages of data collection; pre-test, teaching intervention and post-test. Pre- and post-tests were analysed for errors committed by learners, and the teaching intervention sessions were analysed for opportunities of learning provided. Transcripts were produced from the teaching intervention sessions. They were also analysed to check how students participated in constructing mathematical meanings, and also how effectively their attention was focused on the object of learning. The errors found in learners’ equation-solving were like-term errors, lowest common denominator errors, careless errors, sign errors and restriction errors. The comparison of the number of learners who committed these errors in the pre- and the post-test was insightful. Of 13 learners, 4 committed like-term errors in the pre-test and just 1 in the post-test; 4 committed LCD errors both in the pre- and post-tests; 9 committed careless errors (other errors) in the pre-test, and 6 learners in the post-test; 7 committed sign errors in the pre-test and 1 in the post-test; and 12 committed restriction errors in the pre-test, and 9 in the post-test. These findings suggest that teaching intervention is a necessary pedagogical technique, and needs to be employed when addressing learners’ errors and misconceptions in mathematics. Reduction in learners’ errors and misconceptions was evident after the teaching intervention suggesting that the mathematics clinic provided learning opportunities for participants.Item Algebraic filtrations of the modal m-Calculus(2016) Cromberge, Michael BenjaminIn this thesis we analyse the issue of decidability for two modal logics which contain least binders. Towards this goal, we begin the work with a brief survey of modal logic, PDL, the modal -calculus and algebraic filtrations as exposited by Conradie et al. The first such modal logic we analyse is the fragment of the modal -calculus corresponding to PDL; the second logic is the equational theory of the class of -algebras (motivated by the least root calculus of Pratt). We offer a new, algebraic, proof for the decidability of PDL by showing that PDL has the finite model property with respect to the class of dynamic algebras. We then show that the equational theory of the class of -algebras has the finite model property with respect to the class of -algebras; this is based on the proof of Pratt but differs in an important detail. The finite model property results for these two modal logics are achieved by an algebraic filtration method based on that of Conradie et al.Item Learners' mathematical reasoning when generalizing from number patterns in the general education and training phase.(2011-09-20) Ndlovu, Williams ChapasukaThis study aims to explore GET learners’ mathematical (algebraic) reasoning when generalizing from number patterns. Data was collected in a former model C school in greater Johannesburg area by means of a questionnaire based task involving number patterns. The mathematical reasoning of the grade 9 participants when generalizing from number patterns was examined within a commognitive framework. According to this perspective, thinking is a special activity of communication in which a participant of a discourse engages. The participants’ responses to questions in the questionnaire based task were classified according to particular aspects of the discourse they used, specifically routines (strategies) and visual mediators. The participants’ generalization routines were further classified into one of the three main categories; numeric, figural and pragmatic generalizations. The analysis focused on how the learners’ derived rules for the nth term and their justifications for their responses. The results of this study strongly support the notion that students’ algebraic reasoning when generalizing in number patterns is intertwined with their choices of routines and mediators. Most learners used recursive routines while a few used explicit routines (classified and categorized as numeric routines) and number-mediators. Also, most participants found it easier to informally verbalize their generalizations. However participants’ spoken justifications of their written and spoken responses often did not match their use of routines and visual mediators. As such, an awareness and appreciation (by teachers) of students’ diverse use of routines and mediators when generalizing from number patterns could have direct pedagogical implications in the mathematics classrooms.Item An investigation of learners' symbol sense and interpretation of letters in early algebraic learning(2009-07-06T12:25:20Z) Naidoo, Kona Sagaren KanakasabyResearch in early algebra is critical because a smooth transition from arithmetic to algebra will influence future algebra learning that is central to school mathematics. This study investigated learners’ interpretation of letters in different levels of generalised arithmetic activities. Thirty grade nine learners from one inner city school participated in this study. All learners engaged with seventeen paper and pencil tasks encompassing six different interpretations of letters and six learners were then interviewed. Analysis of the data showed that the overall performance of learners was very poor and most learners have not been successful in making the transition from arithmetic to algebra. Learner responses suggested a strong arithmetical influence and a poor understanding of algebraic letter and basic manipulative skills. Throughout the data a number of misconceptions surfaced which suggested that most learners in this sample were lacking ‘symbol sense’.