M.Ed. (Educational Management) / Please refer to full text to view abstract

D.Phil. (Mathematics Education) / In tertiary education, a statement like ‘Low graduation rates prevail around the world’ is common knowledge since the 1940s, and therefore one does not need any longer to mention references. The factors that contribute to it though, are innumerable. One of those factors is the ability of a student to solve problems. Problem solving has been accepted as a prerequisite for lifelong learning by many governments and it is enshrined in their educational policies. However, problem solving can be associated with academic performance (mastery of content knowledge being a main contributor) as well as application/transfer of content knowledge. Critical thinking on the other hand is embedded in problem solving, acquisition of knowledge and application. Then an investigation into the relationships between all these constructs is warranted. This research aimed at shedding some or more light into this proverbial problem. Problem solving is equated by some authors to learning. Learning while solving problems and solving problems result in learning. Almost all theorists see problem solving as a process and be one of the products of learning. This research concluded that problem solving is a product of its own as a result of a number of complex cognitive processes. The simple argument is: If a problem solver cannot solve a problem successfully then no product is produced by those cognitive processes. In actual fact, the possibility of the existence of misconceptions could be one of the reasons for the failure of solving the problem. If that is true, then the statement: ‘we should be diagnosing rather than teaching’ could be valid. Furthermore, teaching problem solving as a process gives rise for it to be treated as an ‘algorithm’ by students which they try to memorise without having a conceptual understanding of the problem. However if it is treated as a product the students will be encouraged to think of the various cognitive processes that are necessary to solve the problem. This research concluded that cognitive processes such as critical thinking, acquisition of (mathematical) knowledge and application thereof, can lead to a product which was guided by ‘quality control processes’. Therefore problem solving in this research is not explicitly expressed but implicitly. As a result ‘successful problem solving’, the product, is closely associated with academic achievement.

A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. Johannesburg, 2016. / The 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. / LG2017

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 2002. / In this thesis I consider how mathematics students in a traditional firstyear Calculus course at a South African university appropriate mathematical objects which are new to them but which are already part of the official mathematics discourse. Although several researchers have explained mathematical object appropriation in process-object terms (for example, Sfard, 1994; Dubinsky, 1991, 1997; Tall, 1991, 1995, 1999), my focus is largely on what happens prior to the object-process stage. In line with Vygotsky (1986), I posit that the appropriation of a new mathematical object by a student takes place in phases and that an examination of these phases gives a language of description for understanding this process. This theory, which I call “appropriation theory”, is an elaboration and application of Vygotsky’s (1986) theory of concept formation to the mathematical domain. I also use Vygotsky’s (1986) notion of the functional use of a word to postulate that the mechanism for moving through these phases, that is, for appropriating the mathematical object, is a functional use of the mathematical sign. Specifically, I argue that the student uses new mathematical signs both as objects with which to communicate (like words are used) and as objects on which to focus and to organise his mathematical ideas (again as words are used) even before he fully comprehends the meaning of these signs. Through this sign usage the mathematical concept evolves for that student so that it eventually has personal meaning (like the meaning of a new word does for a child); furthermore, because the usage is socially regulated, the concept evolves so that its usage is concomitant with its usage in the mathematical community. I further explicate appropriation theory by elaborating a link between the theoretical concept variables and their empirical indicators, illustrating these links with data obtained from seven clinical interviews. In these interviews, seven purposefully chosen students engage in a set of speciallydesigned tasks around the definition of an improper integral. I utilise the empirical indicators to analyse two of these interviews in great detail. These analyses further inform the development of appropriation theory and also demonstrate how the theory illuminates the process of mathematical object appropriation by a particular student.

NAVORSINGSVERSLAG voorgele ter gedeeltelike vervuiling van die vereiste vir die graad MAGISTER IN NATUURWETENSKAPPE in die FAKULTEIT NATUURWETENSKAPPE aan die UNIVERSITEIT VAN DIE WITWATERSRAND. / There is widespread concern over some of the problems encountered in the teaching of Euclidean Geometry in secondary schools and also over the fairly negative attitudes experienced by pupils towards Geometry. This piece of research was designed to improve attitudes of Std.9 pupils towards Euclidean Geometry by making use of guided discovery and the integration of the history of Mathematics into the teaching method used. The latter was done in order to humanise the subject and to make it more interesting to pupils who otherwise experience it as very rigid and abstract. Active participation of pupils in developing the Geometry was central to the method employed. The outcome of the research was positive. It showed that attitudes towards Geometry can be improved if a deliberate attempt to do so is made. The results of this research suggest guidelines by which the teaching of Euclidean Geometry in secondary schools could be improved. / AC 2018

A research report submitted to the Faculty of Science, University of Witwatersrand, in partial fulfilment for the degree of Masters of Mathematics Education by coursework and research report. Johannesburg, March 2017. / The current mathematics curriculum in South Africa require that learners are provided with opportunities to develop abilities to be methodical, to generalise, to make conjectures and try to justify and prove their conjectures. These objectives call for the use of teaching strategies and tasks that support learners’ participation in the development of mathematical thinking and reasoning. This means that teachers have to be cautious when selecting tasks and deciding on teaching strategies for their classes. Tasks differ in their cognitive and difficulty levels and opportunities they afford for learner to learn mathematics competently. The levels of tasks selected by the teachers; the kinds of questions asked by the teachers during the implementation of the selected tasks and how the questions asked by the teachers and the teachers’ actions at implementations affected the levels of the tasks were the focus of this research report. The study was carried out in one high poverty high school in South Africa. Two teachers were observed teaching and each teacher taught their allocated grades. One teacher was observed teaching Grade 9s while the other taught Grade 11s. Both teacher taught number patterns at the time their lessons were observed. The research was qualitative. Methods of data collection and instruments included lesson observations; collection of tasks used in the observed classes, audio-taping and field notes. Pictures of the teachers’ work and copies of learners’ workbooks also provided some data. The analysis of data shows that the teachers not only selected and used lower-level cognitive demand and ‘easy’ tasks, that did not support mathematical thinking, but also did not lift up the levels and/or maintain the ‘difficulty levels’ of the task at implementation. Teachers were unable to initiate class discussions. Their teaching focused on ‘drill and practice’ learning and teaching practices. / LG2017

For several reasons, Guilford's psychological theory, "The Structure-of-Intellect" (SI), seems a good candidate for relating to the learning of mathematics. The general purposes of this study were to identify SI factors which would be significantly related to achievement in a junior-college mathematics course for non-science, non-mathematics majors and to determine whether semantic factors would be better predictors than symbolic for students classified as having high verbal ability. The two topics in the mathematics course which were selected for study were (1) numeration in other bases and (2) finite systems. / Typescript. / "August, 1975." / "Submitted to the Area of Instructional Design and Personnel Development, Program of Mathematics Education, in partial fulfillment of the requirements of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 151-153).

"This study investigated whether there is a relationship between first grade children's ability to tell a story about a dynamic picture or a sequence of three dynamic pictures and their ability to describe the picture(s) by a number sequence. The artistic variables characterizing the pictures were controlled so as to provide information concerning which types of illustrations best facilitated interpretation of the pictures and perception of mathematical relationships. An 8 x 2 design allowed analysis of the effects of the form of the drawing, the number of pictures, and the response condition. Ninety-six first grade children were individually tested using an instrument designed by the investigator. Statistical analysis revealed that neither drawing style nor the number of pictures had a significant effect on either the level of assimilation within the stories, the perception of motion, or the number sentence responses. Analysis of the response condition revealed a significant difference favoring the force condition on number sentence responses. Also, initially viewing and interpreting sequences provided a learning experience to significantly effect the interpretation of single pictures"--Abstract. / Typescript. / "August, 1976." / "Submitted to the Area of Instructional Design and Personnel Development, Program of Mathematics Education, in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 162-172).

"The present investigation was designed to study the effect of two instructional treatments on the achievement of students of different abilities--Verbal and Spatial. This was achieved by studying the interaction between the two treatments and each of the verbal and the spatial abilities. The instructional treatments were Figural and Verbal programmed units designed to teach concepts related to modulus seven arithmetic. Subjects for the study were 90 students enrolled in the first year mathematics course at Elmansoura College of Education in Egypt for the academic year 1978-1979"--Abstract. / Typescript. / "December, 1979." / "Submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: Eugene D. Nichols, Professor Directing Dissertation. / Includes bibliographical references (leaves 115-117).

"The purpose of this research is the development and testing of an instrument to be used in prediction of success in college freshman mathematics courses"--Introduction. / Typescript. / "April, 1967." / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Doctor of Education." / Advisor: R. Heimer, Professor Directing Dissertation. / Vita. / Includes bibliographical references (leaves 118-120).