This investigation emanates from the realization that Grade 3 children at schools in disadvantaged areas perform poorly in basic mathematics computations such as addition, subtraction, multiplication and division. The aim of the research was to establish the approaches teachers use when teaching mathematics computation. The qualitative approach, together with the research techniques commonly used with it, namely observation, interviews and document analysis was deemed appropriate for the investigation. The outcomes of the investigation revealed that the multilingual Grade 3 classes made it difficult to assist all children who experienced mathematics problems because teachers could not speak all the other languages that were not the language of learning (LoLT) of the school. Another obstacle that prohibited teachers from spending adequate time with children with mathematics problems was the time teachers were expected to spend on intervention programmes from the Department of Basic Education (DBE) aimed at improving schooling in general. Teachers could not make additional time that could afford children the opportunity of individual attention. With regard to the approach used for teaching mathematics, this study established that the teachers used the whole class teaching approach which is not a recommended approach because each child learns differently. It is recommended that teachers use a variety of teaching methods in order to accommodate all children and also encourage children to use concrete objects. It is also recommended that teachers involved in the SBSTs should consist only of members qualified in the subject and once these children are identified, remediation should take place promptly by their being enrolled (children) in the proposed programme. Finally, this study could benefit foundation Phase teachers in teaching mathematics based on the proposed strategy outlined after teachers’ challenges were identified. The outcome of the study could also be of value to the DBE, especially with curriculum designers. / Early Childhood Education and Development / D. Ed. (Early Childhood Education)

This study explores junior secondary students’ schemata on a line reflection construction task, the research of which was conducted in a secondary school in Hong Kong. The theories drawn on in this study come from the literature on theories of schemata and the corresponding knowledge embedded within, namely conceptual knowledge, manipulation and procedural knowledge. The research built on existing theories on schemata and attempted to categorize the different kinds of schemata as well as investigating the relationship between them among four junior secondary students in the construction of a line reflection task. The study also tried to find out how and why students manipulated in a line reflection construction task and the extent to which manipulation could lead learners to successfully tackle the task. This study researched on four junior secondary students, drawing mainly on qualitative data used in the analysis, including task-based interview with the employment of think aloud method in a designed line reflection construction task, as well as study of students’ drawings. The data analysis mainly focused on three areas. First, the analysis of each of the four cases was conducted by looking into the different kinds of schemata possessed by the student informants. Second, analysis of the different knowledge (conceptual knowledge, manipulation and procedural knowledge) embedded in the schema possessed by the student informants was done. Third, synthesis was drawn upon the analysis made in an attempt to answer the research questions posed in this study. Findings from the study confirmed the core role conceptual knowledge plays in the establishment of a learner’s schemata. Findings also revealed that different learners may possess different schemata towards the same concept such as the concept of same distance. When investigating the manipulative actions employed by student informants, it was found that there is a reciprocal relationship between a learner’s conceptual knowledge and his manipulation. This is also apparent in cases where there was a misconception in the learner’s schemata. The research also found that students exercised manipulation very differently and these manipulative actions were largely informed by their corresponding conceptual knowledge. With regard to why they manipulated, the research revealed reasons including manipulation for exploration, manipulation for representation and manipulation for verification. Based on the observation and analysis done in the four cases, it was found that manipulation helped students in the completion of the task to different extents. Learners with weaker conceptual knowledge in line reflection benefited more from the manipulation done in the construction task. These findings have implications for the teaching and learning of line reflection. Teachers are suggested to consider introducing using manipulative tools when approaching the teaching of line reflection, especially when they are dealing with students without rich conceptual knowledge in the area. The effectiveness of having hands-on experience implies that simply teaching definition and inviting learners to rote-learn does not necessarily lead to effective acquisition of knowledge in the Mathematics topic of line reflection. / published_or_final_version / Education / Doctoral / Doctor of Education

published_or_final_version / Education / Master / Master of Education

Mathematics presents a stumbling block to many students, particularly those majoring in scientific fields, business administration, or elementary education. Improvement of student achievement in mathematics at the lower division college level is needed. Seven instructional methods were investigated in terms of student achievement: programmed instruction (P.I.), individualized instruction (I.I.), computer based instruction (CBI), laboratory and discovery methods (Lab), television (TV), audio-tutorial (A-T), and tutoring. The research questions were: (1) What is the relative effectiveness of the innovative instructional methods as measured by student achievement and compared to the traditional lecture method? (2) What is the relative effectiveness of the innovative instructional methods on students of differing ability and course levels. (3) What is the effectiveness of combinations of the innovative instructional methods? A meta-analytical approach was used. Studies comparing an innovative method to the lecture or to another innovative method were located, and the summary data in each were used to calculate an "effect size"--a standardized measure of the effectiveness of the innovative method--to which statistical procedures were applied. The meta-analysis found that (1) Relative to the lecture method, six of the innovative methods produced a positive effect on student achievement. The ranking of the methods in order of decreasing effectiveness was: tutoring, CAI, A-T, I.I., P.I., Lab, TV. (2) The most effective methods by level of course were: (a) Precalculus level: CAI, A-T, and tutoring; (b) Calculus level: tutoring, I.I., P.I., and A-T; (c) Foundations of Mathematics (elementary education majors): P.I.; Descriptive Geometry: TV. The most effective methods by ability level of the student were: (a) High ability: CAI and Lab; (b) Middle ability: CAI, I.I., and P.I.; (c) Low ability: P.I. and A-T. (3) The lack of empirical studies prevent a determination of the relative effectiveness of combinations of the innovative methods. Recommendations include the following: (1) Variation of instructional methods; (2) Incorporation of specific, effective elements of innovative methods into the lower division college mathematics instructor's repertoire; and (3) Empirical investigation of the effectiveness of combinations of methods and of various instructional methods on students of different ability levels.

This study explores the relationship between university mathematics teachers' beliefs about the nature of reading mathematics and their practices regarding reading mathematics. It is a response to the calls for reform in mathematics education, particularly to the assertion made by the National Council of Teachers of Mathematics in 1989 that not all students can read mathematical exposition effectively and that all students need instruction in how to read mathematics textbooks. It presupposes a collaboration between reading and mathematics teachers to help students learn to read mathematics. The objectives were (1) to examine mathematics teachers' beliefs and practices regarding reading, mathematics, and thereby, reading mathematics; (2) to determine whether the theoretical perspectives implicit in those beliefs and practices could be characterized vis-a-vis the theoretical orientations that inform Siegel, Borasi, and Smith's (1989) synthesis of mathematics and reading; and (3) to determine the relationship, if any, that exists between mathematics teachers' beliefs about reading mathematics and their practices regarding reading mathematics. The synthesis presents dichotomous views of both mathematics and reading: Mathematics is characterized as either a body of facts and techniques or a way of knowing; reading, as either a set of skills for extracting information from text, or a mode of learning. The latter view, in each case, can be characterized as constructivist. The researcher was a participant observer in a university sumner program. The primary participants were fourteen mathematics instructors. Interviews were conducted using a heuristic elicitation technique (Black & Metzger, 1969). Field notes were taken during observations of classroom activities and other non-academic summer program activities. The data were coded using a constant comparative method (Glaser & Strauss, 1967) comparative method. Twelve instructors held conceptions of reading that were consistent with their conceptions of mathematics. Of those twelve, two held conceptions that could be characterized as constructivist; ten held conceptions that were not constructivist. Two instructors held conceptions of reading that were not consistent with their conceptions of mathematics. Of those two, one held a constructivist conception of reading but not of mathematics; one held a constructivist conception of mathematics but not of reading. Teachers' practices reflected their theoretical orientations. The study has implications for teacher education: If teachers' beliefs are related to their practices, then teacher education programs should (1) acknowledge the teachers' existing beliefs and (2) address the theoretical orientations implicit in various aspects of pedagogy.

A hierarchy of skills in the measurement topic of area was validated on three-hundred and six students between grades six and nine. The hierarchy of skills was based on the rules underlying the individual skills. When a rule for one skill was considered a component of a rule for another skill, then the two skills were hypothesized to be hierarchically ordered. In addition, if a simple rule for a particular skill was replaced by a more complex rule, resulting in a different skill, then these two skills were hypothesized to be hierarchically ordered. The physical representations of the area tasks, as well as the shapes of the area figures were hypothesized as influencing the skill orderings. The use of Latent-class analysis revealed that seven of the nine skill orderings analyzed were hierarchically ordered based on difficulty level and not prerequisiteness. The other two skill orderings indicated equaprobable partial mastery classes. In addition to Latent-class analysis, the incorrect processes used by the students were coded and tabulated. The results revealed that (1) nonstandard shaped area problems were the most difficult for this sample, (2) the most frequent process associated with incorrect responses involved the addition of numbers shown in area problem figures, (3) the second most frequent process involved some form of multiplication, without regard to the area concepts inherent in the task, and (4) students beyond the sixth grade made more errors involving multiplication processes than errors involving addition processes. The study revealed that the use of rules, representations and shapes as the basis for a hierarchy does appear to have merit. In addition, process analysis revealed that students respond in a large variety of ways when they do not know the correct process for area tasks.

<p>This research provided insights into the mathematical achievement of a cohort of tertiary mathematics students. The context for the study was an entry level mathematics course, set in an engineering programme at a tertiary institution, the Cape Peninsula University of Technology (CPUT). This study investigated the possibilities of providing a bridge between the assessment of students by means of tests scores and a taxonomy of mathematical objectives, on the one hand, and the critical analysis of student produced texts, on the other hand. This research revealed that even in cases of wrong solutions, participant members' responses were reasonable, meaningful, clear and logical.</p>

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