Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 83-98). Issued in print and online. Available via ProQuest Digital Dissertations.

M.Ed. (Mathematics Education) / This dissertation was on mathematical proficiencies that learners displayed in their response to final examination questions on sequences and series, differential calculus, analytical geometry and trigonometry. The main focus was on four mathematical proficiencies, procedural fluency, conceptual understanding, strategic competence and adaptive reasoning. The initial goal was to review examination questions to identify mathematical proficiencies that the examination questions demanded and then an analysis of learners’ scripts to identify the mathematical proficiencies that the learners displayed in their response to examination questions. Review of literature first focused on the theoretical framework of the study, constructivism, its guiding principles and how it contributes towards mathematics learning. The framework for mathematical proficiency was shown with an aim of viewing other underlying issues on mathematical proficiency. The four mathematical proficiencies; conceptual understanding and procedural fluency constitute mathematical knowledge and strategic competence and adaptive reasoning are mathematical skills. Mathematical tasks are reviewed as properties of well-structured mathematical tasks in which examination questions were also mathematical tasks. The taxonomy of learning was discussed with an aim of finding properties of the taxonomy that inform the structure of the examination questions. Examination questions on sequences and series, differential calculus, analytical geometry and trigonometry were reviewed, to find mathematical proficiencies that the examination questions demanded and those that learners display in their responses.

published_or_final_version / abstract / Education / Doctoral / Doctor of Education

Poor mathematics performance in South African schools is of national concern. An attempt to gain insight into the problem prompted a study into the possibility of a relationship between visual perceptual skill and mathematic ability. A theoretical review revealed that inherent limitations of traditional psychological theories hinder an adequate explanation for the possible existence of such a relationship. The theory of situated cognition seems to be better suited as an explanatory model, and simultaneously clarifies the nature of both visual perception and mathematics. A small exploratory study, with a sample of 70 Grade 6 learners, provided empirical evidence towards the plausibility of the relationship. Specifically, it proved the hypothesis that visual perceptual skill positively correlates with scholastic mathematics achievement. The results of the study, interpreted within the situated cognitive framework, suggest that a conceptual emphasis in mathematics education – as opposed to a factual emphasis – might improve mathematic ability, which may credibly reflect in scholastic performance. / Psychology / M. Sc. (Psychology)

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

Poor mathematics performance in South African schools is of national concern. An attempt to gain insight into the problem prompted a study into the possibility of a relationship between visual perceptual skill and mathematic ability. A theoretical review revealed that inherent limitations of traditional psychological theories hinder an adequate explanation for the possible existence of such a relationship. The theory of situated cognition seems to be better suited as an explanatory model, and simultaneously clarifies the nature of both visual perception and mathematics. A small exploratory study, with a sample of 70 Grade 6 learners, provided empirical evidence towards the plausibility of the relationship. Specifically, it proved the hypothesis that visual perceptual skill positively correlates with scholastic mathematics achievement. The results of the study, interpreted within the situated cognitive framework, suggest that a conceptual emphasis in mathematics education – as opposed to a factual emphasis – might improve mathematic ability, which may credibly reflect in scholastic performance. / Psychology / M. Sc. (Psychology)

This study used Polya’s problem-solving model to map the level of development of the algebraic problem solving skills of Grade 12 learners from the Oshana Region in Northern Namibia. Deficiencies in problem solving skills among students in Namibian tertiary institutions have highlighted a possible knowledge gap between the Grade 12 and tertiary mathematics curricula (Fatokun, Hugo & Ajibola, 2009; Miranda, 2010). It is against this background that this study investigated the problem solving skills of Grade 12 learners in an attempt to understand the difficulties encountered by the Grade 12 learners in the problem solving process. Although there has been a great deal of effort made to improve student problem solving throughout the educational system, there is no standard way of evaluating written problem solving that is valid, reliable and easy to use (Docktor & Heller, 2009). The study designed and employed a computer aided algebraic problem solving assessment (CAAPSA) tool to map the algebraic problem solving skills of a sample of 210 Grade 12 learners during the 2010 academic year. The assessment framework of the learners’ problem solving skills was based on the Trends in International Mathematics and Science Study (TIMSS), Schoenfeld’s (1992) theory of metacognition and Polya’s (1957) problem solving model. The study followed a mixed methods triangulation design, in which both quantitative and qualitative data were collected and analysed simultaneously. The data collection instruments involved a knowledge base diagnostic test, an algebraic problem solving achievement test, an item analysis matrix for evaluating alignment of examination content to curriculum assessment objectives, a purposively selected sample of learners’ solution snippets, learner questionnaire and task-based learner interviews. The study found that 83.8% of the learners were at or below TIMSS level 2 (low) of algebraic problem solving skills. There was a moderate correlation between the achievement in the knowledge base and algebraic problem solving test (Pearson r = 0.5). There was however a high correlation between the learners’ achievement in the algebraic problem solving test and achievement in the final Namibia Senior Secondary Certificate (NSSC) examination of 2010 (Pearson r = 0.7). Most learners encountered difficulties in Polya’s first step, which focuses on the reading and understanding of the problem. The algebraic strategy was the most successfully employed solution strategy. / Mathematics Education / M. Sc. (Mathematics, Science and Technology Education (Mathematics Education))

This study investigated the problem-solving skills of Grade 12 learners in probability. A total of 490 Grade 12 learners from seven schools, categorised under four quintiles (socioeconomic factors) were purposefully selected for the study. The mixed method research methodology was employed in the study. Bloom’s taxonomy and the aspects of probability enshrined in the Mathematics Curriculum and Assessment Policy Statement (CAPS) document of 2011 were used as a framework of analysis. A cognitive test developed by the researcher was used as an instrument to collect data from learners. The instrument used for data collection passed the test of validity and reliability. Quantitative data collected was analysed using descriptive and inferential statistics and qualitative data collected from learners was analysed by performing a content analysis of learners’ scripts. The study found that the learners in this study were more proficient in the use of Venn diagrams as an aid in solving probability problems than in using tree diagrams and contingency tables as aids in solving these problems. Results of the study also showed that with the exception of Bloom's taxonomy synthesis level, learners in Quintile 4 (fee-paying schools) had statistically significant (P-value < 0.05) higher achievement scores than learners in Quintiles 1 to 3, (i.e. non-fee-paying schools) at the levels of knowledge, comprehension, application, analysis and evaluation of Bloom’s taxonomy. Contrary to expectations, it was revealed that the achievement of the learners in probability in this study decreased from Quintile 1 to Quintile 3 in all but the synthesis level of Bloom's taxonomy. Based on these findings, the study argued that the quintile ranking of schools in South Africa may be a useful but not a perfect means of categorisation to help improve learner achievement. Furthermore, learners in the study demonstrated three main error types, namely computational error, procedural error and structural error. Based on the findings of the study it was recommended that regular content-specific professional development be given to all teachers, especially on newly introduced topics, to enhance effective teaching and learning. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)

1 online resource (xii, 171 leaves) : illustrations (chiefly color), color graphs / ADHD negatively impacts academic performance, and the traditional classroom setting conflicts with the symptoms of ADHD. This research examined the potential of teaching through play and peer-teaching as alternative teaching methods to improve the mathematical performance of Grade 1 children with symptoms of ADHD; by answering, would adapting teaching methods to include teaching through play and/or peer-teaching, in the South African classroom, improve the mathematical performance of children with symptoms of ADHD? A pre-test-post-test control group design was employed in this comparative experimental study. Participants were purposively selected then randomly assigned to one of three intervention groups. An eight-week intervention was implemented as teaching through play or peer-teaching. Pre-test and post-test scores were analysed using a dependent t-test, a Wilcoxon Signed Rank test, and a Kruskal Wallis test. Teaching through play and peer-teaching have the potential to improve the mathematical performance of Grade 1 children with symptoms of ADHD. Special precautions were taken in the process of minor research participants, adhering to the ethical principles of beneficence and non-maleficence, justice, and autonomy. / Psychology / M. Sc. (Psychology (Research Consultation))

Text in Afrikaans / Die doel van hierdie navorsing was om vas te stel of leerders in Graad R die nodige wiskundige vaardighede deur gestruktureerde spel bemeester het wat as grondslag vir verdere wiskunde-onderrig kan dien. In die literatuurstudie is die verskillende wiskunde konsepte in diepte ondersoek. Daar is ook gefokus op die oordra van wiskunde konsepte aan jong leerders. Verder is daar in die literatuurstudie ondersoek ingestel na die doel, waardes, tipes, ontwikkeling van spel asook spel in die moderne era. Vir die empiriese ondersoek is 103 Graad R-leerders van ses verskillende skole betrek. Daar is gebruik gemaak van ʼn ewekansige steekproef by die skole wat bereid was om aan die studie deel te neem. Die ondersoek het plaasgevind deur die gebruik van die positivistiese kwantitatiewe navorsingsmetode. Daar is gebruik gemaak van vraelyste om te bepaal wat die bemeestering van wiskundige vaardighede by Graad R-leerders is. Gestruktureerde spel is as hulpmiddel gebruik vir die insameling van die navorsingsmateriaal. Die resultate is bekom deur die vraelyste statisties te verwerk. Daar is ook vraelyste aan die onderwysers gegee om meer agtergrond oor hulle onderrigmetodes asook die onderwysers se eie gevoel teenoor en kennis van wiskunde te bekom. Die navorsingsresultate bevestig dat die meerderheid van die Graad R-leerders oor ʼn goeie wiskundige grondslag beskik. Dus is daar deur die studie gevind dat dit belangrik is dat jong kinders baie moet speel om nuwe vaardighede aan te leer, maar ook die geleentheid moet hê om die vaardighede te beoefen. / The purpose of this research was to determine whether learners in Grade R mastered the necessary math skills through structured play that could serve as a basis for further mathematics training. In the literature various math concepts were studied in depth. The main focus was on the transfer of math concepts to young learners. The literature study further focussed on the purpose, values, and types of play, development of play and play in the modern era. An empirical study using random sampling was done using 103 Grade R learners from six different schools who were willing to participate in the study. The study was done using the positivism quantitative research methodology. Questionnaires were used to determine the mastery of mathematical skills by Grade R learners. Structured play was used as a tool to collect the research material. The results were obtained by processing the questionnaires statistically. Questionnaires were also completed by the teachers to confirm their background knowledge, teaching methods as well as to determine the teachers‟ own feelings towards and their knowledge of mathematics. The research results confirm that the majority of Grade R learners have obtained a good math foundation. The study also confirmed that it is important that young children need to play to learn new skills and must have the opportunity to practise the new skills. / Psychology of Education / D. Ed. (Sielkundige Opvoedkunde)