Abstract
Abstract : The teaching and learning of rational numbers in the schooling system has long been arduous and problematic for both teachers and learners for a long time. The aim of this study was to gauge the level of cognitive understanding of rational numbers (specifically the fractions-decimals-percentages triad) of the one hundred and seventeen (117) 2015 Foundation Phase first-year student teachers. These students from different school environments and contexts in learning mathematics, encompassing a range of demographics and equally varying levels of competencies, before this cohort, lacked this background knowledge as well. I believe it is better to know their strengths and weaknesses to provide effective support structures in their first-year second-semester mathematics class. The secondary aim of this study was to ascertain the validity of the instrument that was used to elicit their mathematical cognition and answer the question βWhat are the most common misconceptions and their associated errors that student teachers at foundation phase display when studying fractions for teaching?β There are attributes of difficulty, in part, to the need to mentally represent a common fraction like π ππ . The brain recognises the whole numbers 7 and 20 as local values and 0.35 as global value (Gabriel, Szucs, & Content, 2013). For many it was not automatic or easy for the learners and adults to cross the bridge from whole numbers to conceptualise fractions. The rational number knowledge of student teachers, in particular, the relationships and equivalence of fractions-decimals-percentages, and including their comparison and ordering, is the focus of this thesis. Each of the triad concepts has associated misconceptions that appear to emerge from working with whole numbers and seem to interfere with boundaries associated with these new concepts. The research design involved constructing an assessment instrument that included apposite sixty-seven (67) questions with ninety-three (93) items deemed to be varying in soliciting the proficiency levels of the students in triad. The collected data is analysed through the use of two lenses. Mapping relationships and learning progress is done in the Structure of the Observed Learning Outcome (SOLO) plotting levels of cognitive demands as it becomes more complex (Biggs & Collis, 1982). Secondly, the application of the Rasch Model provides an indication of the successful, or otherwise, construction of this v proficiency-based measuring instrument, thereby enabling a reflective approach to the construct being tested and the instrument. Student responses were analysed into five main categories and allocated question items that solicited relevant and most applicable fluencies to facilitate the descriptive analysis process. A majority of misconceptions were displayed in categories of manipulating fractions symbols (operations); comparing and sequencing; as well as in alternative forms (equivalence). The category solving mathematical word problems with fraction elements was difficult; while the understanding of the triad concepts was managed well by the students. A five-member focus group of students along a scale of ability levels enabled additional insights through interview discussions into the misconceptions and associated errors at two selected focus points that are the operations as well as on the ordering and sequencing. The identified knowledge gaps can be filled by re-teaching the concepts and using the instrument to authenticate the acquired knowledge. We were able to conclude that a refined instrument, applied to university first-year students, can inform the teaching and learning of the triad domain of rational number concepts.
Ph.D. (Mathematical Education)