3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Solving multiplication and division word problems: strategies used by Grade 6 learners(2019) Ramaoka, TumeloThis study was aimed at exploring the models and strategies a class of Grade 6 learners used to solve multiplicative word problems prior to, and following, a series of four one-hour intervention lessons based on the literature on multiplicative reasoning using a Realistic Mathematics Education (RME) – based theoretical – framework. The sample of the study were a group of 36 learners from a fee-paying government school in Johannesburg East district that has been classified as a low performing school based on the Annual National Assessment (ANA) for Mathematics. Outcomes based on pre- and post-tests indicated substantial gains related to the content of the intervention lessons as learners were able to make better sense of multiplicative situations and use more efficient strategies to solve multiplicative situations. Pre-test data pointed out that learners struggled to make sense of multiplicative situations especially division items and used a limited range of models and strategies. Post-test data indicated that there were substantial shifts to using a broader range of multiplicative models and strategies including the emergent use of ratio models. This use of a broader range of models and strategies increased success in learners’ multiplicative problem-solving performance.Item Teaching and learning in multilingual Mathematics classrooms through the use of an instructional Mathematics Application Programme(2019) Katabua, EvalisaMathematics classrooms are often characterized by various teaching aids including, more recently, handheld devices that are often loaded with a Mathematics Application (App) so as to provide assistance in enhancing learners’ Mathematical understanding. However, the same App can be a hindrance if the Language of Learning and Teaching (LoLT) is not carefully considered in the App design stage. This study investigated what possible language issues might exist that currently could be overlooked by focusing on one Mathematics Application called onebillion©. Engelströms’ (1999) Expansive Activity Model was the framework chosen for this study. In collecting data, mapping the App to the curriculum documents, interviews were conducted with Grade 1 teachers and learners, along with two classroom observations. The study established that, there were no major language issues in regard to the isiZulu language as used in the App as compared to curriculum language and as per teachers’ own experiences, in particular taking into account the context where the participants were situated. It is a recommendation that during translation stage, the official curriculum documents be thoroughly consulted so as to ensure minimal language inconsistencies. Regarding future research, I would propose that other language studies be carried out (on the App) for the other African languages in different contexts, as language nuances such as pronunciation, dialects etc. have different implications for different African languages.Item Linear equations :grade 9 learners’ progress and their difficulties over time(2019) Halley, AndrewThis study investigated Grade 9 learners’ progress in solving linear equations over a six month period. The notion of progress is taken to encompass shifts in learners’ abilities to solve equations with fluency, and shifts in communicating their reasoning, Data were collected with six learners in a Johannesburg secondary school, by means of four task-based interviews each.Item Teachers' use of examples to promote mathematical proficiency in statistics(2019) Zulu, SibongileExamples are a prevalent and a powerful tool used in the teaching and learning of Mathematics. In statistics, where data sets used are normally extracted from real life situations, the use of examples is indispensable. This study examines what examples are used and how teachers use them to promote (or fail to) mathematical proficiency in statistics as the outcome of the learning process. Variation theory is employed through the patterns of variation to assess whether the use of examples leads to learning. To ascertain that the use of examples results in the attainment of mathematical proficiency, the strands of mathematical proficiency by Kilpatrick, Swafford and Findell, (2001), are incorporated in the theoretical framework. This study involves observing lessons interactions between Grade 10 teachers and learners, and interviewing teachers after observing the lessons at a school in Johannesburg. An analytical framework derived from variation theory and strands of mathematical proficiency is then used as an assessment tool. Findings show that in and of themselves examples do not necessarily help to achieve the intended object of learning nor mathematical proficiency. More poignant, whatever the teacher’s intention to convey the object of learning through the use of examples, and no matter how the teacher enacts it, learners could still fail to grasp the lived object of learning and attain mathematical proficiency.Item From fraction to ratio : Exploring the features of f irst-year s tudents ’ percent discourse(2019) Luksmidas, Jaqueline MarquesPercent is a familiar, yet complex topic that is found to be difficult for both adults and children. The question of why percent has been persistently difficult has spurred much research, the most notable of which was conducted in the early 1990’s. Those studies have adopted a cognitive perspective. This study adds a commognitive perspective to the discussion by proposing a model for the development of percent discourse (PD-Model). The model rests on Sfard’s premise that learning mathematics is synonymous with modifying and extending one’s discourse. I begin by employing a cognitive framework of percent for the design of written tests to identify the areas of percent that first-year university students experience difficulty with. The quantitative analysis of the written tests shows that less than half the students obtained a score of 50% or more. Later, in search of the features of students’ discourse that hinder their access to percent discourse, I examine the discourse of two pairs of students in interview sessions. I illustrate the application of the PD-Model as an interpretive analytical tool that offers an explanation for the insufficiency in their objectification of percent as a comparative ratio. This study confirms the results of Parker’s (1994) study, that is: percent is difficult for students to work with. The key findings of the discursive analysis show that students’ discourse of percent is narrow and deeply rooted in a percent-as-fraction notion. The students’ discourse is predominantly additive in nature and does not show signs of recognising the underlying multiplicative structures of percent tasks. As such, a fully-fledged objectification of percent as a comparative ratio is not evident in the students’ discourse of percent.Item Developing early number learning using maths recovery principles(2018) Morrison, Samantha SarahNational, regional and international comparative data all paint the same dismal picture of South African learners’ low levels of numeracy (Schollar, 2008; DBE, 2012; Mullis, Martin & Fay, 2008) which has a detrimental knock-on effect on their mathematics learning in higher grades. One of the recommendations made by Spaull (2013, 2016) and others for the remedying of this mathematics ‘crisis’ in our country is for the development and implementation of a structured, evidence-based intervention programme in the midst of a lack of teacher knowledge on the ground related to how children learn early number. Bearing this recommendation in mind, this doctoral study was aimed at improving a sample of South African children’s early number learning using principles based on the Mathematics Recovery (MR) programme (Wright et al., 2006). I opted to investigate the efficacy of a short-term intervention programme using Mathematics Recovery because this programme sets out a detailed, research-based trajectory for children’s early number learning while also providing explicit teaching and assessment guidelines. The scale of the problem on the ground, and my belief that learning mathematics is a social enterprise, led me to adapt the MR programme for grouped intervention. To this end, I used the Emergent Approach (Cobb & Yackel, 1996) which incorporates both cognitive and sociological perspectives of learning as the theoretical underpinning of my study. I employed a teaching experiment using 20 mid-attaining Grade 2 learners that were split into a matched intervention and control group. The 10 intervention learners were then further divided: 2 learners received individual intervention and 8 learners received grouped intervention (two groups of 4 each). The different layers of cognitive analyses allowed me to determine two things. Firstly, learners who received intervention based on MR principles moved further along the early number learning trajectory across all aspects of MR’s Learning Framework in Number (LFIN) than the control group who continued with their normal classroom teaching; and secondly, comparable learning gains were made between learners who participated in different formats of the same intervention (individual and grouped). From the sociological perspective, analysis based on aspects of interaction – which focused on learners’ oral discourse and solution strategies used across different representational settings – provided insight into the emergent and more specific focus on ‘base-ten thinking’, which is one part of the LFIN framework. This emergent specificity led to further development of the Base Ten aspect of the LFIN related to progressions and reversions in learners’ use of base-ten as inferred from their oral discourse and flexibility in enacting base-ten strategies across various representational settingsItem Metacognitive skills of second year extended and main stream University mathematics students: a case study(2017) Moolman, RuanMany universities have introduced so called extended degrees where students’ first year workload is spread over two years to prevent the decline of graduates in mathematics and science. It has been put forward that extended degree courses should include the explicit training of mathematics students in the use of metacognitive skills. This is based on research that shows that successful students in mathematics are able to apply such metacognitive skills and that these skills play an important role in mathematical problem solving. Such skills are concerned with the actual regulation, coordination and control of one’s own learning activities and cognitive processes. Given that extended degree students generally perform weakly in mathematics in comparison to main stream students (non-extended degree students) this research study sets out to consider the differences in the use of metacognitive skills of these two student groupings. A qualitative case study was used to investigate collaborative solving of mathematical problems of one student pair. Students were trained in the use of metacognitive skills by using the metacognitive intervention method called IMPROVE. The student pair was video-recorded during talk-aloud protocols twice before explicit training in the IMPROVE method, and after instruction in order to evaluate students’ development in the use of metacognitive skills. Video recordings were transcribed noting students’ verbal and non-verbal actions and the coding of transcriptions in conjunction with content analysis was used in determining differences in students’ metacognitive skills. Since students worked collaboratively, instances where students acted as so-called social triggers of each other’s metacognitive skills, were also investigated. With student-researcher interaction during observations, the researcher was also regarded as a social trigger of students’ metacognitive behaviour. Apart from these social triggers, environmental triggers of students’ metacognitive skills were also scrutinised. Environmental triggers included the effect of task difficulty and the intervention of the IMPROVE method on students’ metacognitive skills. This study on the social and environmental triggers of individual’s metacognitive skills contributes to the relatively young field in viewing metacognition as cognitive activity that operates on multiple levels during collaborative problem solving, and that metacognition cannot solely be explained in terms of individualistic conceptions but also by social and environmental triggers. Results from the study show that, in general, the main stream student exhibited a greater number of metacognitive skills compared to the extended degree student. Furthermore, it seems that the IMPROVE method as an environmental trigger, had an effect on the development of both students’ metacognitive behaviour. Research findings of the study also reveal that the researcher’s intervention mainly resulted in the students acting as social triggers for each other’s metacognitive behaviour. Furthermore, it was found that there were a greater number of occurrences in which the main stream student acted as social trigger for the extended degree student’ metacognitive behaviour. The level of task difficulty also seems to have acted as environmental trigger for students’ metacognitive behaviour. As an exploratory study, the findings of this study are not generalizable.Item Teacher's use of exemplification and explanations in mediating the object of learning(2017) Barkay, DanielleThis study examined teachers' use of exemplification and explanations in the teaching of algebraic expressions. In particular the focus was on the selection and sequencing of examples as well as what a teacher does with these examples in terms of their explanations to maintain the focus or object of learning. Adler and Ronda's (2015) Mathematical Discourse in Instruction (MDI) framework was the foundation of this study's conceptual and analytical framework and was complemented with the work of Stein (2000) and Moschovich (1999, 2015). Data was collected from two Grade 8 teachers through the use of video-recordings and transcripts. The data was then analysed based on the themes that emerged from the conceptual framework. The findings revealed that the examples themselves had the potential to restrict the object of learning and together with the teacher's corresponding explanatory talk could reduce or shift the object of learning from translating algebraic expressions to focusing on procedures. The findings show how each component of MDI worked separately and then together to mediate the of the object of learning, but this study has additionally highlighted how the components themselves, namely exemplification and explanatory talk, have a direct effect on each other.Item Tasks used in mathematics classrooms(2017) Mdladla, Emmanuel PhathumusaThe 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.Item Using errors and misconceptions as a resource to teach functions to grade 11 learners(2017) Malahlela, Moloko VictorThis research report focussed on the teaching of the function concept directed at the errors the learners make as well as the misconceptions which could be associated with those errors. The study was conducted at a secondary school in Johannesburg, South Africa. This was a qualitative error analysis study which also had a form of interventional or remedial teaching. The research was driven by the following research questions: (1) What errors and misconceptions do grade 11 learners show on functions?, (2) What learning affordances and constraints can be created if teaching is directed at learners’ errors and misconceptions? and, (3) To what extent can the learners’ achievement on the topic functions be boosted if teaching is directed at learners’ errors and misconceptions? The study used a purposive sample of six grade 11 mathematics learners from a group of 34 learners. To answer these questions, I structured the study to encompass numerous phases of data collection using different instruments. Firstly, I constructed a test instrument and used it on this group of grade 11 learners. These learners had been taught functions earlier in the year, so the test was diagnostic to measure the cognitive levels of the learner on the concept and also to establish the errors made and misconceptions they carried onto the section from other sections or picked up from the function concept. The study was mainly based on the constructivism theory of learning and teaching, but also had other theories to link to it such at the socio-cultural theory, the APOS (actions, process, object and schema) theory, the concept image and concept definition as well as the variation theory. The errors I picked up from the pre-test I classified and analysed using the conceptual framework grounded on the abovementioned theories. It was this analysis which enabled me to structure the desired intervention program together with the teacher after which I conducted a post-test with the subjects. Other forms of data collection such as the interview and observation were employed during the study. I used the interview to get clarity from the learners’ pre-test questions responses, whilst the observation I used during the intervention lessons the teacher had with the learners. Indeed the findings were that, while there was a substantial improvement on learner performance on the post-test, it appeared clearly that cognitive levels of the learners on the function concept had been enhanced. This improvement of performance was a result of the teaching that was directed at the errors, which also interprets to having created a favourable environment which could be interpreted as learning affordances to boost the learners’ understanding of the function concept.