With the introduction in 2006 of the school subject Mathematical Literacy (ML) in the further Education and Training band, there have been expectations that such a subject will develop responsible citizens, contributing workers and self-managing people. The extent to which the subject can meet these aims is dependent on the ways in which the subject is assessed which influences the focus of ML in the classrooms. With this in mind, this study set out to explore the ways in which a class of Grade 12 learners engaged with a preparatory examination designed and administered by the KZN Department of Education. This is a qualitative study carried out with seventy-three grade 12 mathematical literacy learners from an urban school in North Durban. The purpose of this research is to explore the learners’ engagement with the examination tasks, thereby identifying possible factors which influence learners’ success in these items. Data were gathered from a document analysis of the 2009 KZN Trial Examination question paper and marking memorandum; 73 learners’ written responses to the examination tasks, and interviews with ten of these learners. The purpose of the document analysis was to identify contexts in which learners performed well or poorly, as well as to assess the design of the instrument. Semi-structured interviews were conducted individually with ten learners and video recorded. The purpose of the interviews was to explore some of the factors which influenced their written responses. The findings revealed that the task design was problematic for learners in terms of the order of the questions and the placement of the crucial information necessary to answer the questions. Some tasks also contained errors. The complexity of the scenario in terms of the amount of information, the language used, and the presence of distracters further influenced the way in which learners responded to the task. Learners’ personal experience of the context also affected the way they interpreted and responded to the task. Factors that constrained learners’ success in the examination task included poor conceptual understanding, misconceptions and language-related misinterpretation. It was also found that learners did not consider it a priority to make sense of the context: they focused on identifying formulae or information that could be used to present answers with little concern about the reasonableness of their responses. Some strategies used by learners in responding to the task included number grabbing, guessing without checking, scanning for crucial information and assumption-making. The study recommends that provincial examination papers be subject to the same stringent moderation requirements of the national examinations. It also recommends that should diagrams be used, they must be relevant to the context and should not conflict with the subject matter. The use of contexts should cater for alternate answers and multiple approaches and the marking memorandum should be flexible to accommodate these multiple approaches. Care must be taken in the presentation and placement of crucial information, so that learners do not miss the information they need to answer the questions. When familiar contexts are being used, task designers should also consider whether learners’ everyday experiences may conflict with these scenarios. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2011.

This was a qualitative study carried out with one grade 8 multicultural, multiethnic, mathematics class. This research study began with the idea of finding out whether the learners home language (especially Zulu Xhosa) could be linked to algebraic misconceptions. The 40 learners (participants) in my study had just been introduced to algebra. I chose the school and participants through “convenience sampling”. This made sense since I am an educator at this particular school. I had explained the meaning of the word "variable" in depth. The concepts "like terms" and "unlike terms" had been explained. The index laws for multiplication and division of the same bases had been discussed. It was within this context that the algebra worksheet was given to the learners, in the first cycle. I examined the algebra errors made by the grade 8 learners after marking the worksheets. I linked the errors to past literature on algebraic misconceptions as well as to Bernard's (2002b) error classification list. The conclusion was that the learners were making common errors which were not affected by their home language. I spent time on reflection since the outcome was not exactly what I had anticipated (that is, I had harboured strong suspicions that English second language learners would commit more algebraic errors than the English home language learners). I then considered a possible link between culture and algebraic misconceptions. Videotaped lessons were used for this purpose. However, observations of these videotaped lessons did not produce much data. I honestly could not reach a conclusion. This formed the second cycle of my action research. Prompted by the obvious lack of interaction in the video recordings from my teaching, I changed my focus to what I, the teacher, did during the lessons, and how these actions may or may not have supported some of the algebraic misconceptions. I reflected on my teaching method and recognized the need to change to a more interactive teaching style. I needed to give the learners the space to think for themselves. I would merely facilitate where necessary. In the third cycle, I drew up a set of problems which matched the new teaching style (interactive teaching).The lessons during which the new set of problems were discussed and solved, were videotaped. These videotaped lessons were analyzed and a completely different picture emerged. The learners were absolutely responsive and showed a side of them that I had not seen before! This study came to be an action research study because I went through three cycles of reflecting, planning, acting and observing and then reflecting, re-planning, further implementation, observing and acting etc. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2010.

Mathematical Literacy (ML) has only recently been introduced to learners, and research in South Africa concerning learners’ conceptual understanding in ML is not widely available. However an important predictor of learners’ success or difficulties in concepts is the success or difficulties that in-service teachers experience themselves. It is therefore important for us as mathematics educators to identify areas in Mathematical Literacy that teachers are struggling to learn and apply. With this in mind, the study sets to explore teachers’ perceptions about, and performance in Mathematical Literacy tasks based on algebraic concepts. This study is located within the principles of the qualitative research case study approach. The combination of data collection techniques has allowed me to identify broad trends across the group as a whole as well as differences within the participants of the group itself. The participants of the study were a class of 17 students who were completing the ACEML programme at UKZN. Four sources of data were used. Firstly, data was generated from teachers’ reflections about certain tasks, the solution of which required the use of algebra. A second data collection instrument was an open-form questionnaire and the third instrument was two unstructured interviews with two teachers. The final instrument was the analysis of the teachers’ examination scripts. For this study, teachers from this group were classified along the lines of whether they were qualified to teach mathematics or not. The theoretical framework for the study was derived from the OECD/PISA (2003) cycle of mathematisation which specifies 5 aspects of mathematisation, together with the theory of reification. For the purpose of this research, a participant was considered as a “mathematics specialist” if s/he studied mathematics up to tertiary level, while a participant was considered as “non-mathematics teacher” if s/he studied mathematics only up to Grade 12 level. The findings reveal that although the teachers conveyed varying understandings of the ML curriculum, they believed that knowledge of basic algebra was necessary and adequate for them to deal with ML problems. Furthermore the teachers believed mathematical teaching experience contributes to improved problem solving in ML and that ‘practice and familiarity’ helped teachers improve their problem solving skills in ML. They also voiced a concern that the pace of the programme constituted a barrier to their success. Within the group, it was found that Mathematics specialist teachers performed better than the non-Mathematics teachers. All teachers found the mathematisation aspects of solving the mathematical problem and of reinterpreting the mathematical solution to make sense of the real-life problems, challenging, while the non-Mathematics teachers experienced problems with all five aspects of mathematisation. The findings of the study suggest that teachers need help in moving from lower levels to higher levels of mathematisation. Opportunities for mathematical modeling experiences need to be incorporated in the part-time in-service contact courses like ACEML. Further research is needed to inform education authorities about whether the use of teachers with only grade 12 mathematical knowledge to teach ML is advisable. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2010.

This thesis replicates and explores some of the recent findings by Robert Siegler regarding the development ofaddition skills in grade one children. Siegler states that children employ a number ofdifferent strategies to solve single digit addition problems, these strategies coexist and compete, and cognitive variability is an essential aspect of cognitive development. He also advocates the use ofthe microgenetic approach in order to explore cognitive development. Many of Siegler' s observations were replicated while the microgenetic approach produced valuable information. Consideration of Siegler's work resulted in two research questions being formulated, both concerning the actual selection of strategies. First, a prediction analysis was employed to test the hypothesis that children attempt to match the most appropriate strategy to the problem presented according to a principle ofleast effort (defined as the attempt to maximise benefit and minimise cost). The predictions were stipulated prior to the analysis and were based on the arithmetic development literature. It was predicted that children would tend to retrieve the answers to small problems and tie-problems or calculate the answer by counting on from the larger addend by the amount indicated by the small addend (which involves reversing the order of the addends when the first addend is the smaller of the two). The strategy selections (n=229) made by a sample of 12 grade one learners on 21 single digit addition problems were categorised and compared to the predictions. The prediction analysis reduced the expected error by 63%, supporting the least effort model of strategy choice. The result is statistically significant (2=10.231, p / Thesis (M.A.)-University of Natal, Pietermaritzburg, 2000.

This was a qualitative study carried out with learners from a grade twelve Standard Grade mathematics class from a South Durban school in the province of KwaZulu-Natal, South Africa. The main purpose of this study was to explore learners‟ understanding of the concept of the derivative. The participants comprised one class of twenty seven learners who were enrolled for Standard Grade mathematics at grade twelve level. Learners‟ responses to May and August examinations were examined. The examination questions that were highlighted were those based on the concept of the derivative. Additionally semi-structured interviews were carried out with a smaller sample of four of the twenty seven learners to gauge their perceptions of the derivative. The learners‟ responses to the examination questions and semi-structured interviews were exhaustively analysed. Themes that ran across the data were identified and further categorised in a bid to provide answers to the main research question. It was found that most learners‟ difficulties with the test items were grounded in their difficulties with algebraic manipulation skills. A further finding was that learners overwhelmingly preferred working out items that involved applying the rules. Although the Higher and Standard grade system of assessing learners‟ mathematical abilities has been phased out, with the advent of the new curriculum, the findings of this study is still important for learners, teachers, curriculum developers and mathematics educators because calculus forms a large component of the new mathematics curriculum. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2008.

This study explores the ways in which Grade 11 girls from an independent, predominantly / Thesis (M.Ed.) - University of KwaZulu-Natal, Durban, 2007.

This study had two components: 1) Investigating the conceptual understanding of teachers teaching elementary mathematics at primary schools in the province of KwaZulu-Natal, who had been successful in their mathematics modules in the National Professional Diploma in Education (NPDE) teacher upgrading program, and 2) Investigating the influence of their mathematical lifehistories on their understanding and personal philosophies about mathematics. It firstly required the NPDE students from the University of KwaZulu-Natal to complete a questionnaire adapted from the TELT interview schedule used by Liping Ma (1999). This questionnaire was to assess whether these high scoring teachers had an understanding of basic mathematical concepts that could have been regarded as being profound. The second part of the study was designed in order to get these teachers to examine their mathematical life histories and then to look at how their life histories could have influenced their level of understanding. It was found that these teachers were procedurally capable and were aware of the algorithms that could be used to solve the problems posed, but they lacked deep understanding of the concepts and were thus conceptually weak. None of the teachers demonstrated an understanding of the fundamental mathematics concepts that were assessed, that could be regarded as been ‘profound’. The mathematical life history portion of this study revealed that these teachers, having experienced mathematics education very differently due to their Apartheid influenced education, mentioned that there were definite influences that had a marked effect on their outlook on the subject and thus their belief in their ability to do basic/ fundamental mathematics. / Thesis (M.Ed. (School of Education and Development)) - University of KwaZulu-Natal, Pietermaritzburg, 2007.

The purpose of this study was to investigate how language affected the teaching and learning of mathematics for English second language (ESL) learners in five Further Education and Training (FET) schools in Northern KwaZulu-Natal, with a particular focus on word problems (WPs). In 2010, fifteen learners (nine boys and six girls) doing mathematics grade 11 from five different FET schools from Mtubatuba District in Northern Kwazulu-Natal participated in the study. Five teachers teaching the same learners from these five schools were also the participants in this study. The researcher’s teaching experience of eleven years as an FET mathematics teacher had taught her that many English second language learners were not able to correctly translate word problems into mathematical equation. This was what motivated the researcher to conduct a study on the impact of English to the teaching and learning of mathematics, especially Word Problems. The study was mostly framed around theory of Social Constructivism. The research instruments used in the study were: learner worksheets, learner interviews (individual and group interviews), teacher questionnaires and lesson observations. Some common challenges in the teaching of WPs were drawn from the analysis of the teachers’ responses: Many learners are unable to translate English statements into mathematical equations. The manner in which WPs are phrased generally pose some problems for many learners. There is lack of mathematics vocabulary such as ‘consecutive’, ‘twice as much as’, ‘doubled and then added to’, ‘squared’. From the learners’ responses, the following could be deduced as challenges in learning WPs: There is very little exposure of learners to word problems. Failure to write English statements mathematically. Less exposure to English due to teachers accepting the use of isiZulu more than English during teaching and learning. Too much wording in the WPs which ends up confusing. Little exposure to mathematical terms such as ‘consecutive’, ‘integers’. Both teachers and learners gave some strategies that they thought could help in the teaching of WPs, namely: Giving more time for learners to construct mathematical statements on their own. Engaging in one-on-one teaching with some struggling learners. Code-switching from English to isiZulu when necessary. Letting learners work through the worked examples first for proper understanding. Rephrasing the problem and breaking it into sections. Use of diagrams and illustrations. Giving learners more activities on WPs. / Thesis (M.Ed.)-University of KwaZulu-Natal, Durban, 2013.

As learners progress through the educational system their interest in mathematics diminishes. Although mathematics is designed to challenge learners, it has produced a high number of failures. Mathematics is most often measured by speed and accuracy of learners’ computation with little emphasis on problem solving and pattern finding. Whilst there are not many opportunities for learners to work on rich mathematical tasks that require divergent thinking as well. Such an approach limits the use of creativity in the classroom and reduces mathematics to a set of skills to master and rules to memorise. In doing so, causes many learners to become anxious and apprehensive. Thus their curiosity and enthusiasm for mathematics disappear, as they get older. Keeping learners interested and engaged in mathematics by recognising and valuing their mathematical creativity may reverse this negative tendency. 97 learners from Riverview High School took part in the study. Three different instruments were used to collect data: Mathematics Anxiety Scale (MAS), Personal Report of Communication Apprehension (PRCA-24) and a focus group interview. The MAS questionnaire was used to measure the level of mathematics anxiety experienced by the learners. The PRCA-24 questionnaire is a self-report measure of communication apprehension. The underlying factors were established that were influential in determining the levels of maths anxiety and communication apprehension in individual learners. The results of the study suggest that learner’ ability and attitude played an important role that would lead to the large failure rate in mathematics. Analysis and interpretation of the findings lead to the following conclusions being reached: (a) perceptions of mathematics as being a difficult subject (b) learners negative attitude in mathematics (c) fear for the subject, (d) learners self-efficacy beliefs in mathematics, (e) peer behaviour and (f) teacher behaviour. The research study findings suggest that learner’ ability and attitude played an important role. These attitudes contribute directly to the existence of maths anxiety and communication apprehension in learners which impacts on their academic performance. The results of the study suggest that learners experience varying levels of maths anxiety and communication apprehension that impacts on their performance in Mathematics, which are barriers to learning mathematics. / Thesis (M.Ed.)-University of KwaZulu-Natal, Edgewood, 2011.

The purpose of the study was to examine the effects of developmental instruction on the whole number computational abilities and mathematical attitudes of kindergarten children. Gender differences in mathematical achievement and attitudes were also explored.Ten traditional mathematics lessons were adapted by the researcher from the adopted mathematic program for the school system, Heath Mathematics, Connections, (Mangre, et al., 1992). Ten developmental mathematics lessons were created by the researcher following the guidelines of the NCTM Standards (1989) promoting the notion of a developmentally appropriate curriculum. The research designed both the Attitudinal Scale and Cognitive Abilities Test which were given both before and after the instructional treatment.The school corporation chosen as the site for the research was located in an urban area consisting of two small cities and the surrounding rural areas. The community consisted of people with diverse socioeconomic status and cultural backgrounds. The sample for the study consisted of 62 kindergarten students enrolled in four half-day classes in one elementary school. Complete data were available for 50 students. Four hypotheses were formulated and tested at the .05 level of significance.ResultsThe four hypotheses were analyzed using a 2 (method) x 2 (gender) MANOVA on the gain scores for both achievement and attitude taken together. Gain scores were obtained by subtracting the pretest score from the posttest score for both achievement and attitude.The findings of the study were:1. There was no significant difference between the whole number computational abilities of kindergarten children receiving developmental instruction and kindergarten children receiving traditional instruction.2. There was no significant difference between the whole number computational abilities of kindergarten boys and kindergarten girls receiving developmental and traditional instruction.3. There was no significant difference in the mathematical attitudes of kindergarten children receiving developmental instruction and kindergarten children receiving traditional instruction.4. There was no significant difference in the mathematical attitudes of kindergarten boys and kindergarten girls receiving developmental and traditional instruction. / Department of Elementary Education