Despite the fact that the facilitation of inquiry learning is a core methodology in the General Education and Training (GET) band of the South African National Curriculum Statement, rote learning and memorization of algorithms remains common practice in many mathematics classrooms. The inquiry-based Family Maths professional development programme, offered by the Nelson Mandela Metropolitan University, attempts not only to support the transformative education practices targeted by the South African National Department of Education, but also to extend them beyond the school walls to the community at large. This study investigates the extent to which the Family Maths professional development programme develops facilitators’ ability to implement inquiry-based learning. It also seeks to explore which aspects of the programme are effective in developing an inquiry-based approach. The research undertaken is an empirical study of 39 facilitators and uses both qualitative and quantitative methods. The facilitators’ inquiry beliefs and ability to implement inquiry learning was measured by means of questionnaires, observation schedules and interviews. As the ‘teacher as facilitator of inquiry-based teaching and learning’ is a requirement of all South African teachers, the findings of this research should make a meaningful contribution to the field of mathematics teacher education in the South African context.

The purpose of this research study is to propose guidelines for providing a positive user experience for pupils using Dr Math®. User experience was found to have a positive impact on the acceptance and adoption of a product. Thus the proposed guidelines contribute in maximizing the adoption and acceptance of Dr Math® among pupils. This study begins with an introductory chapter that describes the problem that forms the basis for this research. The chapter defines the objectives that this study is intended to achieve in order to accomplish its ultimate goal. The methodology followed to conduct this research study as well as its scope are also defined here. The results from a preliminary survey revealed that despite its potential accessibility, Dr Math® has a low adoption rate. However, when compared to other mobile learning (m-learning) applications for mathematics learning, Dr Math® is more popular. Thus Dr Math® was selected as a case for study. Chapter 2 of this study provides a detailed description of Dr Math® as a local mobile application for mathematics learning. It was found that the affordability and accessibility of Dr Math® did not necessarily imply a high adoption rate. There are various possible barriers to its low adoption. User experience (UX), which is the focus of this study, is one of them. Thus, a subsequent chapter deals with UX. Chapter 3 discusses UX, its scope, components and definition and places particular emphasis on its significance in the success of any product. The chapter also highlights the characteristics of a positive UX and the importance of designing for this outcome. In Chapter 4, a discussion and justification of the methodology used to conduct this research is discussed. This study primarily employs a qualitative inductive approach within an interpretivism paradigm. An exploratory single case study was used to obtain an in-depth analysis of the case. Data was collected using Dr Math® log files as a documentary source. Gathered data was then analysed and organized into themes and categories using qualitative content analysis as outlined in Chapter 5. Also the findings obtained from the results, which are mainly the factors that were found to have an impact on the user interaction with Dr Math®, are presented here. The identified factors served as a basis from which the guidelines presented in Chapter 6 were developed. Chapter 7 presents the conclusions and recommendations of the research. From both theoretical and empirical work, it was concluded that Dr Math® has the potential to improve mathematics learning in South Africa. Its adoption rate, however, is not satisfying: hence, the investigation of the factors impacting on the user interaction with Dr Math®, from which the proposed guidelines are based.

Systems of writing have existed for thousands of years. The history of civilisation and the history of writing are so intertwined that it is hard to separate the one from the other. These systems of writing, however, are not static. They change. One of the latest developments in systems of writing is short electronic messages such as seen on Twitter and in MXit. One novel application which uses these short electronic messages is the Dr Math® project. Dr Math is a mobile online tutoring system where pupils can use MXit on their cell phones and receive help with their mathematics homework from volunteer tutors around the world. These conversations between pupils and tutors are held in MXit lingo or MXit language – this cryptic, abbreviated system 0f ryting w1ch l0ks lyk dis. Project μ (pronounced mu and indicating MXit Understander) investigated how topics could be determined in MXit lingo and Project μ's research outputs spot mathematics topics in conversations between Dr Math tutors and pupils. Once the topics are determined, supporting documentation can be presented to the tutors to assist them in helping pupils with their mathematics homework. Project μ made the following contributions to new knowledge: a statistical and linguistic analysis of MXit lingo provides letter frequencies, word frequencies, message length statistics as well as linguistic bases for new spelling conventions seen in MXit based conversations; a post-stemmer for use with MXit lingo removes suffixes from the ends of words taking into account MXit spelling conventions allowing words such as equashun and equation to be reduced to the same root stem; a list of over ten thousand stop words for MXit lingo appropriate for the domain of mathematics; a misspelling corrector for MXit lingo which corrects words such as acount and equates it to account; and a model for spotting mathematical topics in MXit lingo. The model was instantiated and integrated into the Dr Math tutoring platform. Empirical evidence as to the effectiveness of the μ Topic Spotter and the other contributions is also presented. The empirical evidence includes specific statistical tests with MXit lingo, specific tests of the misspelling corrector, stemmer, and feedback mechanism, and an extensive exercise of content analysis with respect to mathematics topics.

These studies have focused on identifying the mathematical requirements of first-year BCom students at Nelson Mandela Metropolitan University. The research methodology used in this quantitative study was to make use of interviewing, questionnaire investigation, and document analysis in the form of textbook, test and examination analysis. These methods provided data that fitted into a grounded theory approach. The study concluded by identifying the list of mathematical topics required for the first year of the core subjects in the BCom degree programme. In addition, the study found that learners who study Mathematics in the National Senior Certificate should be able to cope with the mathematical content included in their BCom degree programme, while learners studying Mathematical Literacy would probably need support in some of the areas of mathematics, especially algebra, in order to cope with the mathematical content included in their BCom degree programme. It makes a valuable contribution towards elucidating the mathematical requirements needed to improve the chances of successful BCom degree programme studies at South African universities. It also draws the contours for starting to design an efficient support course for future “at-risk” students who enter higher education studies.

In South Africa, there is concern about the poor achievement by first additional language (FAL) English learners in mathematics, and this is a consequence not solely of the apartheid era but more appropriately, the existing current situation in the classroom. Since 2006, the Department of Basic Education in South Africa introduced mathematical literacy (ML) as another learning area for the Further Education and Training (FET) band. ML, as an alternative choice to mathematics, is envisaged as a key to the understanding of our everyday world filled with numbers. Mathematical literacy FAL English learners and teachers are exposed to a ML curriculum that demands high linguistic skills in English in order to engage with the mathematical concepts through the medium of English which is not their home language. While the Language-in-Education Policy (DoE, 1997) recommends that school language policies promote additive bilingualism and the use of learners’ home languages as languages of learning and teaching (LoLT), there has been little implementation of these recommendations by schools, for the reason that all assessments and learner and teacher support materials (LTSM) are only available in English. Thus it appears that ML creates a language gap when FAL English learners and teachers have to possess a high level of communication and language proficiency of the LoLT English to successfully engage with the mathematical context and content of the ML curriculum since the language itself carries all the meaning. The majority of FAL English learners and teachers struggle with the necessary English language proficiency to successfully interact with the ML curriculum and are often required to use their own home language (code switching) to bring about understanding. In this study, I explore how teachers and learners who are dominantly FAL English speakers, engage in teaching and learning of ML in bi/multilingual classrooms. The main aim of the study is to investigate the FAL English learners’ and teachers’ experiences and perceptions concerning the use of English as the LoLT in bi/multilingual ML classrooms. Qualitative measures were generated through personal interviews (teachers [n=15] and learners [n=25]) from five different secondary schools situated in the rural areas of the Midlands in KwaZulu-Natal. This study is framed by Wenger’s (1991) model of situated learning and Vygotsky’s socio-cultural perspectives, which propose that learning involves a process of engagement in a community of practice and reflects the learners’ sociocultural relationship to school mathematics respectively. The study also describes Cummins’ quadrants and explores the benefits of Cummins’ notion of language use (Cummins, 1984). Analysis of the semi-structured interviews revealed that, despite both teachers’ and learners’ difficulty with the language, English is the more popular language to be used in teaching and learning in bi/multilingual mathematical literacy classrooms since English is accepted as a global language. Worldwide emphasis and dominance of English as a powerful language that gives access to goods and social mobility were also highlighted. However, overall results in this study showed that most participants did express their preference for using their home language isiZulu alongside English through the extensive practice of code switching for the teaching and learning of ML.

Mathematics is regarded as a driving force in economies worldwide. The performance of South African learners in mathematics over the past decade has highlighted that problems are being experienced across all grades. This situation needs to be addressed with urgency. The South African Department of Education stated that quality learning must be the objective for all grades. The implementation of good teaching practices plays a crucial role in improving the quality of education and in guiding learners towards quality learning. To achieve quality mathematics teaching and learning it is imperative to determine what good mathematics teaching practices are. The identification of good mathematic teaching practices will provide a yard stick to measure the mathematics teaching competency of teachers. This study identifies a set of good mathematics teaching practice indicators and evidences applicable to teachers in the Intermediate phase as a first contribution. These indicators and evidences frame the second research contribution: an assessment instrument entitled “A Classroom Observation Tool for Observing Mathematics Teaching Practices in Primary Schools”. As a third research contribution a generic profile of a Grade four mathematics teacher has been built. This generic profile has been built through an analysis of data gathered by means of self-assessment questionnaires completed by the research sample, as well as through applying the observation tool. The value of the generic profile lies in the identification of shared strengths and shared improvement opportunities in the mathematics teaching practice of the sample and as such, it forms the basis of a theory on Grade four mathematics teaching practice. The fourth research contribution is the design and application of an intervention that addresses the shared improvement opportunities. The research study concludes by comparing pre-intervention classroom observation data with post-intervention classroom observation data and reporting on the impact of the intervention.

This study used diagnostic tests, questionnaires and interviews to investigate explore teachers’ subject content knowledge (SCK) and pedagogical subject knowledge (PCK). It also explored teachers’ and learners’ misconceptions within the topic of graphicacy and how teachers’ SCK and PCK possibly affect learner achievement. A small sample of teachers were drawn from the Keiskammahoek region; a deep rural area of the Eastern Cape. These teachers were part of the Nelson Mandela Metropolitan University (NMMU) Amathole Cluster Schools Project who were registered for a three-year BEd (FET) in-service programme in mathematics education. As part of the programme they studied mathematics 1 and 2 at university level and received quarterly non-formal workshops on teaching mathematics at FET level. The findings of this study suggest that teachers with insufficient SCK will probably have limited PCK, although the two are not entirely dependent on each other. In cases where teachers’ displayed low levels of SCK and PCK, their learners were more likely to perform poorly and their results often indicated similar misconceptions as displayed by their teachers. This implies that we have to look at what teachers know and what they need to know in terms of SCK and PCK if we are to plan effectively for effective teacher development aimed at improving learner performance.

Feedback has an important role to play in the performance of learners. This study looks to identify the challenges that the learners are faced with when the teachers provide them with feedback and the ways they would like like it to be used in order to see feedback assisting them in their learning and improve their performance. Feedback contributes a lot to assessment and has a close link with performance. The study used the qualitative approach to identify the challenges the learners experience when they receive the feedback from their teachers. The participants were sampled from grade 12 learners in the form of a focus group (seven to ten per school) and individual respondents. The method used semi-structured interviews and portfolio observations to collect the data from two high schools in East London (EL) district to look at the nature of the feedback provided to learners. The data was later analysed and interpreted. It has been identified that for both schools feedback conveyed certain messages to learners such motivation to do better, a need to do better and, lastly, affirmation that the learners are on the right track or they are neglected and left to figure out how to do the tasks. The study discovered that, according to Hattie and Timperley (2007)’ there are four levels of feedback. It was found that for the task level learners from school A mainly received oral feedback which was often seen as denigrating them; however, in school B learners received both the oral and written feedback. They felt that the feedback assisted them to understand the task at hand. They also saw this as a way of building up their confidence in all the tasks they come across. Secondly, in the process level, learners in school A did not report receiving feedback at this level but only oral feedback which does not show their mistakes step by step in the task, yet in school B they reported that they got feedback from their teacher individually to help them understand the task. Thirdly, for the self regulation level, in school A there was no data to confirm this. Regarding school B, learners were being assisted by the feedback they received from their teacher and this caused them to monitor their progress. The fourth and last level is the self or personal evaluation where in school A learners were not able to evaluate themselves because they did not receive written feedback, whereas in school B learners could do that freely referring to the written comments from their teacher. The research therefore concluded that in one of the schools the four levels that the study was looking at were not all addressed and thus no meaningful feedback was given. For school B the teacher gave them the feedback which has contributed a lot in their learning. The study recommends that feedback should not be used for right or wrong answers but it must also state clearly why the learner has obtained such mark or grade and what to do to correct the wrongs. Teachers should consider that learner errors also assist them to have a broader picture on what more they need to do in their subjects. It is also recommended that teachers should consider various strategies in giving feedback and the learners’ work has to be monitored timeously for the purpose of the learning process. Lastly, a good approach when feedback is provided is also important because it builds high self-esteem and develops the teacher-learner approach accordingly.

This study explores characteristics of students’ repertoires of representations in two mathematical contexts: whole number multiplication and the comparison of common fractions. A repertoire of representations refers to a set of representations which a student can reconstruct as needed. Of particular interest are (1) how multiplicative relationships among units were represented, and (2) whether continuous measurement was an underlying conceptual framework for their representations. In addition, the characteristics of students’ representations and interpretation of units of linear and area measurement were explored. Data were collected through a series of interviews with Grade 5 and Grade 7 students. Some results of the study were as follows. Each repertoire of representations was exemplified by a dominant form of units, either discrete or contiguous. Within a repertoire, all forms of units were related, first through a common system of measurement (either numerosity or area), and second through their two-dimensional characteristic. In the multiplication context, some repertoires were comprised only of representations with discrete units, but others also included some representations with contiguous units. Students sought characteristics in their representations which reflected those based on continuous measurement, however linear or area measurement was not used as a conceptual framework. Instead, all representations were based on the measurement of numerosity. Also, students exhibited different limits in their representation of multiplicative relationships among units. Some represented no multiplicative relationships, but most represented at least a multiplicative relationship between two units. Relationships among three units were seldom constructed and difficult to achieve. Common fraction repertoires were based on the measurement of either numerosity or area, but the physical characteristics of the units varied. Some repertoires had only contiguous representations of units, others also included representations with discrete units, and a few did not represent fractional units at all. Students’ representations reflected characteristics of area-based representations, however area measurement was not necessarily a conceptual framework. In addition, students’ beliefs about what constituted units of area measurement were variable. As a result, they either represented no multiplicative relationships among units, or fluctuated between representing two-unit and three-unit relationships. Linear measurement was notably absent as a basis for representations in both mathematical contexts. The one-dimensional characteristic of linear measurement did not fit students’ dominant framework for constructing mathematical representations. With respect to measurement, students represented linear units in terms of discrete points or line segments. Counting points and interpreting the count in terms of the numerosity of line segments was problematic for nearly all students. When partitioning regions into units of area, a few students also equated the number of lines with the number of parts. The direct relationship of action and result in counting discrete objects was generalized without consideration of other geometric characteristics. When comparing quantities having linear or area units, numerical reasoning was not always used. Alternatively, either quantities were transformed to facilitate a direct comparison, or only perceptual judgements were made. No students consistently used numerical reasoning to compare fractional units of area. In the latter situations, the part-whole relationship among units seldom was observed. In general, there was no direct relationship between the forms of representations used by students in the two mathematical contexts and the characteristics of their representations of units of the measurement contexts. The development of repertoires of representations appears to be context specific. The repertoires were strictly limited in terms of the forms of representations of which they were comprised. / Education, Faculty of / Curriculum and Pedagogy (EDCP), Department of / Graduate

This study is concerned with how children interpret the slope of a line graph. Today with the vast accumulations of data which are available from computers, people are being faced with an ever increasing amount of pictorial representation of this data. Therefore it is of the utmost importance that children understand pictorial representation. Yet in spite of the popularity of graphs as tools of communication, studies show that many adults experience difficulty in reading information presented in a graphical form. The slope of the graph was chosen for this investigation because it is in this aspect of graphing (as shown by the results of the 1981 B.C. Assessment) that children in British Columbia seem to have the greatest difficulty when they reach Grade 8. The study dealt with positive, negative, zero and infinite slopes, combinations of these slopes, curvilinear graphs and qualitative graphs, that is, graphs that have no numerical data shown on the axes. The researcher chose to use a structured individual interview as a means of collecting data about how the students interpreted the slope of a line graph. Graphs used in the interviews dealt with temperature, height, weight and distance. Twenty-two students were chosen for this study. The students were found to have problems mainly with graphs dealing with distance related to time. This problem may be due to the fact that many students read only one axis and when interpreting distance seem to include direction as an added dimension of the graph. Infinite slope graphs were misinterpreted by every student, which may be due to the fact that they ignore the time axis. In general students used two methods of interpreting graphs. In some cases they observed the direction of the graph from left to right, that is, whether the slope went up or down from left to right. In other cases they examined the end points on the graph and drew their conclusions from them. The choice of method varied with the contextual material shown on the graph, which may be due to the children's concept of the parameter in the physical world and whether they see the parameter as being able to increase and decrease over time. From the study the investigator feels that more discussion of graphing by teachers and students is needed if the misconceptions are to be cleared up. Discussion of the parameters of both axes by teachers might help clear up the misconceptions students have about distance travelled over a period of time when this is expressed as a graph. There would be less chance of a graph being read as a map if the relationships between the two axes were demonstrated to students. Teachers also need to be aware of both methods used by students in interpreting graphs. / Education, Faculty of / Graduate