Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

The purpose of the study was to explore the relationship between mathematics teachers’ subject matter knowledge and their teaching effectiveness. A convenient sample of 19 grade 11 mathematics teachers and 418 students were initially selected for the study and took part in some stages of the study. Of this lot, only 11 teachers and 246 students participated in all the stages of the study. Explanatory Mixed methods research design which entails the use of a co-relational study and a descriptive survey design were employed in the study. Data was collected from the teachers using a self report questionnaire, Teacher Subject Matter Knowledge of Trigonometric Functions Scale (TSMKTFS) and peer evaluation questionnaire, and from students using teacher evaluation questionnaire and Student Trigonometric Functions Performance Scale (STFPS). All the instruments had their validity and reliability accordingly determined. Quantitative data gathered was analysed using descriptive and inferential statistics while qualitative data gathered from teachers’ and students’ tests were analysed using task performance analysis. It was found that a positive, statistically significant relationship existed between teachers’ subject matter knowledge and the composite measure of their teaching effectiveness. The relationships between teachers’ subject matter knowledge and students’ achievement and also between teachers’ subject matter knowledge and students’ rating of the teachers’ teaching effectiveness were found to be positive and statistically significant. However, the relationships between teachers’ subject matter knowledge and teachers’ self rating as well as teachers’ subject matter knowledge and peers’ rating of teachers’ teaching effectiveness were not found to be statistically significant though they were positive. Further data analysis showed that there was a difference between the subject matter knowledge of effective and ineffective teachers and also between the students taught by effective teachers and the students taught by the ineffective teachers. / Institute of Science and Technology Education / PhD (Mathematics Education)

A study was undertaken to explore what constitutes instructional capacity in the teaching and learning of mathematics (TLM), with a focus on how schools (as institutions of teaching and learning) integrate resources for a particular configuration of capacity to promote high achievement levels of Grade 12 students in mathematics. Data were collected in ten public secondary schools, mostly in a disadvantaged context, in the Vhembe District in Limpopo Province, South Africa. The study explores strategies for constructing, organising and replenishing instructional capacity in TLM. Five low- and five high-performing schools were selected, based on the pass rate in mathematics in high stakes examinations. The researcher observed lessons and interviewed ten Grade 12 mathematics teachers, ten principals, five curriculum advisors and a sample of forty Grade 12 mathematics students. The research revealed that the capacity to encourage the new curriculum reform practices in TLM within different schools is often inadequate, and largely fails to compensate for organisational effects and arrangements that shape the capacity to create quality instruction in mathematics. However, high-performing schools were somewhat ahead of low-performing schools in terms of encouraging reform-oriented teaching and learning in mathematics. Recommendations include: Principals should initiate the development and implementation of a school-based clinical supervision programme through collaborative decision-making to promote a sense of ownership by all mathematics teachers. Such a supervision programme would enhance commitment and ensure that all efforts are unified towards improving the quality of TLM. There is a need for the DoE in Limpopo Province to coordinate teacher professional development workshops, where effective practising mathematics teachers model how they teach mathematics in the classroom, while other teachers observe. Context-based strategies to enhance student outcomes in mathematics should be devised, such as modelling good practice by effective teachers in terms of: lesson preparation; subject knowledge; pedagogic approach; assessment and monitoring of classroom practice, including direct observation of teaching by HoDs and principals. It is proposed that attention to these issues, amongst others, would limit the impact of an unpromising context on student achievement levels in mathematics in high stakes examinations in the Vhembe District and elsewhere. / Mathematics Education / D. Phil. (Mathematics, Science and Technology Education)

This study investigated the relationships between diverse variables and secondary school pupils' Mathematics achievement. It also dealt with the relative contribution of each variable to Mathematics achievement and the significance of differences in Mathematics achievements when pupils' gender and home background as well as teachers' experience, gender, education, in-service education, homework assignment and testing frequency are taken into account. A questionnaire was administered on 163 standard 8 pupils. I The most important findings of this study were: {1) There was a significant relationship between pupil variables and Mathematics achievement. (2) Pupil variables, particularly self-concept, contributed significantly towards the variance in Mathematics achievement. (3) Pupils who wrote tests often (more than once per term) achieved significantly less than students who wrote tests less often / Psychology of Education / M. Ed. (Psychology of Education)

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

This study focused on teaching strategies preferred and used by grade 8 mathematics teachers, what they thought was most effective for learning mathematics as well as students’ perspectives of their mathematics classroom. The aims of this study were to investigate the teaching strategies used in the South Korean mathematical classroom and to find out how they attain a high performance in mathematics, in comparison with other countries. The target population was chosen from within the Chungcheongbuk Province and ten schools were selected for the study. In order to determine what teaching strategies are used in the South Korean mathematics classroom, a case study using both quantitative and qualitative research methods was adopted. Data collection methods included questionnaires for the students while interviews were conducted with the teachers. The questionnaire contained fifty closed-ended questions divided into different sections to obtain data on teaching strategies used, on preferred learning styles from the students and on how they felt about mathematics and the mathematical classroom. The interview consisted of ten open-ended questions to get feedback from the mathematics teachers on what teaching strategies they used in the classroom and on what they thought were the best strategies with regard to teaching grade 8 mathematics. From the ten sampled schools there were two hundred and two students who participated in this research, and six teachers were interviewed. The results of the study showed that in the South Korean mathematics classroom a combination of direct instruction, practice and teacher guidance helps the students to learn problem-solving skills and to master mathematics. The students indicated that the teachers mostly used chalkboard instruction and that they practiced solving problems using worksheets, past exam paper questions and through homework or private study. The average student studied mathematics for about six hours a week and most attended afterschool academies for further studying mathematics. Although the South Korean students attained a high performance in mathematics it was evident that they indicated a low interest in the subject. The teachers stated in the interviews that they thought the students needed to see examples on the chalkboard, and then the students need to practice with guidance from the teacher. It was evident that the students focus a lot on guided practice, since they study for about six hours a week. The teachers also felt that the curriculum is overloaded and that there was a gap between the better and the poorer level of students in the mathematics classroom, this gap grew bigger as students lost motivation. The responses to the questionnaire showed that 65% of the students were not interested in mathematics; in spite of this South Korea is placed among the best performing countries in the world. The teachers also indicated that mathematics was very highly valued in South Korea and that parents and universities put a lot of pressure on students to perform well in this subject. This study provides better insight into what is happening in the South Korean mathematics classroom, what methods are used and how the students felt about the mathematics classroom and the strategies that are used. Apart from commenting on teaching strategies, there was also an indication of what teaching style the students preferred. The information in this research study can provide answers to questions regarding South Korean mathematics instructional practices and will be useful for future comparative studies regarding the teaching of mathematics in other countries. / Mathematics Education / M. Ed. (Mathematics Education)

The study constituted an investigation for concept images and mathematical reasoning of Grade 11 learners on the concepts of reflection, translation and stretch of functions. The aim was to gain awareness of any conceptions that learners have about these transformations. The researcher’s experience in high school and university mathematics teaching had laid a basis to establish the research problem. The subjects of the study were 96 Grade 11 mathematics learners from three conveniently sampled South African high schools. The non-return of consent forms by some learners and absenteeism during the days of writing by other learners, resulted in the subsequent reduction of the amount of respondents below the anticipated 100. The preliminary investigation, which had 30 learners, was successful in validating instruments and projecting how the main results would be like. A mixed method exploratory design was employed for the study, for it was to give in-depth results after combining two data collection methods; a written diagnostic test and recorded follow-up interviews. All the 96 participants wrote the test and 14 of them were interviewed. It was found that learners’ reasoning was more based on their concept images than on formal definitions. The most interesting were verbal concept images, some of which were very accurate, others incomplete and yet others exhibited misconceptions. There were a lot of inconsistencies in the students’ constructed definitions and incompetency in using graphical and symbolical representations of reflection, translation and stretch of functions. For example, some learners were misled by negative sign on a horizontal translation to the right to think that it was a horizontal translation to the left. Others mistook stretch for enlargement both verbally and contextually. The research recommends that teachers should use more than one method when teaching transformations of functions, e.g., practically-oriented and process-oriented instructions, with practical examples, to improve the images of the concepts that learners develop. Within their methodologies, teachers should make concerted effort to be aware of the diversity of ways in which their learners think of the actions and processes of reflecting, translating and stretching, the terms they use to describe them, and how they compare the original objects to images after transformations. They should build upon incomplete definitions, misconceptions and other inconsistencies to facilitate development of accurate conceptions more schematically connected to the empirical world. There is also a need for accurate assessments of successes and shortcomings that learners display in the quest to define and master mathematical concepts but taking cognisance of their limitations of language proficiency in English, which is not their first language. Teachers need to draw a clear line between the properties of stretch and enlargement, and emphasize the need to include the invariant line in the definition of stretch. To remove confusion around the effect of “–” sign, more practice and spiral testing of this knowledge could be done to constantly remind learners of that property. Lastly, teachers should find out how to use smartphones, i-phones, i-pods, tablets and other technological devices for teaching and learning, and utilize them fully to their own and the learners’ advantage in learning these and other concepts and skills / Mathematics Education / D.Phil. (Mathematics, Science and Technology Education)

This study explored the influence of irrational beliefs on adolescent secondary school learners’ Mathematics achievement in Zimbabwe. Learner, home and school factors which influence secondary school learners’ Mathematics achievement were discussed and relevant studies were scrutinised. The theoretical views of Albert Ellis regarding the characteristics, effects, acquisition and maintenance of irrational beliefs were discussed together with the major irrational beliefs and their possible relationship with learners’ Mathematics achievement. A sample of 306 randomly selected adolescent Mathematics learners comprising 182 girls and 124 boys in the 14 to 18 year age range participated in the study. A composite questionnaire with subscales on learners’ irrational beliefs, socio-affective variables and perceptions was used during the empirical investigation. Six major hypotheses were tested. The study established that learners’ irrational thoughts about Mathematics correlate negatively with their Mathematics achievement. Learners’ irrational thoughts about Mathematics correlated negatively with motivation, self-concept, parental involvement, and teacher-learner relationships and positively with stress, anxiety and faulty perceptions. Regression analysis proved that learners’ irrational beliefs, socio-affective variables and perceptions jointly explain a greater proportion of the variance in Mathematics achievement than any one of these factors on its own. Therefore, learners’ Mathematics achievement is affected by irrational beliefs together with their socio-affective variables and perceptions. Practical recommendations were given to Mathematics education stakeholders such as teachers, school counsellors, parents and learners to minimise poor Mathematics achievement attributable to irrational beliefs and the allied variables explored in this study. / Psychology of Education / D. Ed. (Psychology of Education)

Over the past several decades there has been an emphasis on educational research pertaining to learners’ performance in Mathematics and on finding methods to improve learner performance in this subject. In South Africa, Grade 12 learners’ results in Mathematics from 2010 to 2013 were unsatisfactory as shown in DBE, 2013a. The teachers are challenged to find new teaching methods that will make the subject more interesting and appealing to the learners (Oliver & Makar, 2010 in Goos, 2010). The purpose of this study was to investigate the effect of using computers in the teaching and learning of Mathematics with special reference to the topic of linear functions in order to improve learner performance. The literature reviewed shows that the use of computers not only improves the learners’ performance but also changes their attitude towards Mathematics (Bester & Brand, 2013). The quantitative research approach was used to gather the data, namely the quasi- experimental, non-equivalent control group pre-test-post-test design. Two intact classes formed part of the research study, that is an experimental group (n=50) and control group (n=50). The experimental group learnt the concept of linear function using GeoGebra software. The control group learnt the same concept through the traditional pen and paper method. The data were analysed using the SPSS on ANOVA. The results indicated that there was a significant difference between the mean scores of the experimental group (μ=70.5) and the control group (μ=47.5). From the results it was evident that the use of computers had a positive effect on learners understanding of linear functions as reflected in their performance and on their attitude towards Mathematics, as seen in the questionnaire responses. / Mathematics Education / M. Ed. (Mathematics Education)

The quality of education in South Africa has come increasingly under scrutiny for various reasons and learners ‘performance in mathematics is one of the main areas of concern. Learners in secondary schools in Nzhelele East Circuit in Limpopo Province, South Africa, are not performing well in mathematics. Learners who want to pursue careers such as civil engineering, medicine and other qualifications where mathematics is a prerequisite find it difficult to follow these careers. The study used focus group discussions with 20 Grade 12 learners, face-to-face semi-structured interviews with six teachers and four SMT to investigate factors affecting grade 12 learners performance in mathematics in two secondary schools. The study highlighted socio-cultural and psychological factors that are seen to be barriers in mathematics performance. The purpose of this study was to explore factors that affect Grade 12 learners’ performance in mathematics. As there are many factors, this study's main emphasis was on the influence of the home environment, teachers’ competence and attitudes in teaching mathematics, learners’ attitudes towards mathematics and strategies that can be used to improve learners’ performance in mathematics. The afore-said were named themes each of which related to questions contained in the semi-structured interview schedule. The study included literature review from Kenya, Ghana and South Africa. Qualitative research method was used. Interviews were conducted with SMT, teachers and learners. The researcher set out to answer the following question What are the dominant and most profound socio-cultural and psychological factors of the Grade 12 mathematics learners affecting performance in the Nzhelele East Circuit, Vhembe District of Limpopo Province? The following points will also be taken into consideration during this research. How do learner performance trends in mathematics education look like among Grade 12 mathematics learners in the Nzhelele East Circuit, Vhembe District of Limpopo Province, and South Africa? What are the teacher-based factors established to have been contributory to the state of learner performance in mathematics in the selected participant schools? What are the School Management Team (SMTs) factors established to have been contributory to the state of learner performance in mathematics in the selected participant schools? THE AIMS AND OBJECTIVES Aim This study investigates the dominant and most profound socio-cultural and psychological factors affecting learner performance in mathematics among Grade 12 learners in the Nzhelele East Circuit, Vhembe District of Limpopo Province, and South Africa. Objectives The objective of this study was to explore the socio-cultural and psychological factors in the home and school environment that affect Grade 12 learners performance in mathematics at Nzhelele East Circuit, Vhembe District of Limpopo Province. Based on the above question the following were findings of the research. • Mathematics is believed to be a critical school subject in most regions of the world in general, and Sub-Saharan Africa in particular. • A plethora of governments – especially those in developing economies where governments are targeting industrialization and technological development believe mathematics is crucial for facilitating development and advancement of the general populace of their regions. • Mathematics is a compulsory subject in most education systems around the world. • There are a variety of complexities and constraints affecting the teaching and learning of mathematics in most regions of the world. • There is poor learner performance in mathematics. This is indicated by high failure rate in mathematics during end-of-year learner assessments. • The reasons for the poor performance of learners in mathematics in the selected schools were vast and intertwined. • Efforts have been made at international level to intervene with regard finding solutions to complexities and constraints affecting teaching-learning environment in mathematics. • The complexities and constraints affecting the teaching-learning environment of mathematics in South Africa stem from South Africa's past era of apartheid education. • Mathematics has been a major target of improvement and transformation by the post-apartheid government in South Africa. • Regardless of the efforts made by the post-apartheid government in South Africa to improve and transform teaching-learning environment in mathematics, there are perpetual complexities and constraints still hindering progress, and this affects learner output in mathematics. • Mathematics is one of the poorly performing subjects in the post-apartheid education system in South African schools – especially those schools which are based in formerly disadvantaged areas such as homelands and townships. • There is a growing body of literature investigating the complexities and constraints affecting the teaching-learning of mathematics in South Africa. Apart from the conclusions based on general observations, the study furthermore concluded that critical learner-based factors, teacher-based factors and SMT-base factors contributed to the state of affairs in the selected participant schools. This study therefore concluded thus: • Learner-based factors Mathematics learners were poorly prepared in the lower grades for senior grades. In other words, learners lacked proper foundation and background in mathematics. Learners were not well taught the basics of mathematics in previous grades. Furthermore, mathematics learners lacked assistance with homework for example because the majority of parents were illiterate and therefore not involved in supervision of learners beyond school environment. Learners showed negative attitude towards their teachers and the subject. Learners were not self-motivated in mathematics. • Teacher-based factors Teachers did not have enough workshops on how to teach mathematics, and also on content knowledge improvement. Less creative teachers were teaching mathematics. Teachers lacked support from their principals. The majority of teachers lacked passion in mathematics. • School Management Teams (SMTs) factors This study concludes that mathematics teachers were not receiving adequate support from the School management Teams (SMTs). Lack of teacher support by SMTs stems from poor organisation of supervisory roles expected from SMTs. Teachers' work is not being properly monitored and supervised. RECOMMENDATIONS Based on the conclusions drawn from the study, the following recommendations are crucial, and they have to be implemented: • Recommendations to the Department of Education This study recommends that the Department of Education should monitor or revisit the methods of teaching and learning of mathematics. Assessment should also be revisited to ensure that the mathematics paper is not too long for the learners during examinations. Enlisted service providers who facilitate teacher workshops for mathematics must be conversant with content requirements of mathematics. Considerations should be made to simplify the mathematics question paper. The Department of Education should also provide necessary resources such as textbooks timeously. Knowledgeable mathematics specialists and advisers could be enlisted to visit regular visits to schools to assist teachers and learners throughout the year. • Recommendations to the School Management Teams (SMTs) The SMT should ensure that their educators are adequately qualified to teach mathematics. as a subject as well as being able to teach their learners in a way that they understand the subject. In addition, teachers who are knowledgeable in teaching mathematics should be appointed to teach mathematics. The SMTs should also ensure that compulsory workshops are provided for mathematics teachers at school level. Furthermore, the SMTs should ensure that educators are able to identify learners that are under-performing, monitor the utilisation of study time and assist in resolving problems regarding the poor performance of such individual learners. School Management Teams shall facilitate additional after-hours lessons and sessions to assist learners identified as lacking in home environment assistance for example. Moreover, SMTs should also encourage all Grade 12 mathematics learners to participate in supervised extra mathematics lessons and extra-mural activities in order to develop and build their self-confidence and positive self-esteem. In addition, the SMT members should be more supportive to their mathematics teachers. Continuous moderation of teachers' work should be encouraged. School Management Teams (SMTs) should also develop motivational programmes and instruments to encourage and motivate learners in their respective schools. For example, performing learners could be awarded with prizes in public – especially during school organised functions which might include learners, teachers and parents. • Recommendations to the teachers The teachers should make it a point that they attended mathematics workshops as organised by various stakeholders. Teachers' interest in mathematics could be improved by improving their knowledge of subject content in mathematics, and personal recognition by authorities. Performing teachers could also be recognised and awarded with prizes publicly at school gatherings and meetings. Teachers should also be encouraged to obtain further qualifications and skills in mathematics. • Recommendations to the learners Efforts should be made to improve learners’ attitudes towards mathematics, and in addition to their mathematics teachers. Learners should be made aware of national target for mathematics education and the implications thereof. This could be achieved by utilizing educators – especially those who teach Guidance as a subject to learners to provide more convincing evidence of the spin-offs of obtaining good passes in mathematics in South Africa. SUGGESTION FOR AREAS NEEDING FURTHER STUDY There is a need to conduct follow-up research on: • applied strategies and instrument to improve learner attitude towards mathematics. • Establishing whether the complexities and constraints affecting learner performance in mathematics affect both girl and boy learners the same considering that their circumstances and conditions might materially differ. / Curriculum and Instructional Studies / M. Ed. (Curriculum Studies)