The study aimed to explore misconceptions that Grade 9 learners at a school in Soweto had concerning the topic of fractions. The study was based on the ideas of constructivism in a bid to understand how learners build on existing knowledge as they venture deeper into the development of advanced constructions in the concept of fractions. A case study approach (qualitative) was employed to explore how Grade 9 learners describe the concept of fractions. The approach offered a platform to investigate how Grade 9 learners solve problems involving fractions, thereby enabling the researcher to discover the misconceptions that learners have/display when dealing with fractions. The research allowed the researcher to explore the root causes of the misconceptions held by learners concerning the concept of fractions. Forty Grade 9 participants from a township school were subjected to a written test from which eight were purposefully selected for an interview. The selection was based on learners’ responses to the written test. The researcher was looking for a learner script that showed application of similar but incorrect procedures under specific sections of operations of fractions, for example, multiplication of fractions. Both performance extremes were also considered, the good and the worst performers overall. The written test and the interviews were the primary sources of data in this study. The study revealed that learners have misconceptions about fractions. The learners’ definitions of what a fraction is were neither complete nor precise. For example, the equality of parts was not emphasised in their definitions. The gaps brought about by the learner conception of fractions were evident in the way problems on fractions were manipulated. The learners did not treat a fraction as signifying a specific point on the number system. Due to this, learners could not place fractions correctly on the number line. Components of the fraction were separated and manipulated as stand-alone whole numbers. Consequently, whole number knowledge was applied to work with fractions. A lack of conceptual understanding of equivalent fractions was evident as the common denominator principle was not applied. In the multiplication of fractions, procedural manipulations were evident. In mixed number operations, whole numbers were multiplied separately from the fractional parts of the mixed number. Fractional parts were also multiplied separately, and the two answers combined to yield the final solution. In the division of fractions, the learners displayed a lack of conceptual knowledge of division of fractions. Operations were made across the division sign numerators separate from the denominators. This reveals that a fraction was not taken as an outright number on its own by learners but viewed as one number put on top of the other which can be separated. Dividing across, learners rendered division commutative. A procedural attempt to apply the invert and multiply procedure was also evident in this study. Learners made procedural errors as they showed a lack of conceptual understanding of the keep-change-flip division algorithm. The study revealed that misconceptions in the concept of fraction were due to prior knowledge, over-generalisation and presentation of fractions during instruction. Constructivism values prior knowledge as the basis for the development of new knowledge. In this study, learners revealed that informal knowledge they possess may impact negatively on the development of the concept of fractions. For example, division by one-half was interpreted as dividing in half by learners. The prior elaboration on the part of a whole sub-construct also proved a barrier to finding solutions to problems that sought knowledge of fractions as other sub-constructs, namely, quotient, measure, ratio and fraction as an operator. Over generalisation by learners in this study led to misconceptions in which a procedure valid in a particular concept is used in another concept where it does not apply. Knowledge on whole numbers was used in manipulating fractions. For example, for whole numbers generally, multiplication makes bigger and division makes smaller. The presentation of fractions during instruction played a role in some misconceptions revealed by this study. Bias towards the part of a whole sub-construct might have limited conceptualisation in other sub-constructs. Preference for the procedural approach above the conceptual one by educators may limit the proper development of the fraction concept as it promotes the use of algorithms without understanding. The researcher recommends the use of manipulatives to promote the understanding of the fraction concept before inductively guiding learners to come up with the algorithm. Imposing the algorithm promotes the procedural approach, thereby depriving learners of an opportunity for conceptual understanding. Not all correct answers result from the correct line of thinking. Educators, therefore, should have a closer look at learners’ work, including those with correct solutions, as there may be concealed misconceptions. Educators should not take for granted what was covered before learners conceptualised fractions as it might be a source of misconceptions. It is therefore recommended to check prior knowledge before proceeding with new instruction. / Mathematics Education / M. Ed. (Mathematics Education)

With the realization that traditional instructional approach has not yielded satisfactory results, quasi-experimental and descriptive research designs were employed to investigate whether the application of constructivist instructional approach in the learning of stationary points in differential calculus by Grade 12 learners in South Africa would improve conceptual learning. Three Gauteng high schools of 204 Grade 12 learners constituted the research fields – one served as the control group while the other two represented the experimental group. Being a mixed-method research, quantitative data were gathered through pre-test and post-test while qualitative data were collected from classroom observations. Both inferential and descriptive statistical methods of data collection and analysis were used. The results obtained indicate that the experimental group demonstrated a better understanding of the concept of stationary points than the control group. / Mathematics Education / M. Sc. (Mathematics Education)

Since 1994 South African classrooms have become more culturally diverse. In order to teach effectively in such an environment, teachers need to use strategies which meet the needs of all the students. One way of addressing this challenge is to consider learning styles theory. Teachers need to understand how individual students of all cultures learn, and which specific learning styles are significantly related to academic achievement. In order to investigate this relationship, a study was conducted at an independent multicultural senior school in the North West Province of South Africa. The aim of the research was to gather information on the learning style preferences of the students at the school; the relationship between the students’ academic achievements in English and mathematics and their learning style; and finally, the relationship between the students’ nationality, gender, form and age and their learning styles as well as their academic achievements in English and mathematics. A mixed methods research design was used. Data was collected by means of a structured questionnaire that was completed by a sample of 240 students of different forms, genders and nationalities. This was followed by individual interviews with ten top achieving students. The study found that the predominant learning style amongst the students in the school was individual learning. This particular learning style was also most significantly related to academic achievements in English and mathematics. The study further determined that nationality did not significantly influence students’ learning styles but gender and age did. Female students were found to be more inclined to be individual learners. Younger students were also found to be auditory learners to a greater extent than older students. Regarding the studying of English and mathematics, it was found that female students generally used auditory learning styles whilst male students preferred kinaesthetic learning styles. Lastly, the study found that the average achievements in mathematics deteriorated as the students got older and the worst performing form was the Form 6s. To this end, various suggestions were made as to how learning styles could be considered to improve learning. Recommendations for further study were highlighted along with the limitations of this research. / Psychology of Education / D. Ed. (Psychology of Education)

Since 1994 South African classrooms have become more culturally diverse. In order to teach effectively in such an environment, teachers need to use strategies which meet the needs of all the students. One way of addressing this challenge is to consider learning styles theory. Teachers need to understand how individual students of all cultures learn, and which specific learning styles are significantly related to academic achievement. In order to investigate this relationship, a study was conducted at an independent multicultural senior school in the North West Province of South Africa. The aim of the research was to gather information on the learning style preferences of the students at the school; the relationship between the students’ academic achievements in English and mathematics and their learning style; and finally, the relationship between the students’ nationality, gender, form and age and their learning styles as well as their academic achievements in English and mathematics. A mixed methods research design was used. Data was collected by means of a structured questionnaire that was completed by a sample of 240 students of different forms, genders and nationalities. This was followed by individual interviews with ten top achieving students. The study found that the predominant learning style amongst the students in the school was individual learning. This particular learning style was also most significantly related to academic achievements in English and mathematics. The study further determined that nationality did not significantly influence students’ learning styles but gender and age did. Female students were found to be more inclined to be individual learners. Younger students were also found to be auditory learners to a greater extent than older students. Regarding the studying of English and mathematics, it was found that female students generally used auditory learning styles whilst male students preferred kinaesthetic learning styles. Lastly, the study found that the average achievements in mathematics deteriorated as the students got older and the worst performing form was the Form 6s. To this end, various suggestions were made as to how learning styles could be considered to improve learning. Recommendations for further study were highlighted along with the limitations of this research. / Psychology of Education / D. Ed. (Psychology of Education)

The study was carried out to: (1) determine the relationship between English and mathematical literacy scores at Grade 12 in Moloto Circuit, (2) understand and describe the learning difficulties experienced by learners when English language was used as a medium of instruction, and (3) suggest guidelines that could be used in teaching mathematical literacy. Regression and correlation analyses were carried out to determine the functional and strength of relationship between English language and mathematical literacy in the ten schools of Moloto Circuit. The views of the learners on the use of English language in the learning of mathematical literacy and the use of technical terms were sourced and analysed. The views of the educators about the use of English language as the medium of instruction were also analysed. A mixed approach methodology was used since both quantitative and qualitative methods were employed. The target population consisted of 305 learners who wrote the Grade 12 public examinations in 2016, 585 Grade 12 learners and 10 educators who completed questionnaires in January 2017. A census approach was carried out because everyone in Moloto Circuit doing Mathematical Literacy at Grade 12 level and their educators were studied. Grade 12 results for English and Mathematical Literacy for the 2016 academic year were collected and analysed. Questionnaires with closed and open-ended items were administered on Grade 12 learners and educators for the 2017 academic year in January 2017. Results in the ten schools showed that there was a positive relationship between performances in the two areas. In all cases the computed correlations were significant. This suggested that English influenced performance in Mathematical Literacy. This was supported by coefficients of determination calculations which ranged from 15% to 40%. Most responses indicated that learners found Mathematical Literacy difficult when English language was used as the medium of instruction. The learners preferred that Mathematical Literacy be taught in their mother tongue and that educators explain technical terms associated with mathematics. The educators said that learners had problems in Mathematical Literacy because of the use of English as the medium of instruction. They also said that learners found it difficult to relate Mathematical Literacy questions to real life situations and that the learners lacked adequate practice. The educators recommended the use of code-switching in their teaching in order to enhance understanding of Mathematical Literacy. / Mathematics Education / M. Ed. (Mathematics Education)

This study examined “problem solving skills in circle geometry concepts in Euclidean Geometry. This study was necessitated by learners’ inability to perform well with regards to Euclidean Geometry in general and Circle Geometry in particular. The use of naturalistic observation case study research (NOCSR) study was employed as the research design for the study. The intervention used for the study was the teaching of circle geometry with Polya problem solving instructional approach coupled with social constructivist instructional approach. A High School in the Northern Cape Province was used for the study. 61 mathematics learners (grade 11) in the school served as participants for the first year of the study, while 45 mathematics learners, also in grade 11, served as participants for the second year of the study. Data was collected for two consecutive years: 2018 and 2019. All learners who served as participants for the study did so willingly without been coerced in any way. Parental consent of all participants were also obtained. The following data were collected for each year of the research intervention: classroom teaching proceedings’ video recordings, photograph of learners class exercises (CE), field notes and the end-of-the- Intervention Test (EIT). Direct interpretations, categorical aggregation and a problem solving rubric were used for the analysis of data. Performance analysis and solution appraisal were also used to analyse some of the collected data. It emerged from the study that the research intervention evoked learners’ desire and interest to learn circle geometry. Also, the research intervention improved the study participants’ performance and problem solving skills in circle geometry concepts. Hence, it is recommended from this study that there is the need for South African schools to adopt the instructional approach for the intervention: Polya problem solving instructional approach coupled with social constructivist instructional approach, for the teaching and learning of Euclidean geometry concepts. / Mathematics Education / M. Sc. (Mathematics Education)

The present study investigated Grade 9 learners’ perceptions of factors influencing their academic performance in Mathematics in Tshwane municipality, South Africa. The study was based on positivism paradigm. The quantitative research approach was used while the research design was a descriptive survey design. The sample was made up of 400 learners (280 females and 120 males). Convenient sampling method was employed as a result of the nature of data. A self-designed questionnaire was administered to collect data. Data were analysed using quantitative approach with the statistical package for the social sciences, version 24.0, and the few open ended data on strategies for improving learners’ performance were analysed deductively. Frequency tables and ratios were calculated to establish the rating of each item. A Chi-square test was used to establish relationships of the variables on learners’ Mathematics performance. The study revealed that the Grade 9 learners perceived the factors influencing their performance in Mathematics were teacher-learner relationship, school environment and influence of learners’ attitude towards their performance in Mathematics, while peer relationship and home background (parents’ relationship and support) were perceived as not having any relationship with learners’ academic performance. The following recommendations were made to improve the Grade 9 learners’ performance in Mathematics, they are: policies that may consider Mathematics as a compulsory subject in the place of maths numeracy; there should be a regular in-service training for Mathematics teachers; parents should give adequate attention and care to their children in their Mathematics assignments; there should be provision of additional classrooms and infrastructures by the government and other stakeholders; learners should be encouraged to cultivate a more positive attitude towards Mathematics as a subject. Lastly, the study proposed a model for improving Grade 9 learners’ academic performance in Mathematics in Tshwane municipality, South Africa. / Psychology of Education / D. Ed. (Psychology of Education)

Mathematics Education / M. Ed. (Mathematics Education)