Mathematics and Computer Science Education

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An APOS analysis of the teaching and learning of factorisation of quadratic expressions in grade 10 mathematics classrooms.
(2021) Vilakazi, Aubrey Sifiso.; Bansilal, Sarah.
The South African Curriculum and Assessment Policy Statements (CAPS) document, for the Further Education and Training Phase (FET) Mathematics Grades 10-12 (2011) shows that the factorisation of algebraic quadratic expressions or equations pervades the mathematics of the secondary school. As a result, for learners to be successful at mathematics in Grade 12, they need to know a great deal of algebra, particularly the factorisation of quadratics. It is therefore important for us as mathematics educators to identify areas in the factorization of quadratics that teachers and learners are struggling to learn and apply. With this in mind, the study sets to embark on an APOS analysis of the teaching and learning of factorisation of quadratic expressions in Grade 10 mathematics classrooms. Following on from the research questions, this study is located within the principles of the mixed methods case study approach. The combination of methodologies has allowed me to identify broad trends across the groups of learners and those of educators as a whole as well as differences within the participants of the groups themselves. The participants of the study were the groups of Grade 10 learners from the two participating schools, as well as the Grade 10 mathematics teachers from the two circuits of Ilembe District. Five sources of data were used. Firstly, data were generated from 25 teachers from the two circuits who participated in the teachers’ questionnaires. A second data collection instrument was the classroom lessons’ observations of the six teachers. A third data source was the learner group activity and learners’ interviews administered to 12 learners. A fourth data source was the unstructured interviews with six teachers. The final instrument was the analysis of the 205 Grade 10 mathematics 2019 March common paper learners’ scripts. This study was guided by the theory of constructivism and more specifically Action, Process, Object, Schema (APOS) theory which views learning as changes in conception. As an individual engages with a concept, the conception changes from an initial external view towards seeing the concept as a totality upon which other Actions and Processes can act. This study has found that, firstly, teachers and learners tend to rely too much on the use of rules in factoring certain quadratics. In so doing, a prototype of the quadratic expression concept is perceived which consists of isolated and disconnected concepts. As a result, most learners were not able to factor the trinomial quadratic of 𝑎≠1, since they perceived the factoring of 𝑎𝑥2+𝑏𝑥+𝑐 with 𝑎=1 and that of 𝑎≠1 as two different procedures. Secondly, there are also students whose mental constructions (conception) are limited to Action levels in terms of APOS theory. The findings of the study suggest that teachers and learners should be able to consider quadratic expressions as one big idea and follow the fundamental considerations when factoring the quadratic expressions. Furthermore the use of multi-methods in factoring quadratics is encouraged and needed for students to better understand the connections between different methodologies for conceptual development.
The application of Rasch measurement theory to improve the functioning of a mathematics assessment instrument.
(2021) Ngirishi, Harrison.; Bansilal, Sarah.
Assessment is an integral part of the teaching and learning process. Concerns about student performance in assessments often drive the teaching and learning. In South Africa there has been numerous concerns about poor learning outcomes in mathematics and this has led to calls for all stakeholders to work together to try and find solutions. This study focuses on the assessment of mathematics with particular interest in the KZN provincial Grade 12 mathematics trial examination paper 2. The study explored the use of Rasch analysis in improving the functioning of the mathematics assessment instrument. The aim of the study was to use the Rasch analysis to report on the functioning of the test instrument in measuring proficiency in mathematics, checking on the targeting and reliability of the test instrument, explain anomalies where data did not fit the Rasch model, investigate differential item functioning (DIF), response dependency and multidimensionality. The study also sought the teachers’ views about the findings of the Rasch analysis. A sequential explanatory design was used in this study, where the Rasch analysis provided the theoretical framework for the analysis of the quantitative data. The qualitative analysis of the teachers’ responses helped to get more understanding of the results of the quantitative analysis of the leaners’ responses. The study found that the assessment instrument was difficult for this particular cohort, some items displayed DIF for language and response dependency due to some teachers not applying continuous accuracy marking. The study revealed that some teachers were not applying the continuous accuracy marking process. Items which carried more than two accuracy marks, showed misfit to the Rasch model. Teachers cited not applying continuous accuracy marking due to time constraints and large number of learners in classes. Teachers blamed poor performance on learners’ lack of basic understanding, adequate preparation and motivation, societal influences, poor understanding of proof type questions, allocation of many accuracy marks on one item and the language barrier. The recommendations of this study if implemented may help teachers in the teaching process and examiners in producing fair assessment instruments. The recommendations may lead to improvement of mathematics results.
Exploring pre-service teachers’ understanding of similarity and proofs in Euclidean geometry.
(2022) Mbatha, Mduduzi Mhlengi.; Bansilal, Sarah.
This qualitative study explores pre-service teachers’ understanding of similarity and proofs in Euclidean geometry in one South African university from KwaZulu-Natal (KZN) province. Such insight is vital for addressing pre-service teachers’ geometric knowledge, which has been found lacking. The research participants were 34 pre-service teachers (PSTs) in their first year of study towards a Bachelor of Education degree specialising in mathematics at the FET phase. A pen and paper test and semi-structured interviews were employed in gathering the required data for this study. The Van Hiele levels of geometric thought were used as a theoretical framework, which formed the basis for the analysis and discussion of findings. The findings indicated that most pre-service teachers performed adequately on familiar items but struggled with those unfamiliar, which were not typical grade 12 examinable questions. A follow-up of semi-structured interviews was conducted with seven PSTs of mixed abilities to probe the originality of their written responses. Although all interviewed PSTs indicated an improvement when responding to research items verbally than in writing, they did not reach the expected acquisition necessary to teach geometry effectively. Overall, this study found that many PSTs displayed poor levels of understanding similarity and proofs, including (1) limited understanding of the definition of similarity to triangles; (2) poor understanding of how to prove two figures are similar; (3) the haphazard use of geometric theorems in devising proofs; (4) a display of higher Van Hiele levels of understanding for familiar items but lower levels of understanding for unfamiliar items. These findings raised concerns about this group of PSTs teaching geometry, especially if certain concepts require more complex skills that are slightly beyond the secondary school curriculum. It is recommended that professional teacher education training offered to pre-service teachers should include aspects such as (1) Improving PSTs’ geometry content knowledge, (2) Teaching geometry for understanding and (3) Improving PSTs’ written mathematical responses. These factors may be pivotal in improving pre-service teachers’ geometric knowledge beyond the scope of the secondary school curriculum.
Exploring teachers’ instructional practices, confidence and beliefs in teaching mathematics and statistics.
(2018) Umugiraneza, Odette.; Bansilal, Sarah.; North, Delia Elizabeth.
An important contributor to the quality of teaching mathematics is the knowledge of mathematics teachers. In this study, I explore mathematics teachers’ instructional practices, their confidence and beliefs about the teaching of mathematics and statistics concepts. The reason for focusing on mathematics as well as statistics teaching is that in several schools’ mathematics teachers also teach statistics (because statistics is a part of mathematics). This inspired me to undertake a study in order to investigate teachers’ instructional practices in teaching both mathematics and statistics among learners from grade 4 and upwards in KwaZulu-Natal (KZN) schools. The use of KZN as a research location provides an advantage of identifying issues of mathematics teachers’ practices in developing countries. The study was conducted with 75 mathematics teachers from KwaZulu-Natal (KZN) in South Africa who agreed to participate in the study while they were enrolled in an in-service course designed to improve their understanding of statistics. The teachers were invited to participate by filling in a detailed questionnaire, which was adopted from the study of Beswick, Callingham and Watson (2012) which was conducted with teachers from Australia. The detailed questionnaire consisted of open ended, Likert scale as well as yes-no responses. The instrument surveyed the teachers about various aspects of their teaching practices such as the formulation of lesson objectives, the use of the different approaches to introduce mathematics and statistics topics, the use of various teaching and assessments strategies to teach different topics as well as their descriptions about learners’ possible understanding or misunderstanding of the topics. The study also elicited from the teachers their reflections about how they would improve mathematics and statistics teaching and learning. In addition, the study examined the teachers’ beliefs about using mathematics and statistics in everyday life as well as in the classroom, and their confidence in relation to teaching the various mathematics and statistics topics. In addition, the study explored how teachers integrate technology in teaching and learning maths/stats topics. Furthermore, their content knowledge was put under the spotlight through the examination of their solutions to mathematical tasks. The findings revealed that 65.3% of the participants managed to set appropriate lesson objectives. Moreover, these teachers reported that they mostly use practical examples, real life approaches and explicit instruction when teaching the topics. It was also reported by most teachers that they tend to focus on a single approach when they introduce a concept in the classroom. Furthermore, less than half the teachers reported that their learners showed an understanding of mathematics and statistics concepts. For the methods and assessments, teachers generally use a single method and more than one type of assessment. I also found that teachers mostly focus on teacher-led instructional methods and formal assessments. Furthermore, the findings revealed that teachers’ demographic factors such as teaching experience, gender and participation in professional development courses are associated with the choice of a variety of teaching and assessments methods (p-value<0.05). For the use of curriculum, the findings revealed that 19% of teachers had no idea about how they would integrate topics across the curriculum in teaching and learning. With respect to the teachers’ reflections about improving teaching and learning mathematics and statistics, teachers said that developing learners’ interest in learning these conceptions, developing grouping and learner-centred approaches for teaching, applying investigation, practical and real life examples would contribute to improvements. Furthermore, the findings suggest that teachers should use the curriculum in the teaching process and upgrade their studies by doing postgraduate courses in education as the factors that would influence them to make a continuous improvement in the teaching process. The findings showed that participating in professional development courses is a factor that motivate teachers to use curriculum (p-value<0.05). For their content knowledge about solving specific tasks, the findings revealed that teachers demonstrated more understanding in finding the correct answer for the problem of using percentage than for fraction and pie chart. However, they struggled to provide justifications for their answers. This indicated a lack of specialised content knowledge, which refers to ability to give the detailed mathematics explanations to teach the given task and weigh up and analyse unconventional solution methods of their students. Teaching experience becomes an important factor to help teachers develop their content knowledge and solve mathematical tasks appropriately (p-value<0.05). In terms of their confidence in teaching various topics, the finding revealed that teachers were confident in teaching fractions, decimals, percentages, histograms and pie charts, patterns and measurements; however their confidence was lower with respect to teaching aspects requiring connections between mathematics and statistics to other learning areas. In relation to their beliefs, teachers reported a positive view towards the need to be mathematically and statistically literate in everyday life, as well in their teaching practices in general. With regards to the use of technology in teaching mathematics and statistics, the findings indicated that almost all the teachers reported that they never use computers in mathematics and statistics discourse. Although the teachers reported that they do not use computers in teaching and learning, about 80% of the participants conveyed a positive view that using technology improves learners’ understanding of mathematics and statistics. The findings further indicate that the teachers’ propensity to use technology in instructional practice is associated with demographic factors of age, experience and gender (p-value<0.05). The study suggests that teachers should attend more professional development programmes which would improve existing teaching strategies.
Mathematics teachers’ constructions and enactments of learner-centred practices.
(2019) Ndlovu, Henry Cedric Mfanuzile.; Bansilal, Sarah.
An important aspect of effective teaching is the personal understandings that teachers have of theories about teaching and learning. In this qualitative case study, I sought to explore Mathematics teachers’ constructions of learner-centred practices and the extent to which their personal enactments of “learner-centred” practices enabled meaningful learning at Grade 6 level in Eswatini. The sample was convenient, involving three grade 6 Mathematics teachers from three urban schools in the Shiselweni region of Eswatini. Data was collected through lesson observations, semi-structured interviews and field-noted observations. The three teachers were observed teaching a Mathematics topic. All their lessons were video-recorded and were followed by a one-on-one interview with each teacher. The interviews were based on the teachers’ observed lessons. The audio-recorded interviews with the teachers were transcribed verbatim and thereafter analysed thematically. The study is broadly informed by a socio-cultural framework and Meaningful learning theory. Furthermore, the study is located within an interpretative paradigm to gain an insight into the teachers’ constructions of learner-centred practices and the extent to which their personal enactments of learner-centred practices enabled meaningful learning. The study found that the three teachers shared some common understandings that within learner-centred teaching, the teacher takes on the role of a guide. To them, guiding learners basically involved walking around the class and watching the groups working on a problem without making any meaningful intervention, while encouraging them to participate. The study also found that the teachers’ enactment of what they considered as “learner-centred practices” prioritised the outward forms and sidestepped the main function of learner-centred practices which is to enable meaningful learning. The teachers stressed the importance of group work in their personal enactments of “learner-centred” practices. Their belief was that engaging learners in group work would enable meaningful learning in their learner-centred teaching without attending to the matters underlying it. The study recommends that teacher professional development programmes be introduced by the in-service department to ensure that teachers get the required training on the important ideas that underpin learner-centred practices in order to enable meaningful learning. .
Students’ and lecturers’ perceptions of factors which influence the creation of a sustainable e-learning environment in a University of Technology.
(2022) Msomi, Alfred Mvunyelwa.; Bansilal, Sarah.
By virtue of their designation, South African universities of technology need to be pioneers in the use of technology in learning and teaching. Many of these universities have knowledge and expertise in establishing and maintaining an ICT-driven environment for instruction. Accordingly, this study is an investigation into how the integration of technology into learning and teaching practices at the University Technology in South Africa, has been experienced by students and lecturers. This study aims specifically to comprehend access and technology usage difficulties, as well as infrastructure and training levels in ICT-challenged environments. In doing so, this study created a conceptual framework for encouraging instructional ICT in universities. The present study conducts a thorough assessment of one historically underprivileged University of Technology in South Africa using a case study methodology. The following individuals made up the study's target population: 835 students across first, second, and third-year levels; and 97 lecturers across the faculties of Engineering, Management Sciences, and Natural Sciences from whom data were collected using questionnaires. In addition, the researcher in the current study, performed semi-structured interviews with Deans of the faculties (2), Heads of Department (5), senior staff members in the Teaching and Learning Development Centre (TLDC) (2) and senior staff members in the Information and Technology Network Department (ITN) (3). Hence data were collected from various sources to gain multiple perspectives regarding the creation of a sustainable eLearning environment within a University of Technology. A mixed-methods approach was used in the current investigation. The use of technology for social empowerment to bridge the digital divide is the focus of a mixed-methods approach, which gathers, analyses, and interprets quantitative and qualitative data in a single study via the theoretical lens of creating a sustainable e-Learning environment. Informed by a continual literature review of the use of emergent learning technologies, the lived experience of e- Learning students, lecturers, and involved stakeholders was utilised as the basis for the first data collection. When choosing participants, deliberate sampling was utilised. From a standpoint of methodological interpretivism and positivist viewpoints as a technique of inquiry, the researcher's function was that of participant observer, interviewer, and human instrument. Thematic analysis was utilised in this study because, in contrast to descriptive statistics and inferential statistics, it is effective in identifying patterns in participant-reported qualitative data. Using the test-retest reliability approach, the instrument's reliability coefficient was calculated using Cronbach's alpha. The data gathered using quantitative approaches had response rates of 84 and 84.3 percent, whereas the data gathered using qualitative methods had a response rate of 100 percent. While processing qualitative data with the ATLAS.ti's package, the quantitative data was analysed using the R statistical computer program, 2020, version 3.6.3, to provide descriptive and inferential statistics. By following the university under study ethical guidelines, the study's ethical component was successfully achieved. The results of this study have shown that integration of technology in an educational institution of higher learning is dependent on many factors. The clearly distinguished factors from this study were: (i) poor access to internet connectivity, (ii) lack of continuous training for both students and lecturers on how technology could be integrated in the teaching and learning practices, and (iii) unaccommodating technology infrastructure. These factors were indicated as having a potential in facilitating or hindering technology integration for lecturers and students. The implications of this finding require the immediate development of capacity building plans and strategies for the adoption and integration of students and lecturers to an e-Learning platform. This study also suggests that for lecturers to effectively incorporate ICT into educational practice, they must have suitable pedagogical abilities in addition to fundamental ICT knowledge and skills. University policies and strategic level agendas are critical for success using the framework developed in this study for ICT-challenged environments. Based on these findings, it is recommended that the university management for the participating institution in the present study understands that the e-Learning system has great potential to improve the teaching and learning environment, provided in-depth ICT policy and strategies are put in place and suitable technology infrastructure is available.

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