An exploration of preservice teachers’ Mathematics knowledge for teaching in Trigonometry at a higher education institution.

Doctoral Degree. University of KwaZulu-Natal, Durban.This study investigates the extent of final-year preservice teachers’ understanding and development of the mathematics knowledge for teaching in trigonometry. Teachers’ lack of adequate mathematics knowledge to teach mathematics effectively is one of the major source of low mathematics attainment in South Africa. On this basis, the readiness of prospective teachers to teach mathematics must be established at the point of exit. The purpose of the present research study is to explore preservice teachers’ understanding and development of content knowledge and pedagogical content knowledge in teaching trigonometry. The review of literature revealed that many preservice teachers lack the conceptual understanding of school mathematics. Thus, preservice teachers exit teacher education and enter the world of teaching with limited skills and abilities of teaching mathematics. The content test, task-based interview, lesson planning and lesson observations were used to gather data on preservice teachers’ understanding of content knowledge in trigonometry in response to three research questions. The sample of the study was composed of fifteen mathematics final-year preservice teachers who were registered for a Bachelor of Education degree programme at a rural-based institution of higher learning in South Africa. The sample was selected purposively. The mathematics knowledge for teaching conceptual framework by Ball, Thames and Phelps was used to structure the present study and provided lens for data analyses. The analysis of the content test results revealed that preservice teachers’ mastery of content knowledge in trigonometry was inadequate. The results from the task-based interview, lesson plan and lesson observation analyses indicated that the preservice teachers’ mastery of pedagogical content knowledge in trigonometry was limited. Moreover, the extent of preservice teachers’ development of mathematical knowledge for teaching based on results from classroom practices was sub-standard. The traditional teaching methods and learner-misconceptions never left preservice teachers all through the four years of teacher education. Therefore, more needs to be done by the higher education institution to accelerate growth of content knowledge and pedagogical content knowledge through the provisions of methodology, content and teaching practices courses. The interplay of the three, methodology courses, content courses and teaching practice form the basis of an ideal preservice teacher.Abstract also available in isiZulu

Creation of Innovative Teaching Spaces with Gamma Tutor: A Techno-Blended Model for Rural Mathematics Teaching

Innovative techno-blended teaching methodologies are needed for 21st-century classrooms. This paper reports on the introduction of a techno-blended device for mathematics teaching in South African senior secondary classrooms. The research sample included 12 third-year mathematics education student teachers from a rural university. A qualitative case study design was employed. Participants were purposively selected. Data collection methods included non-participatory observation and in-depth individual interviews. The TPACK model and the Mathematical Knowledge for Teaching Framework guided the analysis of the data. The study found that the mathematics student teachers successfully implemented GammaTutor in their classrooms, thus demonstrating their proficiency and aptitude in utilising technology in the learning environment. The GammaTutor device also aided in teaching and learning mathematics by simplifying mathematical ideas for learners. Furthermore, it enabled the development of learner discourse as a crucial component for developing learners' problem-solving skills. Since the mathematics student teachers engaged the learners with a variety of mathematical exercises using the GammaTutor technology, the integration of GammaTutor in the mathematics classroom exhibited learner-centred provisioning. The study proposes a teaching model for creating innovative teaching spaces in rural schools in South Africa

Students’ Cognition of The Induction Step in Proving Inequality Propositions

The induction step in proof by induction requires some clever tricks in order to get the expected formula for , especially in statements with inequalities. The purpose of this study was to determine undergraduate students’ cognition regarding the induction step in proving inequality propositions in order to create opportunities to teach the principle of mathematical induction better. A class of 67 students participated in the study on learning proof by induction using the problem-based approach. The Action-Process-Object-Schema theory was used to structure the study and the cyclic activities-class-discussion-exercises instructional approach was used to teach proof by induction. Data for the study was comprised of individual students’ written responses to a task of two questions and the transcriptions of the semi-structured interviews. The findings revealed that students at most showed indication of partial understanding of proof by induction. Executing the induction step sits at the heart of proof by induction and necessitates logical reasoning at the object-level conception. Inadvertently, the implication was the most challenging aspect in proof by induction. The majority of students made inroads in setting up the proof properly but could not succeed in proving that . Some students had challenges of where to begin a proof, so much that they chose to start with direct substitution. In line with that, most students also concluded without deriving the expected formula required to draw a conclusion. Keywords: induction step; proof by induction; APOS-ACE teaching cycles; inequalities