Abstract Geometry teaching and learning ought to mirror and embrace the social diversity found in the geometry learning environment as well as the increasingly connected world. For that reason, ethnomathematics approaches that relate geometry teaching and learning to the learners' cultural experiences and background should be used when teaching geometry. The aim of this study was to find out the teachers' teaching approaches in geometry as well as their views towards the incorporation of ethnomathematics into the geometry teaching. A convergent mixed methods design was used in this study. Focus group discussions and questionnaires were used as data gathering instruments. The sample comprised of 40 in-service mathematics teachers. Findings show that both teacher-centered and learner-centered approaches were used in geometry teaching and learning. The study also revealed that teachers had the opinion that ethnomathematics approaches should be integrated into geometry teaching. The study recommends that teachers should be trained to use ethnomathematics approaches when teaching geometry.

Resumen La enseñanza y el aprendizaje de la geometría deben reflejar e incluir la diversidad social que integra el aprendizaje de la geometría en el contexto de un mundo cada vez más conectado. Por esa razón, enfoques etnomatemáticos que vinculan la enseñanza y aprendizaje de la geometría con las experiencias culturales y los antecedentes de los alumnos deben ser utilizados. El objetivo de este estudio fue conocer las perspectivas de enseñanza de los profesores en geometría, además de sus concepciones hacia la incorporación de la etnomatemática en la enseñanza de la geometría. Para hacerlo, se utilizó un diseño de métodos mixtos convergentes. Los debates y cuestionarios de los grupos focales fueron utilizados como instrumentos de recopilación de datos. La muestra consta de cuarenta profesores de matemáticas en servicio. Los resultados demuestran que tanto los enfoques centrados en el maestro, como aquellos centrados en los estudiantes se utilizaron en la enseñanza y el aprendizaje de la geometría. El estudio también reveló que los maestros opinaron que los enfoques etnomatemáticos debían integrarse en la enseñanza de la geometría. El estudio recomienda que los profesores deben ser entrenados para utilizar enfoques etnomatemáticos en la enseñanza de la geometría.

The derivative concept is studied in first-year university mathematics. In this study, we focused on students' ability to correctly apply the rules for derivatives of functions with the different structures that they encounter in their university studies. This was done by investigating the online responses of first-year students at the University of KwaZulu-Natal to online quizzes that contributed to their assessment. Based on this investigation, we then interviewed eight students to gain an insight into the thinking behind their responses. We report on the analysis of students' responses to five items on the online quizzes based on the derivative concept. The categories in which those items were based are: condition for existence of derivative at a point; rules for derivatives of standard functions; application of chain rule to different function structures; the application of multiple rules; and application of derivatives to optimise a function. Our findings indicate that students had difficulty in detecting that multiple rules for derivatives were required to differentiate certain types of functions represented in symbolic form. Furthermore, students had difficulty in finding the derivative of a function when more than one application of the chain rule was required. However, there were students who had the ability to apply the rules for derivatives of functions without difficulty. In particular, most of the students were able to correctly recall the differentiation rules for functions with standard structures f(x)=x n, h(x)=e kx and y=[g(x)]n, n = 0 and k is a non-zero constant. Students were also able to correctly apply the chain rule to an exponential function with base e, raised to 4x. The majority of students were able to correctly apply the chain rule together with differentiation rules for logarithmic and exponential (with bases a >1) function structures, and function structures that required the application of the product rule together with the chain rule. Most of the students were able to apply derivatives to optimise a function. Significance: A significant percentage of students who took online quizzes experienced difficulties with applying multiple differentiation rules in the context of a single function. The difficulties stemmed from their inability to detect from the structure of the function which rules should be applied and also the order in which those relevant rules should be applied

A review of relevant literature revealed that research indicates an alarming number of secondary school students who enter tertiary institutions display a serious lack of mathematical basics required for the study of university mathematics. This so-called change from the secondary school to tertiary study of mathematics has become an important field of mathematical learning research. In this context, at the University of KwaZulu-Natal, online diagnostics for pre-calculus knowledge and skills were made available to first-year mathematics students. We report here on some findings from an analysis of the provision of diagnostic questions for basic arithmetic and algebra. The results show that the diagnostic questions which were in the form of an online quiz assisted students to cope with their mathematics module in the sense that it positively impacted on the final results of students who attempted the quiz. SIGNIFICANCE: • Students who completed an online quiz on basic arithmetic and algebra to identify their strengths and weaknesses indicated that the quiz prepared them for the material they had to study for their mathematics module.

We describe an approach to develop higher-order thinking skills (HOTS) among first-year calculus students. The ideas formulated by Brookhart to develop HOTS were used to identify from the literature three core abilities that should be targeted. Then eight expected learning outcomes for the development of HOTS were documented, in the context of the study of first-year university calculus. Those expected outcomes were used to formulate sample tasks that were designed to target the development of the eight abilities. A pilot study was done to determine whether the tasks had the high mathematical demand envisaged. It was found that about 37% of the participants did not give any response to the tasks. Further it was found that about 31% of the participants were able to critically evaluate a given possible solution to a problem and make a value judgement. It is recommended that to promote HOTS among students, the formulation of tasks should focus on developing the following abilities: interpreting a general definition or statement in the context of a given model; translating a worded or graphically represented situation to relevant mathematical formalisms; identifying possible applications of mathematics in their surroundings; identifying linkages between groups of concepts and interpreting these linkages in the context of a model; working systematically through cases in an exhaustive way; critically evaluating one's and others' presented solutions to a problem; interpreting and extending solutions of problems; and using with reasonable skill available tools for mathematical exploration. SIGNIFICANCE: • A large proportion of the student intake at university level is unable to answer mathematics questions that focus on HOTS. There is therefore a need to deliberately focus on and promote HOTS amongst the average students in the context of calculus.

There is a view that many first-year students lack the basic knowledge and skills expected of them to study at university level. We examined the expected work habits and pre-course diagnostics for students who choose to take a course on differential calculus. We focused on the lecturer pre-course expectations of a student in the context of work habits, knowledge and technical skills. In particular, we formulated outcomes and then sample diagnostic questions to test whether the identified learning outcomes on expected work habits and learning are in place. If students are made aware of the expected learning outcomes and if they take the diagnostic test, they should be able to achieve greater success in their studies. The validity of this assumption will be the subject of a future paper which will report on the implementation of the learning outcomes and diagnostic questions that we formulated for pre-course diagnostics in differential calculus.

This article reports on a study which used the APOS (action-process-object-schema) theoretical framework to investigate university students' understanding of derivatives and their applications. Research was done at the Westville Campus of the University of KwaZulu-Natal in South Africa. The relevant rules for finding derivatives and their applications were taught to undergraduate science students. This paper reports on the analysis of students' responses to six types of questions on derivatives and their applications. The findings of this study suggest that those students had difficulty in applying the rules for derivatives and this was possibly the result of many students not having appropriate mental structures at the process, object and schema levels.

I report on the findings from research and literature on (a) use of symbols in mathematics, (b) algebraic/trigonometric expressions, (c) solving equations, and (d) functions and calculus. From these, some insights and implications for teaching and learning are derived.