Marc Schäfer

The nature of geometry instruction and observed learning-outcomes opportunities in Nigerian and South African high schools

Abstract The purpose of this qualitative case study involving six secondary school teachers was to obtain insight into how geometry is taught in selected Nigerian and South African high schools. It also aimed, by making use of the van Hiele model of geometry instruction, to elucidate what possible learning opportunities observed instructional methods could offer learners in the subject. The sample comprised three mathematics teachers from Nigeria and three mathematics teachers from South Africa, all of whom were selected using purposive sampling techniques. Instructional activities in six geometry classrooms were recorded on videotape. The van Hiele learning phases provided the framework for data analysis. The findings of this study indicate that observed teaching methods in geometry classrooms in the participating schools offer learners scant opportunity to learn geometry. In comparative and relative terms, however, the observed instructional methods in geometry classrooms within the South African subsample offer greater opportunities for the learners to learn geometry than observed teaching methods in geometry classrooms within the Nigerian subsample. The tentative conclusion drawn is that learners whose instructional experiences align approximately with the van Hiele phases of learning demonstrate a better understanding of geometric concepts than their counterparts whose geometry classroom instructional experiences deviate significantly from the van Hiele model. Certain images of teaching evident in the videotaped lessons are discussed and some recommendations offered.

Teacher Code Switching Consistency and Precision in a Multilingual Mathematics Classroom.

This paper reports on a study that investigated teacher code switching consistency and precision in multilingual secondary school mathematics classrooms in South Africa. Data was obtained through interviewing and observing five lessons of each of three mathematics teachers purposively selected from three township schools in the Eastern Cape Province. Elements of Gumperz and Mercer’s work on lesson categories and Dowling’s Domains of Mathematical Practice were used to analyse data. Results showed that code switching frequency in general was inconsistent across different lessons for the same teacher. Code switching frequency by all teachers was, however, consistently highest during questioning and explaining when teaching. All participating teachers used code switching strategies most consistently in the public domain and least consistently in the esoteric domain. Some formal isiXhosa translations of mathematical terms were consistently and precisely used and some were not. Two major forms of code switching emerged, namely borrowing code switching and transparent code switching. Very little transparent code switching, which is critical for supporting students’ understanding and thinking in mathematics, was evident in teacher language. Teachers were consistent in the use of borrowed terms. We conclude that consensual understanding of best practices for code switching is required to promote code switching that is precise, consistent, transparent and thus supportive of teaching for conceptual understanding of mathematics in secondary schools.

A Case Study of Two Selected Teachers as they Integrated Dynamic Geometry Software as a Visualisation Tool in Teaching Geometry

This paper reports on an aspect of a larger research study conceptualised within a teacher development project in Mthatha, Eastern Cape Province. The project was initiated with the objective to develop appropriate skills to use dynamic geometry software (DGS) effectively and strategically as a teaching and learning tool for mathematics. The study reported in this paper aims specifically to ascertain how selected mathematics teachers integrated co-developed technologically aided visualisation tools in the observed lessons. The case study involved two teachers from different schools. The data sources were the classroom observations followed by stimulated reflective interviews with the teachers. The data were analysed to study the use of DGS visualisation tools in relation to Kilpatrick’s framework of teaching proficiency. The lessons evidenced a displayed alignment with the elements of teaching proficiency in the context of teaching geometry. The dynamic visualisation opportunities offered by DGS proved to supplement the teaching repertoire for the participating teachers. Pedagogical practices influence the use of DGS, as evident from the lessons when the participating teachers incorporated collaboratively developed GeoGebra applets into their classrooms. We argue that the collaborative engagement between teachers appears to be a positive way forward in closing the gap between having access to technology and adapting it for effective use in mathematics classrooms.

Enactivism as a Powerful Theoretical Framework for Research and Tool to Reflect on My Own Role as a Supervisor.

Enactivism, as an interesting and useful theoretical underpinning is gaining traction in Mathematics Education research. It forms the central theme of this paper whose aim is two-fold: first to describe and engage with how elements of enactivism informed a PhD study, both on a theoretical and analytical level, and second to reflect on the enacted role of the supervisor of this study. Despite the inevitable embodied relationship between the supervisor and the supervised PhD project, it is not often written about. This paper thus attempts to address this. The PhD study in question used constructs of enactivism such as autonomy, sense-making, emergence, embodiment and experience to design a pre-service mathematics education programme and then explored the growth of student teachers’ mathematical identity and disposition in their development of becoming mathematics teachers. The PhD supervision process was framed by the enactivist notion that learning and the construction of meaning and knowledge is co-created by the lecturer, the student and the particular context. The role of the author of this paper in the study was that of supervisor. The relationship between a supervisor and his/her student is often complex and multilevelled. This paper argues that this relationship can best be described as one of embodiment and co-emergence. The paper thus starts with the author’s own enactivist ontological perspective vis-à-vis this relationship and how elements of enactivism permeated his practice with regard to the PhD he supervised. The study found enactivism to be a powerful theoretical vantage point from which to develop research instruments that enabled deep and meaningful reflection on teacher practice in the mathematics classroom––and on supervision.

Indigenous Technology and Culture

This needs to include the technologies that are used every day by most of the world’s populations in developing countries and not just the culture of advanced industrial countries, as this focus would give only a partial perspective on the nature of technology. The inclusion of ‘indigenous technology and culture’ in the South African curriculum, for example, is one way of developing learners’ sensitivities to the interrelationship between society, the environment, science and technology.