A Hexagon Result and its Generalization via Proof

This paper presents the discovery of a hexagon result on Geometer’s Sketchpad and its generalization via proof for any 2n-gon. The result is : If ABCDEF is a hexagon with opposite sides parallel (not necessarily equal), then the respective centroids G, H, I, J, K and L of triangles ABC, BCD, CDE, DEF, EFA and FAB, form a hexagon with opposite sides both equal and parallel

The definition of the scalar product: an analysis and critique of a classroom episode

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teacher's definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made

The definition of the scalar product: an analysis and critique of a classroom episode

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teacher's definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made

Disputed interpretations and active strategies of resistance during an audit regulatory debate

Purpose: The paper aims to examine disputed interpretations of “key meanings” between the audit regulator and Big 4 firms during a highly contentious regulatory debate, showcasing their use of “strategies of resistance” to achieve their intended outcomes. Design/methodology/approach: A qualitative analysis is performed of the discourse in a South African audit regulatory debate, set within the country\u27s unique political and historical context. The analysis is informed by the theoretical construct of a “regulatory space” and an established typology of strategic responses to institutional pressures. Findings: The study’s findings show how resistance to regulatory intentions from influential actors, notably the Big 4 firms, was dispelled. This was achieved by the regulator securing oversight independence, co-opting political support, shortening the debate timeline and unilaterally revising the interpretation of its statutory mandate. The regulator successfully incorporated race equality into its interpretation of how the public interest is advanced (in addition to audit quality). The social legitimacy of the Big 4 was then further undermined. The debate was highly contentious and unproductive and likely contributed to overall societal concerns regarding the legitimacy of, and the value ascribed to, the audit function. Practical implications: A deeper appreciation of vested interests and differing interpretations of key concepts and regulatory logic could help to promote a less combative regulatory environment, in the interest of enhanced audit quality and the sustainability and legitimacy of the audit profession. Originality/value: The context provides an example, contrary to that observed in many jurisdictions, where the Big 4 fail to actively resist or even dilute significant regulatory reform. Furthermore, the findings indicate that traditional conceptions of what it means to serve “the public interest” may be evolving in favour of a more liberal social democratic interpretation

Flashback to the past a 1949 matric geometry question

Recently I was paging through my copy of the August 1996 issue of the unfortunately now defunct journal spectrum 34(3), and a section on old Mathematics papers on p.63 where the following problem from paper 2 of the 1949 national senior examinations for the union of South Africa caught my attention

A cyclic Kepler quadrilateral & the golden ratio

Proceeding to construct such a ‘Kepler quadrilateral’ ABCD with sides in geometric progression as indicated by the first figure in Figure 1 produces a flexible quadrilateral with a changing shape. No interesting, invariant properties seemed immediately apparent. However, if ABCD is dragged so that the perpendicular bisectors of the sides become concurrent (i.e., so that it becomes cyclic), as indicated by the second figure in Figure 1, it was observed as shown by measurements that not only did it seem that diagonal AC appeared to be bisected by diagonal BD, but also that DG : AG = φ

A generalisation of the Spieker circle and Nagel line

Many a famous mathematician and scientist have described how their first encounter with Euclidean geometry was the defining moment in their future careers. Some of the most well known are probably Isac Newton and Albert Einstein. Often these encounters in early adolescence have been poetically described as passionate love affairs