Density Problems in Sobolev’s Spaces on Time Scales

In this paper, we present a generalization of the density some of the functional spaces on the time scale, for example, spaces of rd-continuous function, spaces of Lebesgue Δ-integral and first-order Sobolev’s spaces.

Graphs

The purpose of this chapter is to introduce the mathematical foundations of graphs and explain their applicability to giving an abstract spatial expression to the brief. As one knows from different kinds of diagrams, visual displays are quite useful for recognizing relationships and summarizing complex information. Graphs allow overview of aspects such as grouping and circulation in a brief. Moreover, they provide connections to designs produced on the basis of the brief, as design representations like floor plans can be directly compared to a requirements graph. The graphs discussed in this chapter consist of spatial entities, for example, spaces in a design or activities in a brief, and relationships between pairs of entities, such as direct access, proximity, or belongingness to the same group. Of the various programs available for developing and working with graphs, one considers in detail Microsoft Visio, a widely available diagramming application.

Exploring links between foundation phase teachers’ content knowledge and their example spaces

This paper explores two foundation phase teachers’ example spaces (a space in the mind where examples exist) when teaching number-related topics in relation to snapshots of their content knowledge (CK). Data was collected during a pilot primary maths for teaching course that included assessments of teacher content knowledge (CK). An analysis of a content-knowledge focused pre-test developed for the larger study indicated a relatively high score for one teacher and a low score for the other. Using Rowland’s (2008) framework, an analysis of classroom practice showed associations between a higher CK and the extent of a teacher’s example space and more coherent connections between different representational forms. Although no hard claims or generalisations of the link between teachers’ example spaces and their level of mathematics content knowledge can be made here, this study reinforces evidence of the need to increase teachers’ CK from a pedagogic perspective in order to raise the level of mathematics teaching and learning in the South African landscape.