The Geometry of Threshold Concepts

This paper proposes a graphical representation for Threshold Concepts that attempts to capture the features and outcomes attributed to them. Typically, conceptual structure is represented as a hierarchy or taxonomy. Hierarchical diagrams emphasise separation. As one examines the structure each path partitions the entries and highlights their differences. As an alternative consider the style used by Garner for the representation of sets and subsets. The same type of relationship is represented but the emphasis is drawn to the integrated nature of the sets. One drawback is that it is flat or two-dimensional. Moving to a three-dimensional style increases the representative power. In this representation we can continue to represent individual concepts in hierarchical form. We draw them as the sides of a three-dimensional pyramidic structure. Extending this representation, inside the walls of the pyramid, unseen, is the threshold concept that integrates the concepts on the pyramid sides. The representation is viewed as a prism and it's behaviour is interpreted to be consistent with that of a prism. Such a metaphor supports the notion that different views of the same structure may yield different outcomes. For example, say a person looks at the structure with a particular Background or disposition. Metaphorically that person's view enters the prism at a particular ‘angle’ or from a particular ‘angle.’ The ‘light’ emitted by the prism will be dependent on that angle. The presence of an incomplete hierarchy on one or more sides is a 'defect' on the surface of that side (e.g. the prism is chipped) but otherwise the prism is still intact. This is interpreted as a section or portion of the conceptual structure that is currently unknown, incomplete or inaccessible. However, the structure is still ‘useable’ albeit not as reliably as if it were well-formed. The prism may have more serious defects. Consider the case of a hole cored right through it or a series of craters on the surfaces. A view entering the prism will be distorted and its passage through the prism and the light emitting from the prism will also be a distortion