ABSTRACT
This chapter offers possibilities of using geometrical thinking to lead into issues of relevance for the planet. Before getting to these activities, it is pertinent to consider what is meant by geometrical thinking. Perhaps the most widely used characterisation of geometric thinking derives from the work of van Hiele (1986). There have been several versions of, and additions to, van Hiele’s hierarchy of levels; the one below comes from a synthesis by Burgher and Shaughnessy (1986):
Level 0 (Visualisation). The student reasons about basic geometric concepts, such as simple shapes, primarily by means of visual considerations of the concept as a whole without explicit regard to properties of its components.
Level 1 (Analysis). The student reasons about basic geometric concepts by means of an informal analysis of component parts and attributes. Necessary properties of the concept are established.
Level 2 (Abstraction). The student logically orders the properties of concepts, forms abstract definitions, and can distinguish between the necessity and sufficiency of a set of properties in determining a concept.
Level 3 (Deduction). The student reasons formally within the context of a mathematical system, complete with undefined terms, axioms, an underlying logical system, definitions, and theorems.
Level 4 (Rigour). The student can compare systems based on different axioms and can study various geometries in the absence of concrete models.
(Burgher and Shaughnessy 1986: 31)
