Interpreting Student Understanding in Geometry: A Synthesis of Two Models
John Pegg University of New England
Geoff Davey Christian Heritage College
There is increasing evidence that many students in the middle years of schooling have severe misconceptions concerning a number of important geometric ideas (see, e.g., Burger & Shaughnessy, 1986; Dickson, Brown, & Gibson, 1984). There are many possible reasons for this. A clear divergence of opinion exists in the mathematics community about the methods and outcomes of geometry, and, as a result, textbook writers and makers of syllabuses have failed to agree on a clear set of objectives. Anecdotal evidence suggests many teachers do not consider geometry and spatial relations to be important topics, which gives rise to feelings that geometry lacks firm direction and purpose.
To some extent these problems may be due to the relatively small quantity of research (as compared with, say, research in number) that has been undertaken into students' thinking in geometry at the school level, which, in turn, may stem from a perceived absence of a theoretical framework. Even though Piaget and his coworkers published two significant works relating to this area, The Child's Conception of Space ( Piaget & Inhelder , 1956) and The Child's Conception of Geometry ( Piaget, Inhelder, & Szeminska, 1960) and these have been followed by various studies in the field of spatial cognition, little impact on classroom practice has resulted. Part of the problem lies with Piaget's "topological primacy theory," on which it has proven difficult to build a school syllabus and about which there have been some fundamental doubts (see Darke, 1982).
It was a combination of these problems and their own classroom experiences in the Netherlands in the 1950s that caused husband-and-wife