Recent research and a critique of theories of early geometry learning: the case of the angle concept

The van Hieles' theory on levels of development has been supported by many studies, particularly in geometry. An important consequence of the van Hiele work has been the realisation that informal experiences assist students to take the necessary first-step of recognising, for example, shapes. Despite this important influence, some authors have cautioned that visual reasoning can be independent of the levels because there are some difficulties in trying to allot students to levels. In order to illustrate developments in our understanding of students' geometry learning and the role of visual processing, studies on the difficulties students have with angle concepts are discussed. Based on a large qualitative study, the ability to notice and analyse angles is explained in terms of aspects of problem solving or investigation. Case studies illustrate how the complex conceptualisations about angles were developing concurrently, rather than in fixed levels, as a result of visual (mental) imagery, selective attention, manipulation of materials, and discussion. Responsiveness, a compound variable resulting from a complex of cognitive processing unique to the individual emerged as important. Responsiveness suggests an empathy or understanding of the problem resulting from cognitive processing, in particular selective attention. Responsiveness affects interactions with materials and with people who, in turn, influence thinking and the continuing cycle of problem solving and understanding of the angle concept