Image-based reasoning in geometry

The goal is to investigate image-based reasoning in school geometry. A theoretical framework is proposed, consisting of a model for conceptualization of image data, four principles of conceptualization, and five specific geometrical skills. An explication of the van Hiele levels in terms of this framework confirms its validity; an interpretation of common errors in geometrical reasoning demonstrates its descriptive utility. It is argued that an axiomatic treatment of geometry in secondary schools is inappropriate, and therefore geometric arguments in school must utilize image data. Use of image data brings into question the generality of geometric results. The proposed framework is applied to understanding this issue. A pitfall of allowing image-based reasoning is that there must be strict limits on permissible conceptualizations from image data. The method of this study is theoretical. However, the theoretical model can be used to frame specific research questions in a future empirical study.