Characterizing the van Hiele Levels of Development in Geometry

This study provides a description of the van Hiele levels of reasoning in geometry according to responses to clinical interview tasks concerning triangles and quadrilaterals. The subjects were 13 students from Grades 1 through 12 plus a university mathematics major. The tasks included drawing shapes, identifying and defining shapes, sorting shapes, determining a mystery shape, establishing properties of parallelograms, and comparing components of a mathematical system. The students' behavior on the tasks was consistent with the van Hieles' original general description of the levels, although the discreteness of levels, particularly of analysis and abstraction, was not confirmed. The use of formal deduction among students who were taking or had taken secondary school geometry was nearly absent, consistent with earlier observations by Usiskin (1982).