Stages of Learning Geometry

Learning geometry can be difficult and studies have shown that students need experience in "thinking" at different levels. The "van Hiele Model of Thinking in Geometry " provides strategies and suggested activities for helping students focus on levels of thinking in geometry and the role of instruction in helping students move from one level to the next.

The van Hiele model of thinking, designed to help students gain insight into geometry, uses five levels to describe student behaviors.

The levels have specific characteristics: the levels are sequential; each level has its own language and set of symbols; what is implicit of a class of shapes at one level is explicit at the next level (e.g., students use the properties of a class of shapes when classifying at Level 0, but they only begin to isolate and describe them when thinking at Level 1); and levels are subject to "reduction" by substituting a rote procedure for thinking.

Level	Description	Example
0	Visual	Judges shapes by their appearances.
1	Analysis	Sees figures in terms of their components and discovers properties of a class of shapes.
2	Informal Deduction	Logically interrelates previously discovered properties.
3	Deduction	Proves theorems deductively
4	Rigor	Establishes theorems in different postulational systems.

Students need to move from one level to the next within each of the topics, van Hieles proposed a sequence of five "phases" of learning. These provide teachers a structure for organizing classroom instruction in geometry as described below:

Level	Description	Example
0	Inquiry	Students discuss and develop questions on a topic to be explored.
1	Directed Orientation	Students explore sets of carefully sequenced activities.
2	Explicitation	Students express explicit views and questions about inherent structures of their investigations.
3	Free Orientation	Students now encounter multi-step tasks and gain experiences in finding their own way of resolving the tasks.
4	Integration	Students form an overview in which objects and relationships are unified and internalized into a new domain of thoughts.

Reference

Fuys, David, Dorothy Geddes, and Rosamond Tischler. "The van Hiele Model of Thinking in Geometry among Adolescents." Journal for Research in Mathematics Education Monograph No. 3. Reston, VA.: National Council of Teachers of Mathematics, 1988.