I found the following article on van Hiele's Levels of Geometric Understanding. What I found interesting is that van Hiele's levels seem to relate very closely to the Cognitive Domain of Bloom's Taxonomy, in that the lower levels speak generally to students' ability to identify figures and their properties, which relates closely to the Knowledge and Comprehension levels in Bloom's. The higher levels in van Hiele seem to relate more closely to the Synthesis and Evaluation levels in Bloom's, and are evidence that students are exhibiting higher order thinking skills.
An interesting set of questions posed in this article relate to how a teacher can be at a different van Hiele Level and still be able to teach at the students' individual levels. To me, this is the essence of what it means to be a professional teacher. One can go through school as a stellar mathematics student, but be unable to teach what they know to students with different learning styles from their own. Or, an "average" mathematics student could be the world's best teacher because they understand that each of their students have a different learning "language" and are able to vary and differentiate their instruction in order to speak to all of their students in their own language.
A revealing quote in this article regarding the implications of van Hiele's theory on instructional practices is this: "... using lecture and memorization as the main methods of instruction will not lead to effective learning. Teachers should provide their students with appropriate experiences and the opportunities to discuss them." Well said!
Even though van Hiele's levels provide broad generalizations of students' learning levels, I believe that they can be best applied to individual students at the individual task level. A student may be at Level 4 (Deduction) for yesterday's task and at Level 1 (Visualization) for today's task. In fact, this should not be surprising at all, since many times mathematics involves mastering a task one day and moving onto a new task the next day. Hopefully, as a student's knowledge builds based on prior knowledge, they can quickly step through the lower levels to the higher levels. And certainly, based on the subject matter, a student can be at different levels of understanding at the same time in their life. There are many possible reasons for this, but it should not be at all surprising.
As I think about van Hiele's model, I view it as an extension of Bloom's Taxonomy which can relate more directly to the study of Geometry. Because I will be teaching geometry for the first time this fall, the timing could not be better for me to be exposed to this model!