As I just learned from Wikipedia, the Van Hiele Model is a theory that explains how students learn geometry. It seems very logical that students first learn to recognize basic shapes without knowing much beyond that and then move on to higher levels of understanding regarding these shapes. Of course, not every student is going to move on to the fourth level, or “rigor.” I don’t even know if I have attained that level of understanding. I remember taking Geometry in High School and not having that solid of a grasp on what the teacher was teaching and I often wondered why because Algebra was so easy for me. I noticed this in other students when I tutored math many years later; namely, that many students that understood algebra did not understand geometry, and vice versa.
Going back to the Van Hiele Model, I don’t think that it only applies to certain groups of students, or particular tasks that students perform. Unless I am missing something, the model provides a logical hierarchy of understanding. Folks may not necessarily reach the highest level of understanding but they are going to attain each level in the order that is layed out in the model.
Can the same person be at different levels at the same time in his/her life? Absolutely. Children may reach different levels for different shapes. Maybe a student will reach level 2, or “abstraction” for triangles, but remain at level 1, or “visualization” for rectangles.
I think this largely depends on the rigor of the instruction provided. I think by now we have realized that all children are capable of learning. Knowing this, attainment of certain milestones is necessarily the responsibility of the instructor. I know many teachers would cringe at the sight of such a statement, but isn’t it logical? I’d love to get feedback.