Van Hiele levels

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Van Hiele levels are a postulated "series of levels of understanding for a geometry topic ... that students must pass through" when learning it.^[1] In general, these levels are a product of experience rather than age. In other words, a child must have enough experiences (classroom or otherwise) with these geometric ideas to move to a higher level of sophistication.

Visualization: children identify prototypes of basic geometrical figures (triangle, circle, square). At this age (or stage), children might balk at calling a thin, wedge-shaped triangle (with sides 1, 20, 20 or sides 20, 20, 39) a "triangle", because it's so different in shape from an equilateral triangle.
Analysis: children can discuss the properties of the basic figueres and recognize them by these properties, but might still insist that "a square is not a rectangle."
Abstraction: learners recognize relationships between types of shapes. They recognize that all squares are rectangles, but not all rectangles are squares. They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus. Students need to be comfortably at this level to be well prepared for a high school geometry course (though many students are not). One of our goals in this class will be to consider ways that we can help children progress to a level 2 understanding of some important geometric topics.
Deduction: learners can construct geometric proofs at a high school level. Learners should be exposed to deduction at a pre-high-school level in the context of level 2 discussions (what properties tell us that all squares are also rectangles), but the primary consideration of level 3 work will not be discussed in this course.
Rigor: learners understand how geometry proofs and concepts fit together to create the structure we call geometry. This is the level at which most college geometry courses (for math majors) are designed.

The Van Hiele theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof e Pierre van Hiele in the Utrecht University, in the Netherlands.

[edit] Further reading

The Van Hiele Levels of Geometric Understanding by Marguerite Mason
Young Children's Developing Understanding of Geometric Shapes by Mary Anne Hannibal