Van Hiele model
In mathematics education, the Van Hiele model is a theory that describes how students learn geometry.
American researchers did several large studies on the van Hiele theory in the late 1970s and early 1980s, concluding that students' low van Hiele levels made it difficult to succeed in proof-oriented geometry courses and advising better preparation at earlier grade levels.
[1][2] Pierre van Hiele published Structure and Insight in 1986, further describing his theory.
The model has greatly influenced geometry curricula throughout the world through emphasis on analyzing properties and classification of shapes at early grade levels.
The five levels postulated by the van Hieles describe how students advance through this understanding.
A child must have enough experiences (classroom or otherwise) with these geometric ideas to move to a higher level of sophistication.
Without such experiences, many adults (including teachers) remain in Level 1 all their lives, even if they take a formal geometry course in secondary school.
Children identify prototypes of basic geometrical figures (triangle, circle, square).
The objects of thought are classes of shapes, which the child has learned to analyze as having properties.
If a figure is sketched on the blackboard and the teacher claims it is intended to have congruent sides and angles, the students accept that it is a square, even if it is poorly drawn.
The objects of thought are geometric properties, which the student has learned to connect deductively.
A student at this level might say, "Isosceles triangles are symmetric, so their base angles must be equal."
Learners can construct geometric proofs at a secondary school level and understand their meaning.
They understand the role of undefined terms, definitions, axioms and theorems in Euclidean geometry.
However, students at this level believe that axioms and definitions are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry.
People can understand the discipline of geometry and how it differs philosophically from non-mathematical studies.
Properties are in fact related at the Analysis level, but students are not yet explicitly aware of the relationships.)
The van Hieles believed this property was one of the main reasons for failure in geometry.
Attainment: The van Hieles recommended five phases for guiding students from one level to another on a given topic:[7] For Dina van Hiele-Geldof's doctoral dissertation, she conducted a teaching experiment with 12-year-olds in a Montessori secondary school in the Netherlands.
Using van Hiele levels as the criterion, almost half of geometry students are placed in a course in which their chances of being successful are only 50-50.
— Zalman Usiskin, 1982[1] Researchers found that the van Hiele levels of American students are low.
[8] Many, perhaps most, American students do not achieve the Deduction level even after successfully completing a proof-oriented high school geometry course,[1] probably because material is learned by rote, as the van Hieles claimed.
[5] Some researchers[9] have found that many children at the Visualization level do not reason in a completely holistic fashion, but may focus on a single attribute, such as the equal sides of a square or the roundness of a circle.
