Van Hiele levels of geometric thinking

2.  Pierre Van Hiele and his
wife Dina Van Hiele were
Dutch researchers and
teachers.
 The theory was originated
in their theses at the
university of Utrecht in
1957.
 Describes how individuals
learn geometry.

3. VAN HIELE LEVELS
OF GEOMETRIC THINKING

4.  Pupils use visual perception and non verbal
thinking.
 They recognise geometric shape as ‘a whole’
and compare the figures with their prototypes
or everyday things.( it looks like door).
 Classify the basic geometrical shapes by
judging appearance.

5.  Focus on students’ thinking on individual
geometric shapes.
For example:
i) Students will confuse to set a triangle with
measures 20 cm, 20 cm, 1 cm or 20cm, 20 cm,
39 cm because these two do not resemble to an
equilateral triangle.
 ii) They will recognise that “ “ this is
a triangle but upside down.

6. TRIANGLES NOT TRIANGLES

7.  Pupils start analysing and naming properties
of geometric figures.
 Objects of thoughts are classes of shapes.
( They learns that shapes are the bearer of
properties.)
 Do not see relationship between properties.
( they think all properties are important.
There is no difference between necessary and
sufficient properties.)

8.  Learn to give reason inductively from several
examples but can’t give reasons deductively
as they do not know how the shapes are
related to each other.
 For example: They will understand that a
square has four equal sides and four equal
angle and diagonals are equal and bisects
each other.
 They may say that a square is not a rectangle.

9. SQUARES NOT SQUARES

10.  Students perceive relationships between
properties and figures.
 They create meaningful definitions.
 They are able to give simple arguments to justify
their reasoning.
 They may say that all squares are rectangles but
all rectangles may not be square.

12.  Since two sides of
isosceles triangles are
equal so base angles
are equal. But they will
give simple reason that
since it is symmetric so
its base angles are
equal. They will
understand the
necessary and
sufficient conditions
and can write concise
definition.

13.  Students can give deductive geometric proofs.
 Objects of thoughts are deductive reasoning.
 They are able to differentiate between necessary
and sufficient conditions.
 They will understand the role of undefined terms
( point, side) , definitions, axioms and theorems,
postulates in the Euclidean Geometry.

14.  Mathematics is understood at the level of a
mathematician. ( new concept, new formula).
 Highest step.
 Students learn about non-Euclidean
geometry. E.g. Hyperbolic function, analytic
function, etc.
 Students understand that definitions are
arbitrary and need not actually refer to any
individual realization.