van Hiele Levels of Geometric Thinking
Objects of
Thought
Level 1 (or Shapes
0)
Visualization Geometric figures
are seen as
entities without
awareness of
parts or
properties or
relationships
between parts of
the figure
Level 2 (or1) classes of shapes
Analysis
Products of
Thought
Classes of
shapes
This is a
rectangle
because it looks
like a door, or
because it looks
like one
properties of
shapes
 Properties are This is a
rectangle
noticed
because is has 4
 They are seen sides, 4 right
as unrelated angles, opposite
1
to one
another
Level 3 (or
2)
Informal
Deduction
Properties of
shapes
 definitions
have meaning
 relationships
seen between
properties and
between
figures – can
deduce
properties
from other
sides are
parallel, it is
closed, opposite
sides are
congruent,
diagonals bisect
each other,
adjacent sides
are
perpendicular…..
Relationships
among
properties
It is a rectangle
because it is a
parallelogram
with right angles
(uses minimal
number of
properties)
2
properties
 Logical
implications
are
understood so
Informal
arguments be
followed
Hierarchical
thinking can be
used
 The role of
deductive
reasoning is
not
understood
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Objects of
Thought
Level 4 (or Relationships
3)
among
Formal
properties
Deduction
 can
construct
proofs
 understand
role of
axioms and
definitions
 meaning of
necessary
and
sufficient
conditions
 can supply
reasons for
steps in a
proof
Products of
Thought
Deductive
systems of
properties
Given that
this is a
parallelogram
and one
angle is a
right angle, I
can prove
that it is a
rectangle
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Questions and Activities:
It is important that the level of the activities
match the level of the student.
Initially, students need to engage in guided,
structured activities
Sorting and classifying shapes
How are they alike, how different?
Put together and take apart shapes in 2
and 3 dimensions
Draw shapes
There needs to be explicit discussion of
objects being studied and properties. But
the language has to match the level.
Focus on properties of shapes rather
than just identifying that you have a
rectangle etc.
Determine properties that are true for
ALL rectangles – look at ALL/Some
types of statements
Later, activities should be more open-ended
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Encourage students to make and test
conjectures – will an observation hold all
of the time?
What properties are necessary and
sufficient to guarantee that you have a
certain polygon – what properties of
diagonals guarantee that you have a
rhombus?
Finally the teacher helps students develop
an overview of the material and try integrate
ideas.
The formal deduction is most likely not
developed until high school
Try to have students at level 2 when
entering high school geometry
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