1) The document discusses the Van Hiele levels of geometric thinking - Visual, Descriptive, and Informal Deduction - which children progress through as they learn geometry. 2) It explains that the Van Hiele theory presents a hierarchy of geometric thinking levels spanning ages 5 through adulthood and that geometry instruction should align to students' levels. 3) The document also outlines five phases of activities - Inquiry, Directed Orientation, Explication, Free Orientation, and Integration - designed by the Van Hieles to promote progression between levels of geometric thinking.

1. How to help School Students Understand Geometry
Playing with shapes ushers children into geometry long before they set foot in the
classroom and formally study it. Though children can naturally think spatially,
deliberate instruction is required as a scaffolding to navigate them through
several levels of geometric understanding and reasoning.
Since children can only memorize geometric facts and rules and don’t think
deductively and are unable to understand the relationships between ideas, it is
inappropriate to teach them geometry, the Euclid way which is learning to
construct the geometric system using the same logical construction of axioms,
definitions, theorems and proofs as taught using this approach.
This surfaced after painstaking fundamental research by a husband and wife
team: Pierre and Dian Van Hiele and supplemented by many other researchers
from around the world. Though this research has been influencing school
curricula worldwide yet, its practical details are largely unknown to teachers.
Van Hiele theory presents a hierarchy of levels of geometric thinking spanning the
ages five through adults. Though originally there were five levels that have been
adapted and renamed by different researchers, now Van Hiele concentrates on
the three levels that cover the period of schooling.
Levels of Geometric Thinking
Visual:

2. This level is akin to non-verbal thinking. Here a child recognizes the words by sight
long before he/she understands individual letter sounds and how they combine to
form words. Shapes are distinguished by their appearance and viewed as a whole
than by any associated component parts. Although basic shape names are known,
there is no explanation offered or any association made. They may say it is a
square because it looks like one or a rectangle because it looks like a box.
Descriptive:
At this level children can identify and describe the component parts and their
associations. They develop appropriate language to cope with new concepts.
However, they can’t yet appreciate the relationships between the properties of
individual components. They will describe that an equilateral triangle has three
equal sides and is different from the rest of the triangles, but they will not be able
to say that an equilateral triangle has three equal angles.
Informal Deduction:
The properties of shapes and their logical relationships are understood at this
level. Students can see one property precedes or follows from the other and
deduce one property from the other. They can apply what they already know to
explain shapes and formulate definitions. They could explain why all squares are
rectangles. The informal deduction such as this forms the foundation for the
formal deduction the place of axioms, definitions and theorems are not
understood.
Activities to Develop Geometric Thinking
Included in the Van Hiele’s theory are the five phases of activities designed to
promote student’s thinking from one level to the next.
Inquiry:
Children thrust into play by the teacher gives them various shapes and structures.
They explore the objects tactilely trying to fit one shape into another and are
encouraged to talk to the other students about the shapes they hold in their

3. hands. While the children are playing the teacher has ample opportunity to
observe and assess the child’s thinking and language.
Direct Orientation:
Activities are presented to the children in a way that a particular shape or
property of the shape is brought into focus. The ideas for this phase arise from
play and pre-planning. In play, the child may combine two shapes to form a third
shape. The teacher then can provoke the students by asking them if all the shapes
can be made from fitting two shapes together. From these activities children gain
specific understanding about shapes. They find out some lengths are equal and
some are not and they can visualize how angles fit together and can flip shapes to
fit appropriate spaces.
Explication:
Tasks and games are incorporated in this phase to deliberately develop the
vocabulary associated with the ideas the students have encountered so far while
still continuing to explore shapes and their properties. Here, the teacher clarifies
the terms students already know and introduces new ones. Students go through
activities that help them to talk and write about their experiences.
Free Orientation:
Children engage in activities and tasks that can be completed in different ways.
The goal in this phase is for the students to take what they already know and
become proficient at it.
Integration:
Now, the children have reached a stage where they can pull together all their
knowledge and think about it as a whole. They will be able to summarize what
they had learnt all through these five phases.
By engaging in activities through all these phases the students develop how to
think geometrically. They would have gone from the simple recognition of shapes

4. to discussing specific geometric properties to making comparisons between them.
They would have progressed from visual to descriptive to informal deduction level
of thinking geometrically.