The methodology of this project
has been based in:
Van Hiele levels of geometry
The work of two Dutch educators,
Pierre van Hiele and Dina van Hiele-Geldof, has given
us a vision around which to design geometry curriculum.
Through their research they have identified five levels
of understanding spatial concept through which children
move sequentially on their way to geometric thinking:
Students can name and recognize
shapes by their appearance, but cannot specifically
identify properties of shapes. Although they may be
able to recognize characteristics, they do not use
them for cognition and sorting.
Students begin to identify properties
of shapes and learn to use appropriate vocabulary
related to properties, but do not make connections
between different shapes and their properties. Irrelevant
features, such as size or orientation, become less
important, as students are able to focus on all shapes
within a class. They are able to think about what
properties make a rectangle. Students at this level
are able to begin to talk about the relationship between
shapes and their properties.
Students are able to recognize
relationships between and among properties of shapes
or classes of shapes and are able to follow logical
arguments using such properties.
Students can go belong just identifying
characteristics of shapes and are able to construct
proofs using postulates or axioms and definitions.
This is the highest level of
thought. Students at this level can work in different
geometric or axiomatic.
In primary we should work in
the three first levels.
Implications of van Hiele for
Geometric taught in the elementary
school should be informal and based in visualization,
analysis and informal deduction.
Such informal geometry activities
should be exploratory and hand-on in order to provide
children with the opportunity to investigate, to build
and take apart, to create and make drawings, and to
make observations about shapes in the world around.
This provides the basis for more formal activities
at higher levels.
Exploration, analysis and communication
In each lesson three aspects
are going to be worked: exploration, analysis and
The first step, exploration,
will focus on an attempt to bring the children in
contact with their natural and built environment,
in order to define and observe its particularities.
This component will allow us to create a link between
the knowledge the children possesses about their environment
and the starting of a formalization process. We try
to attend the children aware of the richness and variety
of structures that make-up their living environment.
The second step, analysis, will
bring a framework as reflection and critique, and
secondly a framework of mathematization. Understanding
geometric can’t be limited to the observation
of similarities or differences. Therefore, understanding
also includes a more structured means of reflection.
Many themes can be addressed: establishing larger
structures from smaller ones, using specific geometric
Finally, the communication component
will focus on developing abilities that will allow
the children to communicate through mathematical discourse
and the various means associated with this discourse.
Communication is an essential element mathematics
and above all, when learning mathematics. Therefore,
communicating in geometry is more than using the appropriate
vocabulary; it also consist of using various means
that allow us to describe the shapes in a space.
Traditionally, approaches to
teaching mathematics have focused in linguistics and
logical teaching methods, with a limited range of
teaching strategies. Some students learn best, however,
when surrounded by movement and sound, others need
to work with their peers, some need demonstration
and applications that show connections of mathematic
to other areas (music, sports, art, architecture),
and others prefer to work alone, silently, while reading
from a text.
The multiple intelligences approach
does not require a teacher to design a lesson in nine
different ways to that all students can access the
material …In ideal multiple intelligences instructions,
rich experiences and collaboration provide a context
for students become aware of their own intelligence
profiles, to develop self-regulation, and to participate
more actively in their own learning.
Knowledge of students’
learning styles assists teachers in developing lessons
that appeal to all learners. Visual learners appreciate
lessons with graphics, illustrations, and demonstration.
Auditory learners might learn best from lectures and
discussions, Kinesthetic learners process new information
best when it can be touched or manipulated; for this
group of learners, written assignments, note taking,
examination of objects, and participation in activities
are valued strategies to consider.
We watch, we listen, we imitate,
we adapt what we find to our own styles and interests,
we build from there.
The idea of visible thinking
helps to make concrete what a thoughtful classroom
might look like. At any moment ,we should ask “Is
thinking visible here? Are students explaining things
to one another? Are students offering creative ideas?
Are they, and the teacher, using the language of thinking?
Is there a brainstorm about alternative interpretation
on the wall? A students debating a plan?
When the answers to all that
questions are consistently yes, students are more
likely to show interest and commitment as learning
unfolds in the classroom. They find more meaning in
the subject matters and more meaningful connections
between school and everyday life.
When thinking is visible in class,
students are in position to be more metacognitive,
to think about their thinking. When thinking is visible,
it becomes clear that school is not about memorizing
contents but exploring ideas.
The programming is based on the Generalitat curriculum.
The children are involved in a weekly geometry lesson,
which lasts for one hour.
For the first 10 minutes of the lesson the class is
involved in mental maths activities (reviewing).
During the next 40 minutes, the main focus of the
lesson is introduced and again involves whole class
teaching. Children study a range of topics about geometry:
2D, 3D, angles, coordinates, perimeters, areas, surfaces,
symmetry, translation, rotation and problem solving.
After the introduction, the children work individually
or in groups, on follow up activities.
The final 10 minutes is called the plenary. Learning
is reviewed and/or homework is discussed.