,MIP2601
Assignment 01
UNDERSTANDING AND APPLYING THE VAN HIELE FRAMEWORK IN
INTERMEDIATE PHASE MATHEMATICS EDUCATION
A: Conceptual - Explain the origin, key ideas and the five levels of the Van Hiele
model. Discuss the characteristics of each level with examples from
geometry.
The Van Hiele model is one of the most important theories in mathematics
education, especially in the teaching and learning of geometry. It explains how
learners develop their understanding of geometric ideas in stages or levels.
Many learners struggle with geometry, particularly when they are expected to
solve problems or write formal proofs. The Van Hiele model helps teachers
understand why this happens and how to support learners better. This essay
discusses the origin of the Van Hiele model, its key ideas, and the five levels of
geometric thinking. It also explains the characteristics of each level with
examples from geometry.
The Van Hiele model was developed in the 1950s by two Dutch educators,
Pierre van Hiele and Dina van Hiele-Geldof. They were both mathematics
teachers in the Netherlands. While teaching geometry, they noticed that many
learners found the subject difficult. Even when teachers explained clearly,
learners could not understand higher-level geometric ideas. Pierre van Hiele
completed his doctoral thesis in 1957, where he described different levels of
geometric thinking. Dina van Hiele-Geldof also conducted important research,
especially on how learners move from one level to the next. Their work later
became internationally recognised and has had a strong influence on geometry
teaching around the world.
One of the key ideas of the Van Hiele model is that learners move through
different levels of thinking when learning geometry. These levels are
sequential, which means that learners must pass through one level before
moving to the next. A learner cannot skip a level. Another important idea is
that progress depends on instruction and experience, not on age. Two learners
, of the same age may be at different levels of geometric thinking, depending on
how they have been taught. The model also explains that each level has its own
language and way of reasoning. Words such as “angle,” “parallel,” or “proof”
may have different meanings for learners at different levels. Finally, the model
shows that proper teaching methods, including exploration, discussion, and
guided activities, help learners move to higher levels of thinking.
The first level of the Van Hiele model is Visualization, sometimes called
Recognition. At this level, learners identify shapes based on their appearance.
They look at the whole shape and recognise it because it looks familiar. For
example, a learner may identify a triangle because “it looks like a triangle.”
However, the learner cannot explain the properties of the triangle, such as the
number of sides or angles. If the triangle is turned upside down, the learner
may not recognise it as a triangle because it looks different. Similarly, a learner
may think that a square is not a rectangle because it looks different from the
rectangles they usually see. At this level, understanding is based only on visual
features, not on mathematical properties.
The second level is Analysis. At this level, learners begin to notice and describe
the properties of shapes. They can talk about sides, angles, and other
characteristics. For example, a learner can say that a rectangle has four sides,
that opposite sides are equal, and that it has four right angles. However, the
learner does not yet understand the relationships between these properties.
They may not realise that a square is also a rectangle, even though it has the
same properties. In other words, they know the facts about shapes, but they
do not connect these facts logically. Learners at this level can describe shapes
in more detail, but their reasoning is still limited.
The third level is Informal Deduction, also called Relational Thinking. At this
stage, learners begin to understand the relationships between properties and
between different shapes. They can make logical arguments, but these
arguments are still informal. For example, a learner may explain that all squares
are rectangles because they have four right angles and opposite sides are
equal. They understand that shapes can belong to more than one category.
They also begin to use “if… then…” reasoning. For instance, they may say, “If a
shape has four equal sides and four right angles, then it is a square.” Although
they can explain their thinking clearly, they are not yet able to write formal
proofs using strict mathematical structure.
Assignment 01
UNDERSTANDING AND APPLYING THE VAN HIELE FRAMEWORK IN
INTERMEDIATE PHASE MATHEMATICS EDUCATION
A: Conceptual - Explain the origin, key ideas and the five levels of the Van Hiele
model. Discuss the characteristics of each level with examples from
geometry.
The Van Hiele model is one of the most important theories in mathematics
education, especially in the teaching and learning of geometry. It explains how
learners develop their understanding of geometric ideas in stages or levels.
Many learners struggle with geometry, particularly when they are expected to
solve problems or write formal proofs. The Van Hiele model helps teachers
understand why this happens and how to support learners better. This essay
discusses the origin of the Van Hiele model, its key ideas, and the five levels of
geometric thinking. It also explains the characteristics of each level with
examples from geometry.
The Van Hiele model was developed in the 1950s by two Dutch educators,
Pierre van Hiele and Dina van Hiele-Geldof. They were both mathematics
teachers in the Netherlands. While teaching geometry, they noticed that many
learners found the subject difficult. Even when teachers explained clearly,
learners could not understand higher-level geometric ideas. Pierre van Hiele
completed his doctoral thesis in 1957, where he described different levels of
geometric thinking. Dina van Hiele-Geldof also conducted important research,
especially on how learners move from one level to the next. Their work later
became internationally recognised and has had a strong influence on geometry
teaching around the world.
One of the key ideas of the Van Hiele model is that learners move through
different levels of thinking when learning geometry. These levels are
sequential, which means that learners must pass through one level before
moving to the next. A learner cannot skip a level. Another important idea is
that progress depends on instruction and experience, not on age. Two learners
, of the same age may be at different levels of geometric thinking, depending on
how they have been taught. The model also explains that each level has its own
language and way of reasoning. Words such as “angle,” “parallel,” or “proof”
may have different meanings for learners at different levels. Finally, the model
shows that proper teaching methods, including exploration, discussion, and
guided activities, help learners move to higher levels of thinking.
The first level of the Van Hiele model is Visualization, sometimes called
Recognition. At this level, learners identify shapes based on their appearance.
They look at the whole shape and recognise it because it looks familiar. For
example, a learner may identify a triangle because “it looks like a triangle.”
However, the learner cannot explain the properties of the triangle, such as the
number of sides or angles. If the triangle is turned upside down, the learner
may not recognise it as a triangle because it looks different. Similarly, a learner
may think that a square is not a rectangle because it looks different from the
rectangles they usually see. At this level, understanding is based only on visual
features, not on mathematical properties.
The second level is Analysis. At this level, learners begin to notice and describe
the properties of shapes. They can talk about sides, angles, and other
characteristics. For example, a learner can say that a rectangle has four sides,
that opposite sides are equal, and that it has four right angles. However, the
learner does not yet understand the relationships between these properties.
They may not realise that a square is also a rectangle, even though it has the
same properties. In other words, they know the facts about shapes, but they
do not connect these facts logically. Learners at this level can describe shapes
in more detail, but their reasoning is still limited.
The third level is Informal Deduction, also called Relational Thinking. At this
stage, learners begin to understand the relationships between properties and
between different shapes. They can make logical arguments, but these
arguments are still informal. For example, a learner may explain that all squares
are rectangles because they have four right angles and opposite sides are
equal. They understand that shapes can belong to more than one category.
They also begin to use “if… then…” reasoning. For instance, they may say, “If a
shape has four equal sides and four right angles, then it is a square.” Although
they can explain their thinking clearly, they are not yet able to write formal
proofs using strict mathematical structure.
