,A: Conceptual
Origin and Key Ideas of the Van Hiele Model
The Van Hiele model of geometric thought was developed in the 1950s by Dutch educators Pierre
van Hiele and Dina van Hiele-Geldof. Their work, completed at Utrecht University, emerged from
research into why many learners struggled with geometry despite being able to perform procedural
tasks. They concluded that learners’ difficulties were not mainly due to age, but rather to the level
of geometric thinking they had reached. According to their theory, progression in geometry
depends more on instruction and experience than on maturation alone (Van de Walle, Karp &
Bay-Williams, 2019).
A key idea of the Van Hiele model is that geometric understanding develops through qualitatively
different levels of thinking. Each level represents a distinct way of reasoning about shapes and their
properties. Learners must master one level before progressing to the next, and they cannot skip
levels. Importantly, teaching that is pitched above a learner’s current level will not result in
meaningful understanding (Burger & Shaughnessy, 1986).
The model identifies five hierarchical levels: Visualisation, Analysis, Informal Deduction, Formal
Deduction, and Rigor. These levels describe how learners’ reasoning evolves from concrete
recognition of shapes to abstract, axiomatic reasoning.
The Five Levels of the Van Hiele Model
Level 0: Visualisation (Recognition)
At the Visualisation level, learners recognise shapes based on their overall appearance rather than
their properties. Shapes are identified holistically, often by comparing them to familiar objects. For
example, a learner might say a shape is a rectangle “because it looks like a door” or identify a
triangle simply because “it looks like a triangle.”
At this stage, learners do not analyse properties such as parallel sides or angle measures. If a
square is rotated, they may fail to recognise it as a square because it no longer matches their
mental image. Learning activities at this level should involve sorting, naming, drawing and
manipulating shapes to strengthen visual recognition (Van de Walle et al., 2019).
Level 1: Analysis (Descriptive)
At the Analysis level, learners begin to identify and describe properties of shapes. They can state
that a rectangle has four right angles or that a square has equal sides. However, they do not yet
understand relationships between properties or between different classes of shapes.
For example, a learner may recognise that a square has four equal sides and four right angles, but
may not realise that a square is also a rectangle because it satisfies the properties of rectangles.
Shapes are still seen as separate categories rather than as part of a hierarchical system. Instruction
at this level should focus on measuring, observing, and listing properties of shapes (Burger &
Shaughnessy, 1986).
, Level 2: Informal Deduction (Relational)
At the Informal Deduction level, learners begin to understand relationships between properties and
between different shapes. They can form logical arguments and make informal deductions. For
example, a learner may reason that “if a quadrilateral has four right angles, then it must be a
rectangle,” or recognise that all squares are rectangles because they meet the defining properties
of rectangles.
Learners start to see geometry as a connected system rather than isolated facts. They can follow
logical arguments and explain why certain statements are true, but they may not yet construct
formal proofs. Activities at this stage include classifying shapes using Venn diagrams and justifying
relationships between figures (Van de Walle et al., 2019).
Level 3: Formal Deduction
At the Formal Deduction level, learners understand the role of axioms, definitions, theorems and
formal proof. They can construct and understand deductive proofs within a geometric system. For
example, they can prove that the angles of a triangle sum to 180 degrees using parallel lines and
corresponding angles.
Learners at this level recognise the structure of a mathematical system and can distinguish
between necessary and sufficient conditions. This level is typically associated with secondary school
geometry, where formal Euclidean proofs are emphasised (Burger & Shaughnessy, 1986).
Level 4: Rigor
The Rigor level represents the most advanced stage of geometric thinking. Learners can compare
different axiomatic systems and understand the formal structure of geometry at an abstract level.
For example, they can analyse differences between Euclidean and non-Euclidean geometries.
At this level, reasoning is highly abstract and symbolic. Learners understand the independence of
axioms and can work within different geometric systems. This level is typically reached at tertiary
level studies in mathematics.
Conclusion
The Van Hiele model provides a structured explanation of how learners’ geometric thinking
develops from simple visual recognition to advanced formal reasoning. Each level is characterised
by a distinct way of thinking, and progression depends on carefully structured instruction rather
than age. Understanding these levels enables teachers to design geometry activities that align with
learners’ current thinking, thereby promoting meaningful conceptual development (Van de Walle
et al., 2019).
Origin and Key Ideas of the Van Hiele Model
The Van Hiele model of geometric thought was developed in the 1950s by Dutch educators Pierre
van Hiele and Dina van Hiele-Geldof. Their work, completed at Utrecht University, emerged from
research into why many learners struggled with geometry despite being able to perform procedural
tasks. They concluded that learners’ difficulties were not mainly due to age, but rather to the level
of geometric thinking they had reached. According to their theory, progression in geometry
depends more on instruction and experience than on maturation alone (Van de Walle, Karp &
Bay-Williams, 2019).
A key idea of the Van Hiele model is that geometric understanding develops through qualitatively
different levels of thinking. Each level represents a distinct way of reasoning about shapes and their
properties. Learners must master one level before progressing to the next, and they cannot skip
levels. Importantly, teaching that is pitched above a learner’s current level will not result in
meaningful understanding (Burger & Shaughnessy, 1986).
The model identifies five hierarchical levels: Visualisation, Analysis, Informal Deduction, Formal
Deduction, and Rigor. These levels describe how learners’ reasoning evolves from concrete
recognition of shapes to abstract, axiomatic reasoning.
The Five Levels of the Van Hiele Model
Level 0: Visualisation (Recognition)
At the Visualisation level, learners recognise shapes based on their overall appearance rather than
their properties. Shapes are identified holistically, often by comparing them to familiar objects. For
example, a learner might say a shape is a rectangle “because it looks like a door” or identify a
triangle simply because “it looks like a triangle.”
At this stage, learners do not analyse properties such as parallel sides or angle measures. If a
square is rotated, they may fail to recognise it as a square because it no longer matches their
mental image. Learning activities at this level should involve sorting, naming, drawing and
manipulating shapes to strengthen visual recognition (Van de Walle et al., 2019).
Level 1: Analysis (Descriptive)
At the Analysis level, learners begin to identify and describe properties of shapes. They can state
that a rectangle has four right angles or that a square has equal sides. However, they do not yet
understand relationships between properties or between different classes of shapes.
For example, a learner may recognise that a square has four equal sides and four right angles, but
may not realise that a square is also a rectangle because it satisfies the properties of rectangles.
Shapes are still seen as separate categories rather than as part of a hierarchical system. Instruction
at this level should focus on measuring, observing, and listing properties of shapes (Burger &
Shaughnessy, 1986).
, Level 2: Informal Deduction (Relational)
At the Informal Deduction level, learners begin to understand relationships between properties and
between different shapes. They can form logical arguments and make informal deductions. For
example, a learner may reason that “if a quadrilateral has four right angles, then it must be a
rectangle,” or recognise that all squares are rectangles because they meet the defining properties
of rectangles.
Learners start to see geometry as a connected system rather than isolated facts. They can follow
logical arguments and explain why certain statements are true, but they may not yet construct
formal proofs. Activities at this stage include classifying shapes using Venn diagrams and justifying
relationships between figures (Van de Walle et al., 2019).
Level 3: Formal Deduction
At the Formal Deduction level, learners understand the role of axioms, definitions, theorems and
formal proof. They can construct and understand deductive proofs within a geometric system. For
example, they can prove that the angles of a triangle sum to 180 degrees using parallel lines and
corresponding angles.
Learners at this level recognise the structure of a mathematical system and can distinguish
between necessary and sufficient conditions. This level is typically associated with secondary school
geometry, where formal Euclidean proofs are emphasised (Burger & Shaughnessy, 1986).
Level 4: Rigor
The Rigor level represents the most advanced stage of geometric thinking. Learners can compare
different axiomatic systems and understand the formal structure of geometry at an abstract level.
For example, they can analyse differences between Euclidean and non-Euclidean geometries.
At this level, reasoning is highly abstract and symbolic. Learners understand the independence of
axioms and can work within different geometric systems. This level is typically reached at tertiary
level studies in mathematics.
Conclusion
The Van Hiele model provides a structured explanation of how learners’ geometric thinking
develops from simple visual recognition to advanced formal reasoning. Each level is characterised
by a distinct way of thinking, and progression depends on carefully structured instruction rather
than age. Understanding these levels enables teachers to design geometry activities that align with
learners’ current thinking, thereby promoting meaningful conceptual development (Van de Walle
et al., 2019).
