Van Hiele Levels of Geometric Thinking – Activities, Examples, and Guide

Geometry, a cornerstone of mathematics, offers a unique realm for learners to visualize, analyze, and understand spatial relationships. Central to the pedagogy of geometry is the Van Hiele model, which delineates the progression of geometric thought in students. But how do these levels influence our teaching methodologies, and at which stages do learners exhibit specific modes of thinking? This comprehensive guide dives deep into the Van Hiele levels of geometric thinking, unraveling its intricacies through activities, examples, and insightful explanations.:

Level 0 – Visualization

Activity: Recognition of shapes based on appearance.

Example: Identifies a shape as a triangle because it “looks like” one, not based on its properties.

Level 1 – Analysis

Activity: Identifying specific properties like number of sides or angles.

Example: Knows squares are rectangles but may not recognize that all rectangles can be squares.

Level 2 – Abstraction

Activity: Understanding relationships between shape properties.

Example: Concludes that a quadrilateral with opposite parallel sides is a parallelogram.

Level 3 – Deduction

Activity: Constructing proofs and formal reasoning.

Example: Can prove that the angles in a triangle add up to 180 degrees using theorems and definitions.

Level 4 – Rigor

Activity: Comparing different geometric systems and understanding axioms.

Example: Can discuss the differences between Euclidean and non-Euclidean geometries, recognizing the role of axioms.

In this guide, we’ll explore:

  • The specific level where learners employ nonverbal thinking in their geometric pursuits.
  • The pivotal role teachers play in nurturing and expanding children’s geometric comprehension.
  • The profound impact of Van Hiele levels on shaping geometric instruction.
  • Whether it’s possible for learners to bypass any stages within the Van Hiele framework, supported by illustrative examples.
  • The criteria at which a learner, based on the Van Hiele model, classifies a shape by its inherent properties.
  • Real-world instances exemplifying geometric thought processes.
  • A deep dive into the wider implications of the Van Hiele model for both teaching and assimilating geometry.
  • A plethora of examples to vividly illustrate each Van Hiele level of geometric thinking.

Embark on this enlightening journey to grasp the nuances of geometric thinking and equip educators with effective strategies to facilitate deeper understanding in their students. Let’s navigate the world of geometry through the lens of the Van Hiele model together!

Van Hiele Levels of Geometric Thinking

The Van Hiele Levels of Geometric Thinking is a theoretical framework that describes the stages through which students progress in their understanding of geometry. This model was developed by the Dutch educators Pierre van Hiele and Dina van Hiele-Geldof in the 1950s. The primary aim of this model is to aid educators in understanding the levels of geometric thought in students and to guide instruction accordingly.

There are five distinct levels in the Van Hiele model:

  1. Level 0 – Visualization (Recognition): At this initial stage, students recognize and name shapes based on their overall appearance rather than by their properties. For instance, they might recognize a shape as a triangle because “it looks like a triangle,” not because it has three sides.
  2. Level 1 – Analysis (Properties): At this level, students start to notice specific properties of shapes, such as the number of sides or angles. However, they might not understand the relationships between these properties. For example, they might recognize all squares as rectangles but might not recognize all rectangles as squares.
  3. Level 2 – Abstraction (Informal Deduction): Students begin to understand the relationships between properties of shapes. They can make logical conclusions about shapes and their properties, even though they might not yet use formal deductive reasoning. For instance, they might deduce that if opposite sides of a quadrilateral are parallel, then it is a parallelogram.
  4. Level 3 – Deduction: At this level, students can construct proofs and use formal deductive reasoning. They understand the properties and relationships of shapes in a formal manner and can work within a geometric system using definitions, theorems, and proofs.
  5. Level 4 – Rigor: Here, students can compare different geometric systems and understand the differences and similarities between them. They recognize the role and significance of axioms, and they can establish connections between different branches of mathematics.

It’s essential for educators to recognize that students’ progression through these levels is not strictly age-dependent. Instructional methods and experiences play a crucial role. Effective teaching tailored to the Van Hiele levels can facilitate a smoother transition from one level to the next, promoting a deeper understanding of geometric concepts.

The specific level where learners employ nonverbal thinking in their geometric pursuits:

At the Level 0 – Visualization of the Van Hiele model, learners predominantly employ nonverbal thinking. During this phase, students recognize and categorize shapes based on their holistic appearance rather than a verbal or analytic understanding of their properties. Their perception is often grounded in visual or tactile experiences. For example, a child at this level might recognize a circle because it “looks round” or resembles the shape of the sun, rather than understanding it as a shape with all points equidistant from the center. The emphasis here is on seeing and not on describing. Hence, instruction at this level often involves activities like sorting shapes, drawing shapes, and identifying shapes in the environment, capitalizing on the students’ intuitive and nonverbal grasp of geometric concepts.

The pivotal role teachers play in nurturing and expanding children’s geometric comprehension:

Teachers are the linchpins in guiding students through the various levels of geometric understanding. Here’s how:

  • Sequencing Instruction: Teachers need to sequence their instruction in alignment with the Van Hiele levels, ensuring that students solidify their understanding at one level before advancing to the next.
  • Providing Concrete Experiences: Especially at the lower levels, hands-on activities, manipulatives, and real-world examples help students anchor their understanding. A teacher might use physical shape tiles, for instance, to help students at Level 0 categorize and recognize different shapes.
  • Facilitating Discussions: Encouraging students to talk about shapes, their properties, and relationships helps in reinforcing and clarifying geometric concepts. Teachers can pose open-ended questions or create group activities where students collaborate and share their geometric insights.
  • Bridging Levels: Teachers play a critical role in transitioning students from one Van Hiele level to the next. This might involve introducing more abstract concepts, fostering logical reasoning, or presenting formal definitions and theorems when students are ready.
  • Differentiation: Recognizing that not all students progress through the Van Hiele levels at the same pace, teachers can provide differentiated instruction tailored to individual students’ current levels of geometric understanding.

The profound impact of Van Hiele levels on shaping geometric instruction:

The Van Hiele model offers a roadmap for educators, providing insights into the sequential stages of geometric understanding. Its impact on geometric instruction includes:

  • Curriculum Development: Educational institutions can design curricula that align with the Van Hiele progression, ensuring topics are introduced in a sequence that resonates with students’ developmental stages in geometric understanding.
  • Instructional Strategies: Teachers can adopt strategies best suited for each level. For example, at Level 1, activities might involve comparing and contrasting shapes based on their properties, while Level 3 might involve more formal discussions around geometric proofs.
  • Assessment: Understanding the Van Hiele levels allows educators to design assessments that are age-appropriate and level-appropriate, ensuring that students are evaluated based on their current stage of geometric comprehension.
  • Professional Development: The Van Hiele model can be a cornerstone in teacher training programs, equipping educators with the knowledge to foster geometric thinking effectively in their classrooms.

In sum, the Van Hiele levels offer a structured framework for understanding the progression of geometric thought, emphasizing the importance of sequencing, teacher intervention, and curriculum alignment in fostering a deeper appreciation and understanding of geometry in students.

The profound impact of Van Hiele levels on shaping geometric instruction:

The Van Hiele levels have fundamentally reshaped how educators approach the teaching of geometry, emphasizing the developmental progression of geometric understanding. Their impact is manifest in several key ways:

  • Sequential Teaching: The Van Hiele model underscores the importance of introducing geometric concepts in a specific sequence. Teachers are encouraged to first establish a strong foundation at one level before progressing to the next. This sequential approach ensures that students have a solid grasp of basic concepts before tackling more abstract ideas.
  • Varied Instructional Strategies: The levels inform the instructional methodologies teachers employ. For instance, at the Visualization level, teachers might use hands-on activities and manipulatives, while at the Deduction level, the emphasis might shift to formal proofs and logical reasoning.
  • Curriculum Design: Curriculum planners can structure the geometry syllabus to align with the Van Hiele progression. By doing so, educational content resonates more with the students’ developmental stages, making learning more effective.
  • Diagnostic Assessment: Teachers can use the Van Hiele model as a diagnostic tool. By assessing which level a student is operating at, educators can tailor instruction to meet individual needs, ensuring no student is left behind.

Whether it’s possible for learners to bypass any stages within the Van Hiele framework, supported by illustrative examples:

The Van Hiele model posits that students must progress through each level sequentially, without skipping levels. This is because each level builds upon the understanding developed in the previous level. However, it’s crucial to note that while the progression is sequential, the speed at which students move through the levels can vary.

Illustrative Example: Imagine two students learning about triangles. Student A, at the Visualization level, identifies triangles based on their appearance. Student B, at the Analysis level, recognizes triangles based on their properties, such as having three sides. If Student B were to skip the Visualization level, they might struggle to identify triangles in complex diagrams or real-world scenarios where the triangle doesn’t fit a stereotypical appearance.

The criteria at which a learner, based on the Van Hiele model, classifies a shape by its inherent properties:

This understanding is primarily achieved at Level 1 – Analysis of the Van Hiele model. At this stage, students move beyond mere recognition and start to understand shapes based on their inherent properties.

  • Properties Understanding: Students begin to notice and describe shapes using terms related to their properties, such as “sides,” “angles,” or “vertices.”
  • Differentiation: A student at this level can differentiate between shapes based on their properties. For instance, they might recognize that a rectangle has four right angles, while a parallelogram, in general, does not.
  • Limited Relationships: While students can identify properties, they might not fully understand the relationships between these properties. For example, they might know that a square has four equal sides, and also that it has four right angles, but might not understand that all squares are also rectangles.

Real-world instances exemplifying geometric thought processes:

Geometric thinking is often grounded in our everyday experiences, even if we aren’t always consciously aware of it. Here are some real-world examples:

  • Navigation and Map Reading: When we use a map or a GPS device, we engage with geometric concepts. Understanding scale, direction, and the relative position of one location to another are all grounded in geometric thinking.
  • Home Decoration and Arrangement: Whether arranging furniture in a room to maximize space or hanging pictures on a wall in a symmetrical pattern, we’re applying geometric principles.
  • Crafts and DIY Projects: Building a birdhouse, sewing, or even simple origami involve understanding shapes, angles, and symmetry.
  • Sports: The trajectory of a ball in games like basketball or golf, or understanding angles to strategize in pool or snooker, are examples of applied geometry.

A deep dive into the wider implications of the Van Hiele model for both teaching and assimilating geometry:

  • Sequential Understanding: The Van Hiele model emphasizes the importance of a sequential approach to learning geometry. One cannot jump into deductive reasoning without a foundational understanding of shapes and their properties.
  • Informed Instruction: Teachers, equipped with the knowledge of the Van Hiele levels, can craft lessons that are developmentally appropriate, ensuring students grasp concepts effectively.
  • Curriculum Design: Educational boards can use the Van Hiele framework to design curricula that progress in a manner aligned with students’ geometric cognitive development.
  • Enhanced Assessment: Tests and evaluations can be tailored based on the Van Hiele levels, ensuring that assessments match students’ current understanding of geometric concepts.
  • Empathy in Teaching: Recognizing that students may be at different Van Hiele levels, even within the same age group, allows educators to approach teaching with empathy, providing additional support where needed.

A plethora of examples to vividly illustrate each Van Hiele level of geometric thinking:

  • Level 0 – Visualization: A child groups shapes together because they “look similar,” not because they understand the properties. For example, they group all round objects together without distinguishing between circles and ellipses.
  • Level 1 – Analysis: Students might say, “All squares are rectangles because they have four sides,” but they might not recognize that “All rectangles are not necessarily squares.”
  • Level 2 – Abstraction: Students recognize relationships between properties. For instance, they understand that a rhombus has all equal sides, and if its angles are right angles, then it’s a square.
  • Level 3 – Deduction: A student might be given the properties of a parallelogram and, using formal reasoning, deduce properties of a subset of parallelograms, like rectangles or rhombi. They might construct a proof to show that the diagonals of a rectangle are congruent.
  • Level 4 – Rigor: At this advanced level, students might compare Euclidean and non-Euclidean geometries. They might explore the implications of changing postulates, like assuming parallel lines can intersect, and delve into the world of spherical or hyperbolic geometry.

The Van Hiele model’s structured approach allows for a nuanced understanding of how geometric thinking develops, with each level building upon the previous one. This progression underscores the importance of a solid foundation in geometry education.


Q: At which level of the van Hiele model do learners use nonverbal thinking?

A: At Level 0 – Visualization of the Van Hiele model, learners use nonverbal thinking. At this level, they recognize shapes based on their appearance rather than by describing or identifying their properties.

Q: Investigate the role of the teacher in developing children’s geometric thinking.

A: Teachers play a vital role in developing children’s geometric thinking. They introduce students to shapes, guide them in recognizing properties, facilitate discussions to foster understanding, provide hands-on experiences, and sequence instruction according to the Van Hiele levels. Teachers also identify where each student is in their geometric understanding and offer differentiated instruction accordingly.

Q: How do Van Hiele levels influence the teaching of geometry?

A: Van Hiele levels provide a structured framework for teaching geometry. They inform teachers about the sequence in which geometric concepts should be introduced, ensuring that foundational concepts are understood before more abstract ones. The levels also guide curriculum development, instructional strategies, and assessment design.

Q: Can a learner skip any of the van Hiele levels? Examples.

A: In general, learners progress through the Van Hiele levels sequentially, building on previous knowledge. Skipping levels can result in gaps in understanding. For example, a student introduced to formal geometric proofs (Level 3) without a solid grasp of shape properties (Level 1) may struggle to comprehend or apply the concepts.

Q: At which level of the van Hiele model does a learner describe a shape based on its properties?

A: At Level 1 – Analysis, learners describe shapes based on their properties. For example, they might describe a rectangle as a shape with four right angles and opposite sides that are equal.

Q: What is an example of geometric thinking?

A: An example of geometric thinking is when someone recognizes the relationship between the sides and angles of a triangle, understanding that the sum of its internal angles is always 180 degrees.

Q: Discuss five implications of the Van Hiele model in teaching and learning geometry.


  1. Sequential Understanding: Concepts should be introduced in a sequence aligned with the Van Hiele levels.
  2. Developmental Appropriateness: Lessons are tailored to the current level of students’ geometric understanding.
  3. Informed Curriculum Design: Curricula can be developed to align with the progression of the Van Hiele levels.
  4. Enhanced Assessment: Evaluations are designed to match students’ current geometric cognitive stage.
  5. Emphasized Teacher Intervention: Teachers play a key role in transitioning students between levels and addressing misconceptions.

Q: Van Hiele levels of geometric thinking examples.


  • Level 0 – Visualization: Recognizing shapes based on their general appearance.
  • Level 1 – Analysis: Identifying properties of shapes, e.g., a triangle has three sides.
  • Level 2 – Abstraction: Understanding relationships between properties, e.g., a square is a rectangle because it has four right angles, but not all rectangles are squares.
  • Level 3 – Deduction: Using formal reasoning and proofs to understand geometric concepts.
  • Level 4 – Rigor: Comparing and contrasting different types of geometries, like Euclidean and non-Euclidean geometry.

Further Reading:

The van Hiele Model of Geometric Thinking

The Effects of Van Hiele’s Phases of Learning Geometry

Understanding Van Hiele Levels for Geometry

Foundation phase teachers’ (limited) knowledge of geometry

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