, PLEASE USE THIS DOCUMENT AS A GUIDE ONLY
1. 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, developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof,
provides a framework for understanding how students develop geometric thinking. This model has
significantly influenced geometry curricula worldwide and helps teachers understand
developmentally appropriate instruction (Van de Walle et al., 2016, p. 513).
Origin and Key Ideas
The Van Hiele theory emerged from research conducted in the 1950s and 1960s, based on the
observation that students often struggle with geometry because instruction is presented at a level
beyond their current understanding. The theory describes five hierarchical levels of geometric
thought that learners progress through sequentially. Key ideas include that levels are sequential
(students must pass through all prior levels), developmental (progress requires appropriate
experiences rather than age), age-independent (students of different ages can be at the same level),
and experience-dependent (advancement comes through geometric experiences, not maturation alone)
(Van de Walle et al., 2016, p. 514).
Level 0: Visualization
At Level 0, students recognize and name shapes based on their global visual appearance rather than
their properties. The objects of thought at this level are individual shapes and what they "look like."
For example, a student at this level identifies a square "because it looks like a square" and may
believe a square rotated 45 degrees is a "diamond" rather than a square (Van de Walle et al., 2016, p.
514). Students sort shapes by appearance, saying things like "I put these together because they are all
pointy." The goal at this level is to explore how shapes are alike and different through observation,
manipulation, and informal classification.
Level 1: Analysis
At Level 1, students focus on properties of classes of shapes rather than individual shapes. The
objects of thought become classes of shapes, and products of thought are the properties themselves.
Students can discuss what makes a rectangle a rectangle (four sides, opposite sides parallel, four
right angles) and understand that irrelevant features like size or orientation do not change the shape's
classification (Van de Walle et al., 2016, p. 515). For instance, a student can list properties of all
parallelograms but may not yet understand hierarchical relationships (e.g., that all squares are
rectangles).
Level 2: Informal Deduction
At Level 2, students develop relationships between properties and engage in if-then reasoning. The
objects of thought are properties of shapes, and products are relationships between these properties.
Students can follow and appreciate informal deductive arguments without formal proof. For example,
a student might reason: "If all four angles are right angles, the shape must be a rectangle. If it is a
square, all angles are right angles. Therefore, a square must be a rectangle" (Van de Walle et al.,
2016, p. 516-517). Students at this level can make conjectures, test hypotheses, and use
counterexamples.
1. 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, developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof,
provides a framework for understanding how students develop geometric thinking. This model has
significantly influenced geometry curricula worldwide and helps teachers understand
developmentally appropriate instruction (Van de Walle et al., 2016, p. 513).
Origin and Key Ideas
The Van Hiele theory emerged from research conducted in the 1950s and 1960s, based on the
observation that students often struggle with geometry because instruction is presented at a level
beyond their current understanding. The theory describes five hierarchical levels of geometric
thought that learners progress through sequentially. Key ideas include that levels are sequential
(students must pass through all prior levels), developmental (progress requires appropriate
experiences rather than age), age-independent (students of different ages can be at the same level),
and experience-dependent (advancement comes through geometric experiences, not maturation alone)
(Van de Walle et al., 2016, p. 514).
Level 0: Visualization
At Level 0, students recognize and name shapes based on their global visual appearance rather than
their properties. The objects of thought at this level are individual shapes and what they "look like."
For example, a student at this level identifies a square "because it looks like a square" and may
believe a square rotated 45 degrees is a "diamond" rather than a square (Van de Walle et al., 2016, p.
514). Students sort shapes by appearance, saying things like "I put these together because they are all
pointy." The goal at this level is to explore how shapes are alike and different through observation,
manipulation, and informal classification.
Level 1: Analysis
At Level 1, students focus on properties of classes of shapes rather than individual shapes. The
objects of thought become classes of shapes, and products of thought are the properties themselves.
Students can discuss what makes a rectangle a rectangle (four sides, opposite sides parallel, four
right angles) and understand that irrelevant features like size or orientation do not change the shape's
classification (Van de Walle et al., 2016, p. 515). For instance, a student can list properties of all
parallelograms but may not yet understand hierarchical relationships (e.g., that all squares are
rectangles).
Level 2: Informal Deduction
At Level 2, students develop relationships between properties and engage in if-then reasoning. The
objects of thought are properties of shapes, and products are relationships between these properties.
Students can follow and appreciate informal deductive arguments without formal proof. For example,
a student might reason: "If all four angles are right angles, the shape must be a rectangle. If it is a
square, all angles are right angles. Therefore, a square must be a rectangle" (Van de Walle et al.,
2016, p. 516-517). Students at this level can make conjectures, test hypotheses, and use
counterexamples.
