The Van Hiele Levels of Geometric Thinking
Mason’s Description(pp. 4-5) / Van de Walle and Folk’s Description
(pp. 327-328)
Level 0 / (Visualization): Students recognize figure by appearance alone, often by comparing them to a known prototype. The properties of a figure are not perceived. At this level, students make decisions based on perception, not reasoning. / (Visualization):The objects of thought are shapes and what they “look like.”
Students recognize and name figures based on their global, visual characteristics – a Gestalt-like approach to shape.
Level 1 / (Analysis): Students see figures as collections of properties. They can recognize and name properties of geometric figures, but they do not see relationships between these properties. When describing an object, a student operating at this level might list all the properties the student knows, but not discern which properties are necessary and which are sufficient to describe the object. / (Analysis):The objects of thought are classes of shapes rather than individual shapes.
Students are able to consider all shapes within a class rather than a single shape.
Level 2 / (Abstraction): Students perceive relationships between properties and between figures. At this level, students can create meaningful definitions and give informal arguments to justify their reasoning. Logical implications and class inclusions, such as squares being a type of rectangle, are understood. The role and significance of formal deduction, however, is not understood. / (Informal Deduction): The objects of thought are the properties of shapes.
Students are able to develop relationships between and amongst properties of geometric objects and to develop relationships between these properties.
Level 3 / (Deduction): Students can construct proofs, understand the role of axioms and definitions, and know the meaning of necessary and sufficient conditions. At this level, students should be able to construct proofs such as those typically found in a high school geometry class. / (Deduction): The objects of thought are relationships among properties of geometric shapes.
Students begin to understand definitions, theorems, corollaries and postulates geometric truth.
Level 4 / (Rigor): Students at this level understand the formal aspects of deduction, such as establishing and comparing mathematical systems. Students at this level can understand the use of indirect proof and proof by contrapositive, and can understand non-Euclidean systems. / (Rigor): The objects of thought are deductive axiomatic systems for geometry.
The object of attention is axiomatic systems themselves, not just the deductions within a system.
Mason, Marguerite (1998). “The van Hiele Levels of Geometric Understanding,” Professional Handbook for Teachers, Geometry: Explorations and Applications.
Van de Walle, John A. and Folk, Sandra. Elementary and Middle School Mathematics: Teaching Developmentally, 2005. Pearson Education Canada Inc.: Toronto, Canada.
Comparing the Van Hiele Levels and the Ontario Mathematics Curriculum
The organization Improving Measurement and Geometry in Elementary (IMAGES) notes that most students in grades K-3 will be at Level 0whilegrades 4-5 may be at Level 1and some possibly at Level 2 ... “It is important for elementary school teachers to provide their students with experiences that will help them move to Level 2 by the end of the eighth grade”.
Level 0 (Visualization) / Level 1 (Analysis) / Level 2 (Informal Deduction)Kindergarten / Use language accurately to describe basic spatial relationships (e.g., above/below, near/far, in/out).
Identify two-dimensional shapes (e.g., circle, square, rectangle, and triangle).
Identify three-dimensional objects (e.g., cans, blocks, balls, cones).
Sort two-dimensional shapes.
Sort three-dimensional objects.
Grade 1 / Describe relative locations of objects or people using positional language.
Identify two-dimensional shapes and three-dimensional figures by their attributes.
Recognize symmetry.
Describe relative locations of objects or people using positional language. (e.g., above, below, behind)
Create symmetrical designs using concrete materials / Describe two-dimensional shapes and three-dimensional figures by their attributes.
Sort and classify common two-dimensional shapes and three-dimensional figures by their attributes.
Compose and decompose two-dimensional shapes and three-dimensional figures.
Build three-dimensional structures and describe the two-dimensional shapes the structures contain
Relate shapes to other shapes, to designs, and to figures.
Level 0 (Visualization) / Level 1 (Analysis) / Level 2 (Informal Deduction)
Grade 2 / Identify and sort two-dimensional shapes and three dimensional figures by their geometric properties (e.g., number of sides and vertices, or number and shape of faces).
Compose and decompose two dimensional shapes and three dimensional figures.
Classify two-dimensional shapes and three-dimensional figures by their geometric properties.
Locate a line of symmetry. / Distinguish between geometric attributes and geometric properties.
Grade 3 / Name prisms and pyramids.
Identify congruent shapes / Sort two-dimensional shapes and three-dimensional figures by their geometric properties (e.g., number of sides and angles, or number of faces, edges and vertices). / Compare two-dimensional shapes and three-dimensional figures by their geometric properties.
Describe relationships between two-dimensional shapes, and between two-dimensional shapes and three-dimensional figures.
Relate different types of quadrilaterals.
Grade 4 / Identify quadrilaterals and three-dimensional figures by their geometric properties.
Construct three dimensional figures using two-dimensional shapes.
Create symmetrical designs by reflecting a shape, or shapes, using a variety of tools and identify the congruent shapes in the designs.
Identifying a straight angle, a right angle, and half a right angle. / Classify quadrilaterals and three dimensional figures by their geometric properties.
Compare various angles to benchmarks (e.g., number of sides, angles, and symmetry).
Classifying prisms and pyramids by geometric properties.
Grade 5 / Identify and construct nets of prisms and pyramids.
Sort three-dimensional figures. / Identify two- dimensional shapes by side and angle properties.
Classify two-dimensional shapes by side and angle properties.
Compare three-dimensional figures.
Distinguish among polygons and prisms.
Level 0 (Visualization) / Level 1 (Analysis) / Level 2 (Informal Deduction)
Grade 6 / Construct polygons and angles.
Sketch three-dimensional figures and construct three-dimensional figures from drawings. / Sorting polygons by lines of symmetry and by rotational symmetry.
Classify polygons and angles.
Sort polygons according to lines of symmetry