Van Hiele levels and learning geometry notes

The Van Hiele's research in math eduction was focussed on the learning of geometry. They postulated (and subsequent research supports their framework), that students must pass through a series of levels of understanding for a geometry topic as they master that content area. Their levels are roughly as follows. Note that there are two alternate numeration schemes (0-4 and 1-5) for Van Heile levels (I am using the 0-4 numeration):

Level 0 (visualization): learners identify a geometric object or concept because it is like a prototypical examples that they have been exposed to.

Children in grades PreK-1 are often at level 0 with respect to recognizing triangles, in that they may not accept thin or obviously scalene triangles as being triangles, because their concept of "triangle" is directly tied to the prototypical example of a point-up equilateral triangle: a triangle is a triangle because it "looks like one" not because of any particular properties.

Students (and people in general) can have conflicting internal definitions, and can go back and forth with what level of understanding they show. For instance, a fourth grader might know the definition of a triangle quite well, and yet say that an equilateral triangle seems more like a "real" triangle than a scalene triangle. Further, students can be working at level 1 successfully during a unit, yet when you return to that topic after a break, they may go back to the level 0 ideas they showed previously.

Level 1 (Analysis): The key feature of a level 1 understanding, is identifying a shape/object by its properties. A triangle is no longer something that looks like a triangle, it is something with a particular set of properties (3 straight lines meeting at 3 corners).

Children are at level 1 in their understanding of shapes such as triangles, squares and rectangles, when they spontaneously identify the properties that those shapes have, and identify shapes by their properties. Some shapes are more difficult than others: for instance, children in grade 4 and 5 are generally still convinced that a square is not a rectangle. Rectangles are particularly tricky, because the definition they are used to (and appears in some dictionaries) specifies that one pair of sides is longer than the other. Switching to the math definition of a rectangle, and internalizing it is a process that takes repeated, spaced out, exposures.

Students working at level 1 generally are not able to distinguish the most important (necessary and sufficient) conditions for identifying a shape (so definitions they create would have either too few conditions, or extraneous detail).

Level 2 (Abstraction): learners recognize relationships between types of shapes.

For example, they should recognize that all squares are rectangles, but not all rectangles are squares.They can tell whether it is possible or not to have a rectangle that is, for example, also a rhombus. They also can make and test conjectures about kinds of shapes, such as: you can make a rectangle out of two congruent right triangles--or--the sum of the angles in a triangle is 180°.

Students need to be comfortably at this level to be well prepared for a high school geometry course (though many students are not). One of our goals in this class will be to consider ways that we can help children progress to a level 2 understanding of some important geometric topics.

Level 3 (Deduction): learners can construct geometric proofs at a high school level. Learners should be exposed to deduction at a pre-high-school level in the context of level 2 discussions (what properties tell us that all squares are also rectangles), but the primary consideration of level 3 work will not be discussed in this course.

Level 4 (Rigor): learners understand how geometry proofs and concepts fit together to create the structure we call geometry. This is the level at which most college geometry courses (for math majors) are designed.

One of the important teaching principles that comes from the Van Hiele's research is that students don't automatically move from one level to another, they gain that abstraction and sophistication as a result of learning experiences they have had.

Notes and sources

Annenberg (Geometry session 10) has another description of the Van Hiele levels, together with a video to analyze. They also have posted an excerpt from Elementary and Middle School Mathematics by John A. Van de Walle with a good description of Van Hiele levels (see the bottom of the page).

I got some of the details for this explanation from the paper The Van Hiele Levels of Geometric Understanding by Marguerite Mason.

I got some of my information about PreK-1 students from the article Young Children's Developing Understanding of Geometric Shapes by Mary Anne Hannibal, published in Teaching children Mathematics Feb. 1999 (available in our library)

Thanks also to information presented by Michael Serra at the 2004 NCTM regional meeting in Minneapolis.