Something that a lot of our lesson plans are missing is an understanding of the Van Hiele levels and how it plays into understanding geometry concepts. Often when our older students aren’t grasping what we are teaching, it is simply because they aren’t ready for it.
When they were younger, they didn’t truly understand the first levels of learning. And this is where the Van Hiele Levels come into play.
What are the Van Hiele Levels?
This theory originated in 1957 by husband and wife team Dina Van Hiele-Geldof and Pierre van Hiele from the Utrecht University in the Netherlands. It helps to describe how students learn geometry. The Van Hiele levels have helped shaped curricula throughout the world, including a large influence in the standards of geometry in the US (source).
How do they work?
Geometric reasoning starts as soon as we can start processing information and in early schooling. However, depending on the individual, the ages in each stage can vary, especially as they progress through school. Basically the level is dependent on the experiences that each student has, no matter what their age.
In Learning Mathematics in Elementary & Middle Schools, Cathcart, et al ”In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2. State standards are written to begin the transition from levels 0 and 1 to level 2 as early as 5th grade “Students identify, describe, draw and classify properties of, and relationships between, plane and solid geometric figures.” (5th grade, standard 2 under Geometry and Measurement) This emphasis on relationships is magnified in the 6th and 7th grade standards.” (source)
Level 0: Visualization
They can recognize shapes by their whole appearance, but not its exact properties. For example, students will think of a shape in terms of what it “looks like.” A rectangle is a door or a triangle is a clown’s hat. And the student may not be able to recognize the shape if it’s rotated to a different standing point.
Level 1: Analysis (Description)
Students start to learn and identify parts of figures as well as see figures in a class of shapes. They can describe a shape’s properties and are able to understand that shapes in a group have the same properties as well. A student in this level will know that parallelograms have opposite sides that are parallel and will be able to group them accordingly.
Level 2: Informal Deduction / Abstraction
A student in this level will start to recognize the relationship between properties of shapes. They will also be able to participate and understand informal deductive discussions about the shapes and their different characteristics.
Level 3: Formal Deduction
At this level students are able of more complex geometric concepts. They can think about properties are related, as well as relationships between axioms, theorems, postulates and definitions. According to John Van Del Walle, students should be able to “work with abstract statements about geometric properties and make conclusions more on logic than intuition.”
Level 4: Rigor
Finally, students will reach the last level of learning geometric reasoning. Even in the absence of concrete examples, they should be able to compare geometric results in different axiomatic systems. Basically, they will see geometry in the abstract. Mostly, this is the level of college mathematic majors and how they think about geometry.
Some students may seamlessly pass through these stages, while others may be get a little left behind. And in the meantime, of course the curriculum keeps going, so without proper attention to the missing links or tutoring they won’t ever be able to full catch up with the lessons.
You can dial in your student’s Van Hiele level understanding by including some extra activities in your classroom. Make sure to check out these ideas from NRICH.ORG.
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