Friday, October 17, 2014

At what level is his thinking?

Pierre van Hiele

My colleague, Jon Hasenbank, and I have been discussing the van Hiele Levels of Geometric Thinking and what they mean for teaching and learning in mathematics. I am particularly interested in finding videos of people sharing their geometric thinking so that we can apply the Levels and evaluate their thinking. If you are not familiar with the Levels, here's how Pierre van Hiele, the architect of the Levels, described the first three Levels in Developing Geometric Thinking through Activities That Begin with Play [PDF].

In my levels of geometric thinking, the "lowest" is the visual level, which begins with nonverbal thinking. At the visual level of thinking, figures are judged by their appearance. We say, "It's a square. I know that it is on because I see it is." Children might say, "It is a rectangle because it looks like a box."
At the next level, the descriptive level, figures are the bearers of their properties. A figure is no longer judged because "it looks like one" but rather because it has certain properties. For example, an equilateral triangle has such properties as three sides; all sides equal; three equal angles; and symmetry, both about a line and rotational. At this level, language is important for describing shapes. However, at the descriptive level, properties are not yet logically ordered, so a triangle with equal sides is not necessarily one with equal angles.
At the next level, the informal deduction level, properties are logically ordered. They are deduced from one another; one property precedes or follows from another property. Students use properties that they already know to formulate definitions, for example, for squares, rectangles, and equilateral triangles, and use them to justify relationships, such as explaining why all squares are rectangles or why the sum of the angle measures of the angles of any triangle must by 180. 

And now for some practice. Given these descriptions, how would you categorize the thinking of the individual in this video?

What evidence do you have to support your categorization of his geometric thinking? (Perhaps there are clues in the task's questions.) If more evidence is required, what questions might you ask to get a better sense of his Level of Geometric Thinking?

As always, your thoughts are welcome in the comments - as long as they are civil.


  1. Wow. That's pretty frustrating after all that buildup, if that's all there is to it. I'm only hearing a visual level justification, though I'm sure he has the math knowledge to go further.

    If it's so easy for everyone to "see", why are they even watching the video? And if it's not, wouldn't it be helpful to explain how you can be SURE it's a trapezoid? And what kind of trapezoid, anyway? Lost opportunities.

    Followup questions:
    How could you convince a skeptic that this is a trapezoid instead of just any old quadrilateral, or a rectangle, or a parallelogram?
    How do you see the symmetry?
    Does it matter how we orient the cutting thing?