In the spring, my prospective elementary teachers move on to the second course in our math content sequence. This one focuses on geometry and measurement.

In my never-ending quest to problematize the routine, I’ve been brainstorming ways to help them with one of our goals for the course: *Distinguishing between properties and defining characteristics of a shape.*

Several things make this difficult:

- The standard domain for examining this is pretty tired-the hierarchy of quadrilaterals.
- Students bring a lot of cultural and educational baggage to this hierarchy.
- Depending on how we choose to define a shape, its defining characteristics and properties change. For example, we can define a square as an equilateral, equiangular quadrilateral. In that case, having right angles is a property. Alternatively, we can define a square as a rhombus with a right angle. Then being
*equiangular*is a property instead of a defining characteris

To rid ourselves of this baggage, I propose to have my students develop a classification of hexagons. Many details to be worked out.

But consider the hexagons below as examples.

These two hexagons have in common the fact that each side is parallel to another side of the hexagon. Sort of like parallelograms. Perhaps we end up calling these parallelogons. And then we’ll have to decide whether we want to subdivide this category further. Do we want to distinguish between these two? Based on what characteristics? And can we agree whether *all regular hexagons are parallelogons?* And can we distinguish between that claim and this one: *all parallelogons are regular*?

Etc.

I’ll have to think all this through.