Interesting conversation on Twitter today with Bryan Meyer, Denise Gaskins and Justin Lanier. It began with these tweets on my part, the result of grading some student work.

Oh dear. Class inclusion is giving some of my elementary kiddos fits. Do we have any post-Piagetian research on this?

— Christopher (@Trianglemancsd) February 7, 2014

I suspect some of it is linguistic and probably appears not just in their math work but in their writing too.

— Christopher (@Trianglemancsd) February 7, 2014

Things quickly got too nuanced for Twitter.

An example of something my students struggle with is answering a question such as, *Is a square a rectangle?*

This type of question asks about class inclusion. Is an element of a subset also an element of the larger set?

Many useful and interesting questions in geometry have to do with whether one class is a subset of another class. Do all isosceles triangles have a pair of congruent angles? Are all quadrilaterals formed by connecting midpoints of other quadrilaterals parallelograms? Are all Stacys concave?

I am trying to sort out the extent to which my students’ struggles with questions of this sort are linguistic, and the extent to which they are about struggles with the idea of class inclusion.

Justin suggested this wording, which I will investigate:

Is a square an example of a rectangle?

Or, more generally:

Is an X an example of a Y?

My suspicion is that this will be helpful for some students when asked in this direction. But I also suspect that asking it in the other direction will be problematic.

Is a rectangle an example of a square?

See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.

I hope these three (and others) will weigh in here where we have more space to work than we do on Twitter. The ideas are really useful. If you’d like to follow the prior discussion, you can follow this link.