It got me realising that if Coxeter uses manipulatives to learn about geometry, then so should I and so should my students.
When I first started teaching I told students to reflect, rotate and shift shapes by "imagining how they would look after the transformation".
Not brilliant advice any way you look at it.
However, when I started giving tracing paper (or acetate sheets) for students to draw the object and then to find the image, immediately there was greater success. I saw this again in a Grade 6 class this week. We gave the students this question from Ontario's Junior EQAO test of 2010:
Initially, students could sort of make out a reflection, a rotation and a shift (or translation... though I myself don't really like that term because of its ambiguity). However, they had trouble describing these transformations.
That was until one student walked to the front of the class to help herself to a small sheet of acetate paper that we had surreptitiously placed. And this is how she used this tool to help her tackle the problem:
Accurately as well.
And when other students saw what she was doing, they immediately wanted to use the acetate too. As one lad said, "It makes my thinking clearer."
* * *
If you want to see engaged students, give them some of M.C. Escher's prints and ask them to find and describe as many transformations as they can. Guaranteed fun.