At the encouragement of one of the teachers, I have resumed the math circle I was running a few years ago, jointly with her. Previous posts can be found here. This time we are targeting some of the youngest primary school children. So far, I’ve attended two sessions. In the first, we played with Polydron and I mainly used the session to observe the children and see what they’re capable of. In the second – described here – we experimented with symmetry.

Before the session, we created a few cardboard templates of three well-defined shapes, numbered 1, 2, and 3 in the picture below.

We told the children that we would be exploring a way of describing shapes as codes, and first had them cut out several copies of the shapes from coloured paper. This was not as easy as I had hoped – the children varied considerably in speed and accuracy of cutting.

We then told them that there were two operations that could be written down in our code, “next to”, written “;” meaning “the thing on the left of the semicolon is placed immediately to the left of the thing on the right” and “F” meaning “flip the shape in a vertical axis”. They seemed to understand this quite easily, and had no problems coming up and seeking clarification of any points. I hadn’t considered that they had not encountered the semicolon before, though, but this was not a problem, even if it was rendered more like a “j” in most of their written formulae.

We asked them to construct a couple of shapes, such as 2;F2 and F1;2, and then experiment with their own. They seemed to have a lot of fun, and the lower part of the photograph shows some that they were keen to show us when we reconvened in a circle on the carpet.

The children noticed that some shapes such as 2;F2 and F2;2 were symmetric, whereas some were not. I pointed out that nobody had tried to construct a shape such as FF1, and asked what that might look like. Several children correctly identified that FF1 would be indistinguishable from 1, so I wrote “FF1 = 1”, and I think I saw a flicker of an “ah ha!” moment in at least one child who seemed excited by the use of equations here.

At this point we ran out of time in our 1hr session. I would like to take this simple algebra further. In particular, I would like to get them to explore:

- the generalisation from the observed FF1 = 1 to ∀x.FFx=x.
- the observation that symmetry in a vertical mirror line corresponds to the statement Fx = x
- the observation that all shapes of the form x;Fx are symmetric, and relate this to the algebraic definition of symmetry
- the property F(x;y) = Fy;Fx

For those of a mathematical bent, these are the properties of a semigroup with involution.

I enjoyed myself, and I think the children did too!

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