Background: Students who have been studying and working with axial symmetry had to come up with a list of instructions, which allow another person (without maths background/training) to determine the number of lines of symmetry in any given shape. A quick test was done before starting, of students ability to determine the number of lines of symmertry in a shape (starting with regular shapes only).

Started with simple instruction to students: Ok, now that you know about this kind of symmetry, how would you go about explaining it to another person? The plan is that the students should attempt to solve the problem. Get stuck. Then I pull out ENV tool, explain it, and let students use it and see if it helps. Gradually allow them some success, then remove certain parameters (e.g. use irregular shapes - which should render their instructions useless/unworkable) and have them go through the problem again from the beginning.

Well there the fun began.

The first response was ...well you just look at it and count them. The students had no appreciation at all of the steps which they were using in order to answer the questions themselves! For me, this was really fascinating.

Step 1. Gave the problem more context: Imagine you are programming a robot or a computer. You will give it/feed it with a shape and the robot/computer will follow your instructions exactly and will tell you the number of lines of symmetry in the shape. 2 competing teams. The test robot is a person from the other team.

So we have the first problem already, the robot cannot simply 'look' at the shape and count the lines of symmetry - it has no way to identify what it has to count. So 'Error' message would result. We had some fun at this point using silly robot voices, which got the students into the problem as a role-play. I let them play around with improving their instructions and then when they were struggling and asking for help I said - Well here's a tool we can use called 'ENV tool' - it's like a screwdriver in a toolbox.

Step 2: Introduced ENV. Explained what ENV meant and asked them to say how we could apply it to the problem before us? So they started examining the parameters of the shapes. Number of sides, number of vertices, width, name, number of lines of symmetry etc. Pretty quickly they noticed that even and odd numbers had a link to the number of lines of symmetry and so on. So one group then started to develop 2 sets of instructions involving folding the shape. Other group worked on different instruction also using folding of the shape.

Step 3: Testing and Fish removal :) : In the end they came up with a very good set of instructions that worked. They gave them to each other to test with random shape pulled from a bank of regular paper shapes I had made.

So I started feeding in irregular shapes... In trouble again... they asked for help and this time I simply reminded them of the ENV tool which they had just used. We reflected at this point also on how the ENV had helped the first time. i.e. rather than look on each shape as a new unique problem look at it as a collection of parameters.

Step1-2: Again: So they looked for other parameters. Some parameters were new - Lengths of the sides etc. -because the shapes were now irregular. This new parameter was the one they used.

Step 3: As they concluded the activity (or so they thought), by testing each others instructions and making recommendations or giving feedback, they realised looking back over their previous versions (which had all been kept) that their instructions had become more sophisitcated and much much shorter. This was a useful observation for them. Students, in maths and science activities in particular, enjoy finding anything that looks like a 'shortcut' or 'trick'. It's insider information really, found through examining the variables and values of the shapes. Their instructions worked. Yay. The instructions included folding the paper shapes.

Back to Step 1!: I forbid them to fold the paper shapes and produced a new bank of laminated shapes :) Cue storm of indignant protest from students. But they still were enjoying themselves as they considered it a challenge and a game by this point (also as another context - I explained about robot war competitions).

Fair play to them! At this point I didn't have to do anything. They went off looking for other parameters on their own initiative. They found some new ones that they hadn't used and made an entirely new set of instructions that would work on any shape at all!

We then moved on to rotational symmetry. It seemed to me that it was easier to explain the 'order of rotational symmetry' element and value.

In the future: I look forward to re-introducing the challenge for solving algebra equations :) As they enjoyed the challenge/game element (they are quite a competitive bunch) I will use the same approach. They will do the task indiviually this time. I will observe and see if they go looking for the parameters by themselves or not.

# Alexander Sokol 2010-12-13 20:04
Deirdre, thanks for a very interesting account. Sounds like it's been a very interesting maths lesson :) I really liked that you introduced the ENV model exactly when the students got stuck. It'd be really interesting to see to what extent the students will remember the tool approach when they face a challenge next time. Do tell us about it.

There are a few things I wanted to ask.

How did you organise the process of 'putting down' the instruction at step 2? If I got you right, all the students had it written as they later on could reflect on it and see the changes. What I am interested in is whether you thought of ways of making these instructions 'findable' next time they need them?

How did the students respond to this meta-level of learning?

If you've got some specific materials you gave the students during the lesson, feel free to upload them under 'materials' section on the site and put a link from your reflective journal. I believe it'd make it easier for some of us to follow.