Reflections on a good turn
Where does it lead?" is a frequently asked question by primary teachers as new topics enter the curriculum, and the question can no doubt be interpreted as "Why are we teaching this?" The two questions are not quite the same. Knowing that the reason for a topic to be taught at primary level is that it is going to be used for something more important later on may to a certain extent convince a primary teacher of its validity, but that in itself is not going to motivate the primary pupil, who is asking: "Why are we learning this?" One wonders anyway about the appearance of some of the topics that persist in the new Order. "Reflection and rotation" appear as transformations at key stage 2. They appear again at key stages 3 and 4 with "opportunities to understand and use the properties of" such transformations "to create and analyse patterns, to investigate the properties of shapes, and to derive results, including congruence". This seems a little far-fetched, given the present traditional nature of the rest of geometry, and one feels that reflection and rotation may be a welcome, if incongruous, hangover from the old days of modern mathematics!
For a primary teacher these later requirements may mean little, other than a vague promise of things to come. To pupils at key stage 2 the next key stages are a long way off anyway, and they want something more immediate to get their teeth into.
Another consideration is the method by which these two ideas are introduced. Most pupils at key stage 2 already have these concepts and what is needed is an interesting and purposeful way to identify and formalise them, preferably in a problem-solving context.
Polyominoes is an old-hat topic, but it is always new to the children. And the degree of interest depends on how they are introduced. I give out some squared paper and scissors, and ask the pupils each to draw a shape which is made from four squares. I ask them to cut out their shapes, and to write their names on them. The latter is important: it means I can refer to each shape by its owner's name, and also I know which way up to pin it to a board!
I suggest to a girl who has drawn shape (a) that it will fall apart when she cuts it out. She cuts it out very carefully so that it does not, so I include hers, as well as other shapes where some squares are joined at corners (b) or where half squares are used (c).
We thus obtain a variety of shapes, which I pin up on the board at random, so that any classification can be decided by the class, not by me. As the first shapes are displayed I ask that any new ones be different from those we already have, and then I ask if there are any other possibilities.
We have a discussion about the ones where half squares are used, or whole squares are not joined at the edges. The class agree to postpone consideration of them for the moment, and I move them to one side.
We look at the rest, and I ask if any of them are the same. "The same" is a fairly vague term, like the "different" I have just used, and it is purposely chosen, because I intend that the pupils shall be free to interpret it.
There are some shapes which are obviously identical, in that they are not only congruent but also have the same orientation. This does not need much discussion, and I remove duplicates.
Sean suggests that these two are the same: (d) "Why?" I ask. "How can you show they are the same?" This is in a subtle way a reference back to the congruent shapes that have the same orientation, and I am aware that we are now using the phrase "the same" with two slightly different meanings.
"Well, if you turn Peter's round, it will look the same as Mary's."
"How must I turn it round?" "It has to be upside down."
Now we found ourselves in one of those strange situations that only happen in classrooms. I know exactly what Sean means, even though he has not expressed it very precisely. It is something that happens all the time in everyday conversations.
We express ourselves in vague, imprecise terms, but we are generally understood, because everyone is used to certain conventions about speech, and these understandings are supported by context and by body language. We do not often say exactly what we mean, and it is rarely necessary to do so.
But this is a mathematics lesson, and circumstances are different. There are certain concepts that I as a teacher am trying to clarify in the minds of the pupils, and language is important in helping to achieve this. So I use my customary ploy of defying convention and pretend that I do not understand. I do this by choosing a legitimate alternative interpretation of what Sean has said. I turn Peter's shape thus: "No," says Sean, "that's not what I meant. It has to be turned so that the top comes to the bottom."
Again I can choose a different interpretation of what he has said.
(f) "No. Start again. Keep the paper on the board, and turn it round." I start to turn it to the left. "No, to the right."
(g) "Yes, but further."
"How far must I turn it?" Gradually Sean, and his classmates, become more and more precise about these two shapes, and then about other pairs. They find better words to describe what they mean, and they develop language that is more mathematical. "Upside down" is replaced by expressions to do with turning; "turn" becomes "turn round" or "turn over"; turning round is qualified by an angle; turning over is qualified by a line about which the shape is turned. It is important that the pupils struggle with their own language to achieve this precision, and link it with ideas they already have about angle and bilateral symmetry. When the ideas are clarified, I can introduce words like rotation and reflection.
There is an implied problem about the four-square shapes. Having removed all the duplicates, we want to know if there are any other possibilities, and how many there are altogether. This is a fairly easy problem to solve once the duplicates are out of the way.
We can also, if we like, relax the restriction of edge-to-edge joining and see what other shapes are possible, or consider shapes made from half-square, or go on to shapes made from five squares or more, and for the sake of completeness look at shapes made from three squares or two squares.
But the identification of the duplicates has been an important part of the original problem, and the ideas of reflection and rotation have been crucial to it. It may not have been the first time the pupils have met these two ideas, and it will certainly not be the last. But they have been introduced, or re-introduced, in the context of a problem, as a necessary part of the problem solving process. So the pupils will not have to wait until key stage 3 in order to find out why they are learning about them.
David Fielker is a freelance lecturer and writer