David Fielker explores children's efforts to grasp geometrical definitions
Sarah Mottley-Harris's class of nine-year-olds had been looking at ways of representing numbers on 2mm-squared paper. The number 243 for example could be represented as two large squares of 100, plus four rows of 10, plus a row of 3. One intention was to reinforce ideas about place value, but if you leave children free to explore different representations then you get more variety than you might expect. Another idea is to concentrate on rectangles made by different numbers, and thus explore their factors.
I asked them what rectangles you could make from 10 squares, and they suggested 2 by 5, 5 by 2, 1 by 10 and 10 by 1.
What about 100? We quickly had 50 by 2, and 2 by 50, and then Sheban offered 10 by 10. "That's not a rectangle," the others said; "it's a square."
At times like this a teacher has to make a quick decision. Should I ignore this objection, or overrule it, and proceed with whatever sketchy lesson plan I had in mind? Or should I go along with what I knew from previous experience was going to be a geometrical discussion about definitions?
If I had ignored it, there would probably not have been such a lively lesson.
Part of my previous experience had been discussions with some of the children on this question, and in fact John remembered it, although not accurately.
"Last year," he said, "you told us that a square is a special sort of rectangle."
I had not told this to the class, but the class had, after a lengthy discussion, come to this conclusion. However, children's perceptions of authority are hard to overcome. I felt that I could tactfully bypass John's mis-perception with a mere "Did I really?" and continue with a similar discussion.
"What's a rectangle?" I asked.
"It has two long sides and two short sides," was a popular idea.
"It's symmetrical," said Molly.
Decisions again. Molly, like several others, was new to the class, and I did not know what her experience of symmetry was. I could pursue it with Molly and find out. I chose not to, and for the moment merely accepted this interesting idea without commitment.
"It has even sides," was another suggestion. I decided to query this incorrect language, and others corrected "even" to "equal". Numbers could be even. Sides could be equal.
"OK," said someone now already at the board and drawing a rectangle, "that and that are equal and that and that are equal," indicating opposite sides.
Jakob was more sophisticated: "It has equal sides, and equal angles - all 90x," he added. Jakob is always perceptive in an original way; it is rarely that anyone describes a rectangle as having equal angles.
I wanted to ask whether a square also had equal sides and equal angles -of 90x, but events were bypassing me as children came out uninhibitedly to the board.
Nicole pointed to the rectangle and said, "You can turn it round," and she motioned a half turn.
"Can you do that with a square?" I asked. Yes, they agreed. But you could turn a square round more ways. Again I was stopped from pursuing the details by further suggestions.
Matt said, "You can fold it in two ways." He demonstrated in the air with an imaginary piece of paper. I hastily tore a piece off a sheet of paper to make it approximately a square, and asked whether that could be folded two ways. "Yes," they said, and after a pause, "but you can fold it in two other ways as well."
"What do we call these fold lines?" "Lines of symmetry," several said.
"So," I said to Molly, "you were right, were you?" There was more lively discussion, the details of which I have forgotten. Somehow we managed to agree - I think - that a square fulfilled any of the various requirements of a rectangle, even if it fulfilled more besides, and therefore was indeed a "special sort of rectangle".
We could now return to Sheban's 10 by 10, and agree that it was a rectangle.
"But you can't turn it round," they said, not as an objection, but as an additional comment. The rectangles which were not squares could be "turned round", for instance 4 by 25 could also be a 25 by 4.
The lesson ended, and Sarah Mottley-Harris could now follow up the numerical ideas, where the idea that a square could not be "turned round", as rectangles in general could, would be important in a consideration of factors of square numbers.
Definitions are difficult for children. I can tell them what a rectangle is, and I can show them that a square fulfils these conditions, therefore - logically - a square is a rectangle. But this is no good if a square does not feel like a rectangle. Feelings are stronger than logic.
It might help if early on children were not presented with pictures saying "this is a rectangle" and "this is a square" (with sides horizontal and vertical) with an implication of mutual exclusivity, though of course that would then preclude this sort of discussion!
Generally, mathematicians prefer to define things inclusively, so that squares are included in rectangles which are included in parallelograms which are included in quadrilaterals, and isosceles triangles include those which are equilateral, because if three sides are equal then it follows that two sides are. But children, and the general public, prefer exclusive definitions.
Another problem is coping with the difference between the meaning of a word in "everyday life" and its meaning within mathematics. Rectangles and squares are different in the "real" world, and furthermore a square turned on its corner becomes a "diamond".
With a class of 10-year-olds we had a similar sort of discussion about the meaning of the word diagonal. In the "real" world this generally means "not horizontal or vertical". In mathematics diagonal is a noun rather than an adjective, and I can turn a rectangle so that its diagonal is either horizontal or vertical: A definition as "a line which joins two corners" worked to some extent. However, when I drew this quadrilateral: and asked how many diagonals it had, the internal one was readily accepted, but not the external one. Somehow it did not feel right!
David Fielker is a freelance writer and lecturer, and teaches part time at the American Community School, Egham. His book Extending Mathematical Ability Through Whole Class Teaching has recently been published by Hodder Stoughton, price Pounds 12.99