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People report that they use mental imagery to reason about a wide variety of problems. Mental imagery helps develop abilities in memory, problem solving, creativity, emotion and language comprehension (Ganis et. al., 2006).
The mental images we construct for ourselves can be visual: will that box fit on the shelf? Auditory: what does the doorbell sound like? Tactile: what does your favourite pair of slippers feel like on your feet? And so on. Is it important to encourage imagery whilst teaching geometry (and maths more generally)? If so, how can we do it so that the depth of understanding is increased? What kinds of tasks can be used and when?
The first question seems to be an obvious yes, but maybe centred around visualisations rather than a full range of senses. However I did once have a pupil describe acute angles as sharp and obtuse ones as blunt: was he thinking about the sensation caused if he touched them? (He was spot on in terms of etymology – see Ed Southall’s lovely talk about mathematical words). Key to developing geometrical thinking is to encourage pupils to strengthen their powers to form mental images and then to express what they are imagining in words and diagrams (Johnston Wilder and Mason, 2005). Here are a few simple tasks to encourage pupils to visualise and then share the images they have constructed.
Imagine a square of coloured paper, one side red and the other green. Fold your square in half with a single crease. What shapes are possible? Draw all the top views of the folded shape that you can. Can you do it so that you can see a bit of each colour? How? (Hint: I said in half but not necessarily along mirror lines)
What if I did this with a rectangle? Rhombus? Parallelogram? More folds?
Here’s a shape I folded in half, what did it start life as?
On a recent holiday I visited Egypt and toured the Great Pyramids of Giza. I took a number of photos whilst there. Sketch a picture in 30 seconds of a photo of a single pyramid I could have taken.
Here are three examples – where was I standing compared to the pyramid when I took each one?
Cutting Folded Paper
Imagine a square of paper. Fold it in half horizontally and then again vertically. Cut the corner off where the folds meet. Open up your paper. What shape hole have you created? Now try it yourself.
Imagine three identical long rectangles. Each rectangle’s horizontal length is much more than it’s vertical.
a) Keeping the folds vertical fold the first rectangle in half, half again and half again. How many creases in total is that? Cut a triangle out of the right edge and a rectangle out of the left hand side of the folded paper.
b) Keeping the folds vertical fold the second rectangle like a fan. Make 6 folds. Cut a triangle out of the right edge and a rectangle out of the left hand side of the folded paper.
c) Roll the third triangle into a tube and then flatten it using your hand. Cut a triangle out of the right edge and a rectangle out of the left hand side of the folded paper.
Predict what each rectangle will look like when you unfold them. Now try each yourself, unfold each, were you right?
Above all we need to give pupils opportunities to think and imagine for themselves, a moment where they picture the problem, visualise it themselves and represent it however they want with no outside influence. Through sharing these visualisations in words, diagrams or models, pupils can perfect their descriptions and begin to work on their mathematical communication.
Can you come up with an interesting task to help pupils construct mental imagery? Tweet it to us @CambridgeMaths.
SOMETHING TO TRY:
KS1: Close your eyes and imagine four balls in a row. The first ball is yellow, the second black, the third is green. And the last one is red. Which balls are next to the green ball? Which ball is next to the yellow ball? What about if we take the black ball away – what changes?
KS2: I folded a piece of paper in half. Its picture is below. What shape was the paper to start with?
KS3: I have a paper cup. If I undo the seams of the cup what shape will the pieces be?
KS4: A cube is hung by one vertex and lowered into a container of water. What shapes does the cross-section form as it is lowered into the water?
KS5: The follow region is rotated around the y-axis. Describe the shape it forms.
Ganis, G., Kosslyn, S. M., and Thompson, W. L (2006) The Case for Mental Imagery, Oxford, Oxford University Press
Johnston-Wilder, S and Mason, J. (2005) Developing Thinking in Geometry, London, Sage Publications Ltd