Imagination takes different forms for different people. In our minds, we can form and manipulate images of past, future or purely abstract objects and occurrences.
In The Mind Teaches the Brain, Gattegno describes the importance of imagination:
The imagination says that we can make the potential actual…
The imagination can supply a future for the present... This future, being possible, does not need to be actualised, or even to be probable, for the self to entertain it and find it valuable in its education…
Imagination is needed to recognize boundaries and their artificiality, and to remove them through invented exercises that bring to the center of awareness what needs to be done...
Work with the imagination may be one of the most important aspects of mathematics education.
But how often are students allowed time to imagine? If a student is sitting looking into space, it is often assumed they are doing nothing. I have often heard teachers advise seemingly 'inactive' or 'stuck' students to 'do something', meaning to write something down.
But this might be counter-productive when solving problems. In his book Creative Thinking, JG Bennett suggests putting pen to paper may inhibit creativity, the ability to think of something new:
Sometimes when people set themselves to understand something… they proceed to get all the relevant information down in front of them. They write it down on a sheet of paper… But this is not at all sufficient. It is necessary that one should do that work of selection, elimination and assembly (as far as possible) inwardly…
If one wants to bring thinking into the sphere of creativity, this preparatory work has to be done in a special way… I sit down and try to put it together in my mind.
Bearing this in mind (!), try to solve the problem below without putting ever pen to paper, even if you are very tempted!
[If possible, you may also find it interesting to notice how you form and manipulate images in your mind]
