Lateral Thinking Read online

  Building up from smaller units.

  Comparing to another figure.

  Modifying another figure by addition or subtraction.

  As before one may have to draw additional diagrams to show what is meant If one cannot understand oneself what the student means then he is asked to explain it himself.

  3. How would you describe the figure shown below?

  Alternatives

  Two overlapping squares.

  Three squares.

  Two L shapes embracing a square gap.

  A rectangle divided into half with the two pieces pushed out of line.

  Comment

  The ‘two overlapping squares’ seems so obvious a description that any other seems perverse. This illustrates how strong is the domination by obvious patterns. Once again it may be felt that ‘two squares overlapping’ is the same as ‘three squares’ since the latter is implied by the former. This is a tendency that must be resisted because often even a minor change in the way a thing is looked at can make a huge difference. One must resist the temptation to say that one description means the same thing as another and hence that it is just quibbling.

  There may be elaborate descriptions which seek to be so comprehensive that they cover all possibilities: ‘Two squares that overlap at one corner so that the area of overlap is a square of side about half that of the original squares’. Such comprehensive descriptions almost reproduce the diagram and hence must include all sorts of other descriptions. Nevertheless these other descriptions must be accepted in their own right Logically a description may be redundant in that it is implied by another but perceptually the same description may make use of new patterns. For instance the idea of three squares is useful even though it is implicit in the overlap description.

  4. How is the pattern opposite made up?

  Alternatives

  A small square surrounded by big squares.

  A big square with small squares at the corners.

  A column of large squares pushed sideways to give a staircase pattern.

  Basic unit made out of one large and one small square.

  Extend the edges of a small square and draw other small squares on these extended edges.

  A line is divided into thirds and perpendiculars are drawn at each third.

  In a grid pattern some of the small squares are designated in a certain way and outlined and then the lines are removed and the spaces filled with big squares.

  Big squares are placed against each other so that the side of each one half overlaps the side of every adjacent square.

  Two overlapped patterns of lines, one at right angles to the other.

  Comment

  There are very many possible variations other than those listed above. The descriptions offered must be workable. The description should clearly indicate how the pattern is being looked at. What is of importance is the variety of ways the pattern can be treated: in terms of large squares only, in terms of small squares only, in terms of both large and small squares, in terms of lines, in terms of spaces, in terms of a grid pattern.

  Activity

  The examples used so far call for different descriptions of a presented pattern. One can mow on from different ways of looking at things to different ways of doing things. This is rather more difficult since with description it is only a matter of selecting what is already there but to do something one has to put in what is not there.

  5. How would you divide a square into four equal pieces? (For this example it is better that each student tries to draw as many different versions as he can instead of just watching the board and offering a new approach. At the end the papers may be collected if the teacher wants to analyse the results or eke left with the students for them to tick off the various versions.)

  Alternatives

  Slices.

  Four smaller squares.

  Diagonals.

  Divide the square into sixteen small squares and then put these together to give swastika or L shapes as shown overleaf.

  Other shapes as shown overleaf.

  Comment

  Many students at first stick to the slices, diagonals and four small squares. One then introduces the idea of dividing the square into sixteen small squares and putting these together in different ways. The next principle is that any line which passes from a point on the edge of the square to an equivalent point on the opposite edge and has the same shape above the centre point as below it divides the square into half. By repeating the line at right angles one can divide the square into quarters. Obviously there is an infinite number of shapes which this line can have. It may be that some students will offer variations on this principle without realizing the principle. Rather than listing each variation one puts them together under the one principle. A variation on this principle involves dividing the square into half and then dividing each half into half again. For each half any division which passes through the centre of that half and is of equivalent shape on each side of the centre point will do. This introduces a whole new range of shapes.

  Since this is not an exercise in geometry or design the intention is not to explore the total possible ways of carrying out the division. What one tries to do is to show that there are other ways even when one is convinced that there cannot be. Thus the teacher waits until no further ways are offered and then introduces the variations suggested above, one at a time. (It may of course happen that all the variations listed above are introduced by the students themselves.)

  6. How would you divide up a square of cardboard to give an L shape with the same area as the square? You can use not more than two cuts. (Actual squares of cardboard can be used or drawings should suffice.)

  Alternatives

  The two rectangular slices (see figure overleaf).

  The cutting out of the small square.

  The diagonal cut.

  Comment

  The requirement ‘use not more than two cuts’ introduces the element of constraint The constraint is not meant to be restrictive, on the contrary it encourages the effort to find difficult alternatives instead of being easily satisfied.

  Since one is used to dealing with vertical and horizontal lines and with right angles the diagonal method is not easy to find. Perhaps the best way to find it is to ‘cut across the square diagonally and then see where that gets one’. In effect one is beginning to use provocative manoeuvres rather than simple analytical ones.

  Non-geometrical shapes

  Having used the geometrical shapes to illustrate the deliberate search for alternatives (and also the possibility of such alternatives) one can move on to more complex situations. In these more complex situations it is not so much a matter of picking out standard patterns as alternatives but of patting things together to give a pattern.

  7. A one pint milk bottle with half a pint of water in it How would you describe that bottle?

  Alternatives

  A half empty bottle of water.

  A milk bottle half filled with water.

  Half a pint of water in an empty one pint milk bottle.

  Comment

  In itself the milk bottle example is trivial. But it does serve to illustrate how there can be two completely different ways of looking at something. It also shows that when one way has been chosen the alternative way is usually ignored. It is of interest that when the bottle is half filled with milk it is more often described as half empty, but when it is half filled with water it tends to be described as half full. This probably happens because in the case of the milk one is working downward from a full bottle but in the case of the water one is working upward from an empty milk bottle. The history of a situation has much effect on the way it is looked at.

  Pictures

  Photographs from newspapers or magazines are the most easily available source of pictures. The difficulty is to make them available to a large group. This could be done by getting individual copies of a newspaper and keeping them until the material is out of date. If sufficiently skilled the t
eacher could actually draw pictures on the board but this is much less satisfactory. The type of material needed has been discussed in the section ‘Use of this book’.

  Pictures can be used in two ways:

  Describe what you think is happening in that picture.

  Describe three different things that could be happening in that picture.

  In method 1 the teacher uses an ambiguous picture and asks each person to make his own interpretation. At the end he collects the interpretations. Variability between individual interpretations shows the alternative ways of looking at the picture. The teacher is careful not to judge which way is best or why one way is unreasonable. Nor does he reveal what the picture was actually about (he can conveniently have forgotten this).

  In method 2 the students are asked to generate a quota of different interpretations. If the students tend to be blocked by the most obvious interpretation and are unwilling to guess at any others then they may be allowed to list the interpretations in order of likelihood. In addition the teacher throws in some outlandish suggestions about the particular picture being used in order to suggest what is required.

  Examples

  A photograph shoving a group of people wading through shallow water. They are not dressed for paddling. In the background appears to be a beach. The following interpretations were received:

  A group of people caught by the tide.

  People crossing a flooded river.

  People wading out to an island or sand spit.

  Wading through flood water.

  People wading out to a ferry boat which cannot come inshore.

  People coming ashore from a wrecked boat.

  Comment

  In fact the photograph showed a group of people protesting at the poor state of the beach. It was not important that anyone should have guessed this since it was not an exercise in logical deduction. What was important was that there were several different interpretations of what was going on. Apart from noting these variations one should have been able to generate them (even if only to reject them).

  Example

  Photograph of a boy sitting on a park bench.

  Alternatives

  Picture of an inactive or lazy boy.

  An empty space on a park bench.

  Part of the bench is being kept dry by the boy.

  Comment

  The description of this picture is quite different from the other example. There is less attempt to say what is happening (e.g. a boy waiting for his pals, a tired boy resting, a boy playing truant from school, a boy enjoying the sun). Instead the description is directed at the scene itself rather than the meaning (e.g. a boy on a park bench, an empty space on the bench). There is also an attempt to look at the picture in an unusual way. This might have gone too far with ‘part of the bench being kept dry by the boy’ but there really are no limits. In any picture there are several different levels of description: what is shown, what is going on, what has happened, what is about to happen. In asking for alternatives the teacher may leave it quite open at first but later on he specifies the level of description within which the alternatives have to be generated.

  Altered pictures

  The trouble with pictures is that too often the obvious interpretation is completely dominant. Not only is it difficult to find other ways of looking at it but these other ways seem silly and artificial. To avoid this difficulty and to make things more interesting the teacher can alter pictures by covering up parts of them. It immediately becomes far more difficult to tell what the picture is about from the exposed part and thus one is able to generate alternative possibilities without being dominated by an obvious interpretation. There is also the added incentive of trying to guess the right answer which will be obvious when the full picture is revealed.

  Example

  Half of a picture is obscured. What is revealed is a man balancing on the edge of a ledge running along the side of some building.

  Alternatives

  A man threatening to commit suicide.

  Rescuing a cat that has got stuck on a ledge.

  Escaping from a burning building.

  Film stunt man.

  A man trying to get into his room, having locked himself out.

  Comment

  The rest of the picture would have shown some student posters which the man was sticking up. The use of partial pictures makes it easier to generate alternatives but ultimately one wants to be able to restructure pictures in which an obvious interpretation makes it difficult to find alternative structurings. It is especially those situations which are dominated by an obvious interpretation that one wants to practise restructuring. One can use the easier partial pictures, however, to acquire experience. Another advantage of the partial picture is that it indicates that the interpretation may lie outside what is visible. This makes one inclined to look about not only at what is in the actual situation being examined but at things outside it.

  Written material — Stories

  Stories may be obtained from newspapers or magazines or even from books that are being used elsewhere in the curriculum. By story is not meant a tale but any written account.

  Stories may be treated in the following ways:

  Generate the different points of view of the people involved.

  Change what is a favourable description to an unfavourable one not by changing the material but by changing the emphasis and looking at it in a different way.

  Extract a different significance from the information given than that extracted by the writer.

  Example

  Newspaper story of an eagle that has escaped from the zoo and is proving difficult to capture. It is perched on a high branch and is resisting the efforts of the keepers to lure it back to its cage.

  Alternatives

  The keeper’s point of view: the bird may fly away and get lost or shot unless it is coaxed back soon. It is uncomfortable having to climb up trees after the bird and one feels a bit of a fool. Someone is to blame for having let it escape.

  The newspaperman’s point of view: the longer the bird stays out the better the story. Can one get close enough to get a good pictures? One ought to find some other interest such as different people’s ideas on how to catch the bird.

  The eagle’s point of view: wondering what all the fuss is about. Strange feeling not to be in a cage. Getting rather hungry. Not sure in which direction to fly.

  The onlooker’s point of view: hoping the eagle will fly away and be free for evermore. Amused to see the strenuous efforts being made to catch the bird. The eagle looks so much better out on its own than inside a cage. Perhaps one could show how clever one was by catching the bird when no one else could.

  Comment

  Whenever there is a story with different people involved then it is a simple matter to try to generate the point of view of everyone concerned. Every student could try to generate the different points of view or else different students could be assigned to generate the different points of view. The exercise is not so much to try to guess what other people are thinking but to show how the same situation can be structured in different ways.

  Example

  A story describing the uncomfortable life in a primitive community where the people cannot read or write and where only a bare subsistence can be obtained by hard work in the fields.

  Alternatives

  Comfort as a matter of what one was used to. If one was used to simple things and could obtain simple things perhaps this was better than expecting complex things and being dissatisfied when one could not obtain them.

  Perhaps reading and writing only upset people by making them aware of the awful things that are happening in the rest of the world. Perhaps reading and writing make people more dissatisfied.

  Most people are usually working hard at something or other; perhaps hard work in the field is more rewarding since one can actually see something growing and one is actually going to eat what one grows.

  Comment

  Th
e alternative point of view does not necessarily have to be the point of view held by the person generating it The person may actually hold exactly the same point of view as the writer. The purpose is to show that one can look at things in a different way. Nor is it a matter of trying to prove one point of view to be better than the other. There is no question of arguing for instance ‘that the simple community may seem pleasant but if one is ill one must just die etc.’ In practice it is difficult to avoid arguing. It is also difficult to put forward a point of view with which one does not agree. The advantage of being able to put forward an opposing point of view is that one then has much more chance of restructuring it.

  Example

  A story may cite the long hair and colourful clothes of young men as an example that they were being demasculinized and becoming effeminate; that one could no longer distinguish between boys and girls.

  Alternatives

  Wearing long hair shows courage, it shows the courage to defy conventions.

  Until quite recently men always wore long hair as in the Elizabethan era and far from being less masculine they were more masculine. As for the colourful clothes these were flamboyant not feminine. They indicated a masculine search for individuality.

  In any case why shouldn’t boys and girls look alike?

  In that way at least girls would get equal rights.

  Comment

  In this type of restructuring no extra information may be introduced. It is definitely not meant to be a presentation of the other side of the case. The purpose is to show that the material put together to give one point of view can also be put together in a completely different way.

  Problems

  Problems can be generated from the inconveniences of everyday living or by looking through a newspaper. Newspaper columns are full of difficulties, disturbances, things that have gone wrong and complaints. Though these may not actually be stated as problems they can easily be rephrased as such. It is enough that a general problem theme be stated; there is no need to set up a formal problem. Any situation where there is room for improvement can be used as a problem and also any difficulty that can be imagined.