We are on a remote sheep farm. A one track road runs between the main road and the farm buildings. There are high hedges on either side of the road. The farmer is driving his sheep from one field to the other. He hears the sound of a van approaching. He can not turn his sheep around easily. He has to make a decision. Twist or turn.
The farmer is well used to country life. He has been driving his sheep along this road for many years. He calculates the speed of the van, looks at his sheep – and whistles to his dog. The dog bounds ahead and edges the sheep into the one little hollow in the road. The sound of the van grows closer. The dog waits, the sheep wait and the farmer waits.
The van drives past – now travelling slowly. The driver glances at the dog and the sheep, and waves to the farmer. The farmer inclines his head. He knows that his delivery of the post will bring good news.
If the sheep had been driven by a townsman it is unlikely that the dog would have been given the right commands. If the sheep had been upset by the speed and the disturbance caused by the vehicle then they may not have been prepared to wait patiently. If the farmer had not trusted his dog then he may have asked the driver to reverse or he may have turned his sheep.
The farmer showed that he had the ability to reason and solve a problem. His reasoning was fortified by his past experiences of the road, his dog and his sheep. The one unknown was the reaction of the driver. The driver had to reason that he too knew the road, and that he was expecting the farmer to be driving his sheep at around the same time every day. The delivery of the post and the passage of the sheep were exclusive activities – but the outcome was united by reasoning.
The farmer, while he waited for the post, was probably doing an eleven plus question in his head. His daughter, you see, was writing her eleven plus. As he watched his sheep waiting patiently he thought to himself:
The sum of the series of consecutive odd numbers, beginning with unity, is always a perfect square. Thus 1, 3, 5 and 7 give 16 – which is the square of four.
The farmer then had a choice. He could urge his sheep to move towards the gate – so that they would be off the road by the time that the post van returned. He could also wait for the postman and then drive his sheep on.
Should he try another trial and see if the same rule held out? 1, 3, 5, 7 and 9 gives 25 – and the square root of 25 is 5.
The farmer then has to do a little more reasoning. He has been able to confirm the numbers rule. He is confident that he can explain the rule to his daughter. He knows that if his daughter can demonstrate true versatility of reasoning she should be able to pass the eleven plus. He knows too that if he makes his daughter go on and on working on this question she will possibly become bored and demonstrate a complete lack of sensitivity. The farmer has to be aware of his daughter’s boredom thresholds. A characteristic of some bright children is a lack of patience.
The farmer waits for the van to pass. His dog drives the sheep into the field. The farmer closes the gate. He walks back to the house, fondling his dog as he walks along. He thinks of a cup of tea. He is happy – he has had another delivery of eleven plus papers. The eleven plus is one step closer.