In the lesson today a boy became very excited.
He felt that his marks had improved – and that he was able to do a lot more on the papers he brought from home.
He explained that he was at the top of the second group and wanted to climb into the top group. He went on to say that there were two groups in the class but he was better at maths than his friend who was in the top group. He also offered, without prompting, that he was more confident.
Naturally the teacher will make the final decision on which group he will enter after the April holidays. The test takes place this Thursday. The class, we understand, will be given their results on Friday and all will know which mathematics group they will be in after the holidays.
We presume that the teacher will mark all the papers, collect the data and then abstract the information that she requires. She will also be able to make her own observations on how well our lad will be able to cope in the top group. There is no doubt that she will go on to make inferences about the probability of him being successful. Inferences, however, are not finite – they can be wrong. We can make an inference, for example, that a woman who wears no ring on the third finger of her left hand is unmarried – but this could be hopelessly inaccurate for many reasons.
We do know why the teacher is referred to as a woman – because this piece of information was offered as part of the conversation!
The teacher could place all the children who are at the bottom of the top group and the top of second group onto one table. She could the gather information about the mathematical output of the children. Was the improved score a `one off’? Has there been steady progress or has do some of the children still seem to lack confidence? We know that the results of the children will vary from day to day so how long does the teacher need to observe the children?
If some of the children in the second group do better, on the same test, as some in the top group can we infer that this position will continue for the rest of the school year? The teacher can look back at performance and results over the whole year – or can take a sample or snapshot of the present position. Logic seems to tell us that the larger the sample the more likely the correct decision will be made. A problem lies in the inherent difficulty of a `one off’ teacher constructed test – the test may not be testing to see whether the child should move from the second group to the top group. The test may be testing something quite different.
If the teacher took two samples or snapshots and looked at the data from both tests then the boy may be able to feel that he had a fairer chance of moving to the top group. If the first sample was very different from the second then the actual act of blending the results may make any decision markedly unreliable.
The obvious excitement of the boy seems to suggest that his enthusiasm is legitimate. We all hope he does well and that he moves up a group. Think, however, of his poor teacher. For him to go up another may need to go down. All we can do is hope and pray that the right decision is made.
One question – can a boy from the top of the second group (of the two groups in the class) improve enough to win a grammar school place? Answer – we have seen it happen!