The old 5-14 assessments used to give robust, consistent answers to specific questions. The questions were of the form "what percentage of P7 students have managed to score at least 15 marks in one of a specific set of nationally generated assessment instruments". We thought that the questions were of the form "what percentage of P7 students have passed level D mathematics", but of course that isn't what we were measuring. We were measuring performance in tests. The mistaking of test performance for evidence of "passing a level" was bad, but we were right in thinking that we compared like with like as we looked at performance data from across Scotland.
Now we have abandoned national tests, and have moved towards measuring "passing a level" based on a combination of a rich bundle of assessments and teacher professional judgement. But what does it actually mean to have passed level 3 in numeracy? CfE documentation talks about assessing breadth…
"If A then B" is logically equivalent to "If (Not B) then (Not A)". For example "if a shape has three sides, then it is a triangle" is logically equivalent to "if a shape is not a triangle, then it does not have three sides". The second statement in quotes is called the contrapositive of the first statement in quotes. Logically speaking, they are identical statements. If one is true, the other must be, and vice versa. In this case, they are both true statements.
A commonly touted inspirational message we deliver to young people can be distilled down to "if you do all the things right that are within your power, then your dreams will come true". Consider its contrapositive.
Several months ago I found myself sitting beside a quite senior staff member from one of our colleges of initial teacher education, who complained to me that her student teachers all seemed to be obsessed with behaviour management.
I was flabbergasted, but failed miserably to put together a reasonable argument for why the students were quite right.
That conversation came back to me today as I taught a maths cover class, who were learning about the graphs of linear equations. The class had some lively characters in it, and they were clearly struggling with the central concept: that a line on a coordinate diagram represents all the points where a particular linear equation involving x and y is true.
As they explored this concept through a series of activities, many of them experienced confusion and frustration. These are normal, healthy emotions for learners. We were able to stick with these challenging experiences partly because the pupils were operating in an environment in which mis…