Weaknesses in 5-14 maths teaching, the subject of unprecedented attention in the past couple of weeks, may be due to the fact that teachers are pressing on too quickly to the next stages.
This is one of the main conclusions of the latest survey from the Assessment of Achievement Programme, published yesterday. It warns that pupils need to be more "secure" in their performance before moving on.
The AAP report tested 15,500 pupils in P3, P5, P7 and S2 in almost 850 schools in May and June of last year.
The results are slightly different this time, testing P3 and P5 pupils instead of P4. Comparisons with previous years are therefore not available, although figures for P7-S2 which are comparable bring no more comfort to ministers other than that "performance is broadly static", according to a Scottish Executive spokesperson.
Some of the figures show a familiar pattern of declining attainment as pupils move up through the 5-14 stages. One in 10 pupils failed to demonstrate basic skills in P5 and P7 at levels B and C respectively, which fell to a fifth of S2 pupils with similarly low attainment at level D (although most should be at level E by the end of S2).
"Pupils' enthusiasm for mathematics learning decreases with increasing age," the report notes.
The commentary with the results calls on teachers to be more "inventive" in how they follow the 5-14 guidelines. It stresses "the need to strive for 'secure' knowledge and 'considerable strengths', favouring depth of treatment in the earlier stages over any pushing on prematurely.
"This calls for teachers at all stages to interpret the 5-14 guidelines in the fullest way to challenge the pupils in their care. There is scope for more inventive and extensive interpretation of the guidelines when it comes to reviewing examples within any strand, such as time, fractions and percentages. The aim should be for 'secure' knowledge and understanding or 'considerable strengths' to be demonstrated."
The AAP defines a "secure" performance as one where a pupil gets at least 65 per cent of the test items correct and "considerable strengths" where 80 per cent or more of the answers are correct.
The report says the findings have shown the need for more staff development to ensure teachers know what "consolidation" of learning might mean. This would ensure that "staff are comfortable with their interpretation of the guidelines, going beyond the stated targets and challenging pupils . . .
and resisting the temptation to accelerate prematurely towards the higher levels".
The survey says this problem is illustrated during the first two secondary years where able mathematicians are said to be well catered for, but the number showing "secure" attainment has declined. Some pupils who struggle and achieve only basic skills at level D (getting at least 50 per cent of the test items right) "find themselves in considerable difficulty as they encounter the greater challenges of level E work".
The results also include those for reading and writing in maths tests and the use of ICT, which is said to be under-utilised. There were no gender differences in pure maths attainment at any stage, but girls did better than boys in reading and the quality of writing at most stages.
* More than 95 per cent of pupils had at least basic skills at level A, just under 90 per cent were "secure" or better at this level and almost 70 per cent showed considerable strengths.
* At level B, the corresponding proportions were more than 60 per cent, around one-third and just under 10 per cent.
* 90 per cent of pupils demonstrated at least basic skills at level B, more than 70 per cent were at least "secure" and just over 40 per cent showed considerable strengths.
* At level C, the corresponding proportions were 65 per cent, 40 per cent and 15 per cent.
* At level D, the figures were around 25 per cent, just under 10 per cent and 2 per cent.
* Just under 90 per cent of pupils gained at least basic skills at level C, almost 75 per cent were "secure" and more than 40 per cent showed considerable strengths.
* At level D, the corresponding proportions were almost 70 per cent, more than 45 per cent and more than 15 per cent.
* At level E, the three categories were 35 per cent, almost 20 per cent and just over 5 per cent.
* 80 per cent of pupils had basic skills at level D, 60 per cent were "secure" and 30 per cent showed considerable strengths.
* At level E, the corresponding proportions were just under 60 per cent, almost 40 per cent and just under 20 per cent.
* At level F, the figures were more than 15 per cent, more than 5 per cent and 2 per cent.