A question of performance
However, monitoring of nationally representative samples of pupils still takes place in Scottish schools at three-yearly intervals under a national assessment of achievement programme (AAP). Four surveys of mathematics have been carried out since 1983. So in Scotland we do have recent data and can draw some conclusions which may throw light on the debate about "standards".
The findings relate only to the primary and early secondary years, but if things are going wrong in school maths, it helps to look at these formative stages.
The pupils targeted by the surveys are in Primary 4, Primary 7 and Secondary 2, aged 8, 11 and 13 approximately. The last one took place in May-June 1994, when about 2,500 pupils in primary and 4,000 in secondary took part in a written component. About half of the 180 primary and 140 secondary schools also did some practical exercises.
The findings are grouped under AAP categories, which relate closely to the 5-14 curriculum guidelines in Scotland. These cover attainment targets, or "outcomes", of information handling, number, money and measurement, shape, position and movement, and problem solving.
So how are we doing in Scotland? In the national guidelines, attainment targets have been written at five levels to cover the progression expected from the age of 5 to 14, with A being the lowest and E the highest.
There are also additional targets for the most able pupils. Each survey booklet contained a few items below the expected level of attainment and one or two advanced ones to see what pupils could achieve.
The national guidelines recommend that "most pupils" should achieve level B in Primary 4, level D in Primary 7 and level E in Secondary 2. In the absence of precise definitions of "most", we have taken it to mean at least two-thirds of the sample.
The findings of the survey revealed that some attainment targets were being achieved comfortably, but others were being managed by only a few pupils, most notably in number, money and measurement, and shape, position and movement.
In Primary 4, most pupils managed almost 60 per cent of the items at the expected level B, and more than a third of the items at the higher level C. In Primary 7, most pupils managed less than 40 per cent of the items at the expected level D, about 75 per cent at level C and just over 25 per cent at the higher level E.
In Secondary 2, most pupils achieved less than 20 per cent of items at the expected level E, less than half at the lower level D and 75 per cent at level C. Most managed just 3 per cent at level E+.
A further breakdown of data showed how pupils were doing in particular categories: in Primary 4, achievement at level B was excellent in information handling (100 per cent of the items were achieved by more than two-thirds of the pupils, but weak in number concepts and basic processes (only 46 and 41 per cent respectively were achieved by more than two-thirds of pupils).
In Primary 7 at level D not one category showed particularly high achievement by most pupils. Performance was weakest in basic processes (only 23 per cent of items were answered correctly by most pupils) and applications of number, money and measurement (only 34 per cent).
In Secondary 2, the highest achievement at level E was in information handling, where 67 per cent of items were answered correctly by most pupils. There was a fairly low level of success for number concepts (only 20 per cent of items achieved by most pupils), basic processes (13 per cent) and applications (10 per cent). For shape, position and movement, only 18 per cent of items were successfully completed by most pupils.
In general, pupils' achievements did not measure up to the standards set in the national guidelines, if our definition of "most pupils" is reasonable. So unless the teaching profession considers that there is a significant mismatch between targets and levels, it would seem that current performance is not good enough.
We compared data with previous surveys from 1991 and 1988: in Primary 4, between 1991 and 1994 performance apparently dropped in five of the 10 categories compared, and between 1988 and 1994 performance on basic processes and applications fell significantly. In Primary 7, between 1991 and 1994 performance fell in only one category (basic processes - decimals, fractions and percentages), but between 1988 and 1994 there was a significant drop in two categories of basic processes - decimals, fractions and percentages, and whole numbers arithmetic, as well as in number concepts.
In Secondary 2, between 1991 and 1994 there has been a significant fall in four out of 14 categories, again in both sub-divisions of basic processes, and in applications of decimals. Between 1988 and 1994, performance dropped in both categories of basic processes, applications of decimals and angle.
It appears that performances in some basic aspects of mathematics have been dropping significantly over time, and that most pupils are not reaching the attainment targets set in many areas of the curriculum.
But specific issues need to be addressed. In the national guidelines in Scotland there are attainment targets for carrying out basic processes with and without a calculator. This survey identified a fall in performance in those ones performed without a calculator.
Calculators are introduced early into Scottish schools - a quarter had introduced them at Primary l and a further half at Primary 3. Most schools had introduced them by the end of Primary 4. So, are there any adverse effects in introducing them so early? Do we need to pay more attention to what they are used for?
By contrast, pupils at all stages, even as young as eight, were extremely accurate in their use of calculators, and targets set in the guidelines for using them seemed unrealistically low. However, appropriate rounding to use the result obtained was not carried out with the same mastery. Pupils found estimation difficult, but few appeared to make errors in using a calculator. So, is it worthwhile continuing to stress rounding and estimation as targets for most pupils?
Pupils in Secondary 2 had difficulty understanding and using fractions and percentages. Only those in common use were well understood. If greater attention was paid to place value in early years, it might be advisable for teachers to adopt a decimal approach to teaching percentages.
Some aspects of number concepts caused particular problems for pupils - are concepts such as ratio, speed and formulae being introduced too early? Algebra also gave some cause for concern. At present, relatively little algebra is taught at Secondary 2. Should more time be spent on this and other areas of difficulty in Scotland?
Finally, we also identified several issues concerning education policy and teachers' practice in the early secondary sector, which need more debate: is more intervention and teaching by teachers required, particularly at Secondary 2? Should a teacher's role as manager of resources be played down and resources relegated to support roles? Should mixed-ability groups be recognised as generally unsuitable for maths teaching? In Scotland, what should be the role of the new education authorities (and indeed, Her Majesty's Inspectorate of Schools) in determining such policies?
Isobel Robertson is a lecturer in the department of mathematics, science and technological education, University of Strathclyde. The AAP maths survey was carried out in the department by Isobel Robertson and Bert Meechan (project director), with Dave Clarke and Jennifer Moffat