1. Mathematics is a hierarchical subject where conceptual
understanding of new ideas is very dependent on mastering earlier ideas.
This is true from the earliest stages of learning mathematics and involves much
more than the procedural knowledge needed to produce correct answers to routine
questions. Mathematics is very different from most other school subjects where
students can make sense of a wide range of topics at different levels of sophistication
and understanding. Given a certain level of literacy anybody can appreciate
to some extent a novel, a description of a historical event or the geographical
features of their locality. Similarly the production of artefacts in design
and technology and art is not wholly dependent on mastering a hierarchy of ideas.
There are strong parallels with science, because it is very dependent on mathematical
understanding, and to a lesser extent with learning foreign languages, where
there is a similar continuous building up of knowledge of vocabulary and structures,
although without such a strong element involving conceptual understanding of
ideas.
2. Assessment procedures, through the style of questions, the frequency of
tests and the importance of the results, have a profound effect on the way in
which mathematics is taught. Clearly at some point some summative assessment
is required to provide appropriate certification for employment and higher education,
but that does not necessitate the current excessive emphasis. Revising assessment
procedures so that they are more conducive to effective long term learning is
vital, but that is not the purpose of this short paper which has the more limited
aim of showing why reducing or eliminating the impact that league tables and
targets have on mathematics should be a high priority.
3. National Curriculum levels are largely described in terms of content. Similarly
the range of content differs in the three tiers of GCSE entry, which are linked
to the range of grades that can be attained. In general the system equates level
of difficulty with the perceived difficulty of particular items of content rather
than with the difficulty of the problems that students are expected to solve
using that content. There is clearly a hierarchy of content difficulty, but
it is also obvious that the same content can lead to problems of widely differing
levels of difficulty.
4. The development of fluency, understanding and application is discouraged
by the intense pressure, massively exacerbated by league tables and targets,
to attain the next National Curriculum level or a higher GCSE grade. This is
particularly acute at the C/D GCSE grade boundary, where, for example, it is
a common for both teachers and students to concentrate on data handling and
neglect algebra, because that is perceived as the best tactic to maximise marks.
Most students would be much more positive about mathematics and find it a much
more rewarding subject if new ideas were not forced on them before understanding
of earlier ideas was really secure. It would be more productive for most students
to spend longer using and applying the ideas at level n rather than moving on
to level n+1 immediately. The phenomenon of moving to new content too quickly
is a feature of mathematics teaching right across the ability spectrum.
5. For example, nearly all students encounter addition of fractions in year
7 and fail to master the ideas because their underlying knowledge from year
6 is insecure. They then meet them again in successive years with further failure
each time leading to increasing distress and disillusion. This repetition is
the common experience of a wide range of students across a wide range of ideas
in the mathematics curriculum at all levels (including at degree level!), whereas
in most other curriculum subjects, including science, students can make some
sense of each new topic at their own level and are not constantly repeating
ideas that they have failed to grasp.
6. The nature of the whole system of assessment, long standing traditional
styles of mathematics teaching and our whole cultural attitude to learning mathematics
all bear upon these problems and cannot easily be changed. However, the effect
of the additional pressures created by league tables and targets is particularly
serious for mathematics because of its hierarchical nature and its role as a
basic entry qualification to employment at all levels.
7. Abandoning league tables and targets would have the virtue of both saving
money and immensely boosting teacher morale, but it would involve a major change
in attitude from those in government. That should not prevent it being a very
strong recommendation of the Inquiry, because progress will not be
made until there is recognition at a political level that league tables and
targets, whatever the intentions behind them, are one of the very real barriers
to progress in raising standards in mathematics. Putting more energy
and resources into the vital task of improving the quality of pedagogy through
CPD and in other ways will only bear fruit if some of the barriers to progress
are removed. The Inquiry should have the courage to push for measures which
require a change in the political climate as well as making recommendations
with more immediate political appeal. Wales has recently abandoned league tables
and targets: why should England not follow this eminently sensible lead?
PS The recent announcement from the Secretary of State with regard to league
tables and targets in primary schools, whilst welcome as an indication that
some of the concerns have been recognised, is inadequate because it does not
remove the problem and it does not apply to secondary schools.
28.5.03.
