4.2 Some possible causes

In recent years English school mathematics has seen a marked shift of emphasis, introducing a number of time-consuming activities (investigations, problem-solving, data surveys, etc.) at the expense of `core' technique. In practice, many of these activities are poorly focused; moreover, inappropriate insistence on working within a context uses precious time and can often obscure the underlying mathematics. Such approaches, if well-directed, have value, but priorities must be agreed. At no stage has there been a serious debate as to which topics or skills are of primary, and which are of secondary, importance for students' subsequent progress. The position was made worse by the recent administrative squeeze which effectively reduced the amount of time available for mathematics by around 20%. This reduction should be urgently reviewed.

During the same period we have also seen implicit `advice' (from HMI (1985) [14], from OFSTED(1994) [22], in the wording of the National Curriculum [20], and from elsewhere) that teachers should reduce their emphasis on, and expectations concerning, technical fluency. This trend has often been explicitly linked to the assertion that ``process is at least as important as technique''. Such advice has too often failed to recognise that to gain a genuine understanding of any process it is necessary first to achieve a robust technical fluency with the relevant content. Progress in mastering mathematics depends on reducing familiar laborious processes to automatic mental routines, which no longer require conscious thought; this then creates mental space to allow the learner to concentrate on new, unfamiliar ideas (as one sees, for example, in the progression from arithmetic, through fractions and algebra, to calculus).

In parallel with these changes in emphasis, evidence that many English pupils were unable to solve standard problems involving, for example, decimals, fractions, ratio, proportion and algebra (Hart (1981) [12], APU (1980-85) [1,2] etc.) was interpreted by many curriculum developers and those responsible for defining national curricula, as meaning that such topics were `too hard' for most English pupils in the lower secondary years. This interpretation ignored two key facts.

(i)

Many of the topics that were postponed, or eliminated, are fundamental to subsequent progress in mathematics, and cannot be neglected without serious consequences.

(ii)

Many countries (where pupils often start school at a later age than in England) managed to teach these topics effectively to a far larger proportion of pupils.

Many `hard' topics must be introduced early if students' subsequent progress is not to be blocked. And it is often the case that, the longer one postpones a first treatment, the more inaccessible a topic will become. Despite recent advances in cognitive psychology, few quotes express the general point more cogently than the following :

``It is not true that the easier subjects should precede the harder. On the contrary, some of the hardest must come first because nature so dictates, and because they are essential to life. The first intellectual task which confronts an infant is the acquirement of spoken language. What an appalling task: the correlation of meanings with sounds! [...] All I ask is that, with this example staring us in the face we should cease talking nonsense about postponing harder subjects.'' (Whitehead (1922) [27]).