Success and Failure in Arithmetic and Algebra

Arithmetic and algebra are central parts of the National Curriculum throughout the British Isles. In both of these there is a known and disconcerting level of failure, in algebra even more than in arithmetic. Why does this failure occur? Or, to look at the more positive side of the coin, why are some fortunate souls able to do these parts of mathematics almost without effort. By the conservation of energy, if there is no effort, there can be no work done. So, by implication, there must be some other source of energy within the more able that drives their success. Research performed recently at Warwick (Thomas 1988, Gray 1991a, Gray 1991b, Tall & Thomas 1991, Gray & Tall, 1991) reveals that there is indeed a qualitative difference between the thinking processes of those who succeed and those who fail, a difference that makes the mathematics easier for the more able and harder for the less able, exacerbating the chasm between them. The more able develop a way of thinking that fires an inner engine with a feed-back loop creating new knowledge from old, the less able seek solace in being able to carry out procedures that may be successful in the short term but are likely to lead to long-term failure.