What might a fraction mean to a child and how would a teacher know?

The teaching and learning of fractions is not only very hard, it is, in the broader scheme of things, a dismal failure. HartÕs (1984) article documents the understanding of fractions by children in the 11 - 16 age range: the results are depressingly negative. What could be at the bottom of this difficulty to learn fractions, or to teach them well? The research literature on fraction learning is extensive. However it is painfully clear that this literature falls into small subgroups, with authors from one subgroup rarely citing the results of others. The situation is reminiscent of that in a branch of biological science described by Crick (1988, p. 158): ÒIt tended to fall into a number of somewhat separate schools, each of which was rather reluctant to quote the work of the others. This is usually characteristic of a subject that is not producing any definite conclusions.Ó This is not to attribute blame to the dedicated and capable people who work in this area: it is our belief that it simply indicates no real progress is being made.