Skills and Knowledge Structures

Suppose that is a set of problems and is a set of skills. A skill function assigns to each problem – i.e. to each element of – those sets of skills which are minimally sufficient to solve ; a problem function assigns to each set X of skills the set of problems which can be solved with these skills (a knowledge state). We explore the natural properties of such functions and show that these concepts are basically the same. Furthermore, we show that for every family of subsets of Q which includes the empty set and , there are a set of (abstract) skills and a problem function whose range is just . We also give a bound for the number of skills needed to generate a specific set of knowledge states, and discuss various ways to supply a set of knowledge states with an underlying skill theory. Finally, a procedure is described to determine a skill function using coverings in partial orders which is applied to set A of the Coloured Progressive Matrices test Raven (1965).