ALAN: A CASE STUDY OF KNOWLEDGE OF UNITS AND PERFORMANCE WITH FRACTIONS

involving units of different sizes in interpreting fractions. His solutions to equivalence problems, produced using inappropriate figure-based algorithms, were unsupported by any coherent method of verification. The mental processes known to be available to Alan for dealing with unit systems are contrasted with the extent to which the processes were reflected in his solutions to fraction problems. This article provides an interpretation of the behavior of a fourth grader, Alan, in his attempts to solve a set of partition problems and a set of fraction problems. The two sets of problems were structurally similar with respect to the types of units in the problems. Evidence of solution strategies that Alan used to relate units when solving partition problems involving discrete quantities was sought in his solutions to certain fraction problems. Underpinning the study of Alan's solutions is the assumption that the individual child actively constructs mathematical knowledge. Such knowledge is acquired through the development and reorganization of cognitive structures that allow children to interpret and control quantitative aspects of their environment. Since basic fraction concepts are the seedbed for many important mathematical ideas-including notions of equivalence, inverse, decimals, probability, ratio, and proportion-the mental mechanisms that support a knowledge of fractions need to be understood. Control over these numbers can allow a child to solve a great variety of quantitative problems. THEORETICAL RATIONALE Integral features of the problems designed to investigate aspects of Alan's knowledge of partitions and fractions were the types of units and their relationships. McLellan and Dewey (1895), in their classic work The Psychology of Number, were careful to distinguish between primary units and derived units, both of which they considered important in constructing a knowledge of number: "The psychical process by which number is formed is