The Logarithmic-To-Linear Shift: One Learning Sequence, Many Tasks, Many Time Scales

The relation between short-term and long-term change (also known as learning and development) has been of great interest throughout the history of developmental psychology. Werner and Vygotsky believed that the two involved basically similar progressions of qualitatively distinct knowledge states; behaviorists such as Kendler and Kendler believed that the two involved similar patterns of continuous growth; Piaget believed that the two were basically dissimilar, with only development involving qualitative reorganization of existing knowledge and acquisition of new cognitive structures. This article examines the viability of these three accounts in accounting for the development of numerical representations. A review of this literature indicated that Werner's and Vygotsky's position (and that of modern dynamic systems and information processing theorists) provided the most accurate account of the data. In particular, both changes over periods of years and changes within a single experimental session indicated that children progress from logarithmic to linear representations of numerical magnitudes, at times showing abrupt changes across a large range of numbers. The pattern occurs with representations of whole number magnitudes at different ages for different numerical ranges; thus, children progress from logarithmic to linear representations of the 0–100 range between kindergarten and second grade, whereas they make the same transition in the 0–1,000 range between second and fourth grade. Similar changes are seen on tasks involving fractions; these changes yield the paradoxical finding that young children at times estimate fractional magnitudes more accurately than adults do. Several different educational interventions based on this analysis of changes in numerical representations have yielded promising results.