A psychometric model called Saltus, which represents the qualitative aspects of hierarchical development in a form applicable to additive measurement, was applied. Both Piaget's theory of cognitive development and Gagne's theory of learning hierarchies were used to establish the common features of hierarchical development: (1) gappiness--the logical construction of the hierarchy which occurs when there is no state between adjacent stages; and (2) rigidity--learning behavior, exhibited by a fixed sequence of progression through stages. Saltus assumes a theory with gappiness expressed through items or tasks, and estimates the rigidity of data, thus testing the hypothesized gappiness. Four experiments tested the Saltus model in elementary school Australian children and with subjects aged 3, 4, 5, 8, 12, and 21 years. Three Piagetian tasks of rule assessment involving proportionality (balance scale, projection of shadows, and a probability task) gave clear evidence of rigidity in the step from the pre-operational stage to the concrete operational stage. The next step, to the formal operational stage, did not show rigidity, although gappiness was evident. The hypothesized existence of a gap to split the concrete operational stage was not supported. The Gagnean data on a subtraction task showed strong rigidity. (Author/GDC)