Concrete, abstract, formal, and systematic operations as observed in a "Piagetian" balance-beam task series

We performed a Rasch analysis of cross-sectional developmental data gathered from children and adults who were presented with a task series derived from Inhelder's and Piaget's balance beam. The partial credit model situates both participants and items along a single hierarchically ordered dimension. As the Model of Hierarchical Complexity predicted, order of hierarchical complexity accurately predicted item difficulty, with notable exceptions at the formal and systematic levels. Gappiness between items was examined using the saltus model. A two level saltus model, which examined the gap between the concrete/abstract and formal/systematic items, was a better predictor of performance than the Rasch analysis (chi square = 71.91, df = 4, p < .01).