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).

Using Rasch scaled stage scores to validate orders of hierarchical complexity of balance beam task sequences.

These studies examine the relationship between the analytic basis underlying the hierarchies produced by the Model of Hierarchical Complexity and the probabilistic Rasch scales that places both participants and problems along a single hierarchically ordered dimension. A Rasch analysis was performed on data from the balance-beam task series. This yielded scaled stage of performance for each of the items. The items formed a series of clusters along this same dimension, according to their order of hierarchical complexity. We sought to ascertain whether there was a significant relationship between the order of hierarchical complexity (a task property variable) of the tasks and the corresponding Rasch scaled difficulty of those same items (a performance variable). It was found that The Model of Hierarchical Complexity was highly accurate in predicting the Rasch Stage scores of the performed tasks, therefore providing an analytic and developmental basis for the Rasch scaled stages.