Hierarchical Complexity and Difficulty DARE Home层次复杂性和困难敢回家.doc

Hierarchical Complexity and Task Difficulty Michael Lamport Commons and Alexander Pekker Harvard Medical School and Harvard University Michael Lamport Commons, Ph.D. Assistant Clinical Professor of Psychiatry Program in Psychiatry and the Law Department of Psychiatry Harvard Medical School Beth Israel Deaconess Medical Center 234 Huron Avenue Cambridge, MA 02138-1328 Telephone 617-497-5270 Facsimile 617-491-5270 Commons@tiac.net Alexander Pekker, Ph.D. Department of Mathematics Harvard University Cambridge, MA 02138 Abstract Theories of difficulty have generally not addressed hierarchical complexity of tasks. Within developmental psychology, notions of hierarchical complexity have come into being in the last 20 years. We show how a model of hierarchical complexity, which assigns an order of hierarchical complexity to every task regardless of domain, may help account for difficulty. The orders correspond to natural numbers, thus insuring that the orders are separated by gaps. The model naturally leads to the existence of performance stages, thereby formalizing many implicit properties of stage theories. Key Words: Task Difficulty, Mathematical Theory, Complexity, Hierarchical Complexity, Distributivity, Rasch, Scales, Stage, Ordinal Measure  If we could understand why people fail at tasks, we might be able to offer a better framework for success – particularly success in areas where under normal circumstances people would have otherwise failed. An understanding of how to succeed could have substantial implications for example, for education reform or improving job performance. Because relative task failure rates can indicate empirically how difficult a task is to complete, a good framework for success would stem from an analysis of task characteristics that contribute to difficulty. Tasks are defined as sequences of contingencies, each presenting stimuli and requiring behaviors that must occur in some non-arbitrary fashion (Anderson, 1980). Tasks may