CMU-CS-98-186Computer Science Department School of Computer Science, Carnegie Mellon University
CMU-CS-98-186
Christian J. Lebiere November 1998 Ph.D. Thesis
CMU-CS-98-186.ps
Keywords: ACT-R, cognitive arithmetic, Bayesian learning,
activation spreading, dynamical systems, parameter analysis, power law,
machine learning, hybrid systems
While specialized models have been quite successful at accounting for many aspects of cognitive arithmetic, this thesis aims to provide an integrated model of the field using a general-purpose cognitive modeling architecture (ACT-R). This model makes minimal assumptions but instead relies on the architecture's Bayesian learning mechanisms to derive the desired results from the statistical structure of the task. The behavior of this model is analyzed using several approaches: separate simulations of each main result, a single simulation of a lifetime of arithmetic learning, a formal analysis of the model's dynamics and an empirical variation of the simulation's parameters. This thesis provides a unifying account of the main results of cognitive arithmetic. Through its parameter analysis, it suggests some practical lessons for the teaching of arithmetic. The constraints of a lifetime simulation of arithmetic learning also expose the underlying assumptions of ACT-R's associative learning mechanism. While a simplifying assumption commonly used in machine learning is shown in this case to be inadequate, a more powerful algorithm closely replicates human behavior. The formal and empirical analyses of the model parameters establish that despite its less-than-perfect performance, human cognition is surprisingly optimal. Finally, the behavior of the simulation through a lifetime of arithmetic learning can best be described as a dynamical system affected not only by its external environment but also by its internal dynamics. 137 pages
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