Computer Science Department
School of Computer Science, Carnegie Mellon University


The Dynamics of Cognition:
An ACT-R Model of Cognitive Arithmetic

Christian J. Lebiere

November 1998

Ph.D. Thesis

Keywords: ACT-R, cognitive arithmetic, Bayesian learning, activation spreading, dynamical systems, parameter analysis, power law, machine learning, hybrid systems

Cognitive arithmetic, the study of the mental representation of numbers and arithmetic facts and the processes that create, access and manipulate them, offers a unique window into human cognition. Unlike traditional Artificial Intelligence (AI) tasks, cognitive arithmetic is trivial for computers but requires years of formal training for humans to master. Understanding the basic assumptions of the human cognitive system which make such a simple and well-understood task so challenging might in turn help us understand how humans perform other, more complex tasks and engineer systems to emulate them. The wealth of psychological data on every aspect of human performance of arithmetic makes precise computational modeling of the detailed error and latency patterns of cognitive arithmetic the best way to achieve that goal.

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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