A computational theory of early mathematical cognition
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The primary focus of this dissertation is a computational characterization of developmentally early mathematical cognition. Mathematical cognition research involves an interdisciplinary investigation into the representations and mechanisms underlying mathematical ability. Significant contributions in this emerging field have come from research in psychology, education, linguistics, neuroscience, anthropology, and philosophy. Artificial intelligence techniques have also been applied to the subject, but often in the service of modeling a very restricted set of phenomena. This work attempts to provide a broader computational theory from the perspective of symbolic artificial intelligence. Such a theory should serve two purposes: (1) It should provide cognitively plausible mechanisms of the human activities associated with mathematical understanding, and (2) it should serve as a suitable model on which to base the computational implementation of math-capable cognitive agents. In this work, a theory is developed by synthesizing those ideas from the cognitive sciences that are applicable to a formal computational model. Significant attention is given to the developmentally early mechanisms (e.g., counting, quantity representation, and numeracy). The resulting model is validated by a cognitive-agent implementation using the SNePS knowledge representation, reasoning, and acting system, and the GLAIR architecture. The implementation addresses two aspects of early arithmetic reasoning: abstract internal arithmetic, which is characterized by mental acts performed over abstract representations, and embodied external arithmetic, which is characterized by the perception and manipulation of physical objects. Questions of whether or not a computer can "understand" something in general, and can "understand mathematics" in particular, rely on the vague and ambiguous term "understanding". The theory provides a precise characterization of mathematical understanding, along with an empirical method for probing an agent's mathematical understanding called 'exhaustive explanation'. This method can be applied to either a human or computational agent. As a case study, the agent is given the task of providing an exhaustive explanation in the task of finding the greatest common divisor of two natural numbers. This task is intended to show the depth of the theory. To illustrate the breadth of the theory, the same agent is given a series of embodied tasks, several of which involve integrating quantitative reasoning with commonsense non-quantitative reasoning.
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