Rendell, Gerard Vincent Alfred (2012) An integrated modeling framework for concept formation : developing number-sense, a partial resolution of the learning paradox. (PhD thesis), Kingston University, .
Abstract
The development of mathematics is foundational. For the most part in early childhood it is seldom insurmountable. Various constructions exhibit conceptual change in the child, which is evidence of overcoming the learning paradox. If one tries to account for learning by means of mental actions carried out by the learner, then it is necessary to attribute to the learner a prior structure , one that is as advanced or as complex as the one to be acquired, unless there is emergence. This thesis reinterprets Piaget's theory using research from neurophysiology, biology, machine learning and demonstrates a novel approach to partially resolve the learning paradox for a simulation that experiences a number line world, exhibiting emergence of structure using a model of Drosophila. In doing so, the research evaluates other models of cognitive development against a real-world, worked example of number-sense from childhood mathematics. The purpose is to determine if they assume a prior capacity to solve problems or provide parallel assumptions within the learning process as additional capabilities not seen in children. Technically, the research uses an artificial neural network with reinforcement learning to confirm the emergence of permanent object invariants. It then evaluates an evolved dialectic system with hierarchical finite state automata within a reactive Argos framework to confirm the reevaluated Piagetian developmental model against the worked example. This research thesis establishes that the emergence of new concepts is a critical need in the development of autonomous evolvable systems that can act, learn and plan in novel ways, in noisy situations.
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