Q-Learning Fraction Sense: Applying Piaget's Operative Theory of Meaning in a Multi Agent System to overcome the Learning Paradox

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When children in the concrete operations state of cognitive development start to tackle fractions from their learned knowledge of whole numbers, they are presented with a set of challenges that in many ways resembles Fodor's notion of the Learning Paradox: how can a structure create a structure more complex than itself. Bereiter, in reviewing Fodor, classifies a solution to this paradox with the idea of bootstrapping cognition. One way that this problem can be investigated is via an experiment that determines how a Software Agent (modelled using reinforcement learning with an evolutionary algorithm that learns discrimination in a PID controller) attempts to simulate Piagetian equilibration in a Number Line World. The problem focuses on the expression of conservation (measurement) as the precursor to an understanding of numbers, and from this, the development of Piaget’s scheme constructions of: Action Coordination (Permanent Object); Semiotic Function; Symbolic Functions (logic of Functions), Causality (Structural Implications). This real world problem presents many technical challenges, but a solution will provide many significant benefits, not least is a way of simulating operative meaning i.e., understanding that the development of meaning of numbers must come from progression (evolutionary bootstrapping) by the Software Agent through all the Piagetian developmental stages (Sensory Motor, Pre-Operational and Concrete Operations). Secondly, it is postulated that operative meaning may provide clues to the identification of the ‘emergence of structure’ in dynamic systems i.e., overcoming the learning paradox.

Keywords: Constructivism, Modelling Cognitive Processes, Machine Learning, Learning Paradox, Understanding Fractions
Stream: Curriculum and Pedagogy; Student Learning, Learner Experiences, Learner Diversity
Presentation Type: Paper Presentation in English
Paper: Learning Fraction Sense

Gerard Rendell

Ph.D Researcher, Computing, Information Systems and Mathematics, Kingston University
Kingston-Upon-Thames, Surrey, UK

Gerard Rendell is a Ph.D. candidate working on a thesis: Pattern Matching in Asymmetric Systems which has the sub-title: The realization of individual meaning making through paradoxical discourse. His supervisor for the research is Chris Tompsett.

The aim of the research is to model Piaget’s operative theory of meaning in a Multi-Agent System to determine if it is possible to overcome Fodor's learning paradox: can a structure create a structure more complex than itself, by examining a microworld of number lines.

The researcher has an MS.c. in Information Systems from Kingston University and is CEO of a requirements engineering company in Atlanta USA that focuses on strongly typed requirements using the Rational Unified Process.

Ref: L08P0389