Learning Fraction Sense: Applying Piaget’s Operative Theory of Meaning in a Multi Agent System to Overcome the Learning Paradox

By Gerard Rendell.

Published by The Learner Collection

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Article: Print $US10.00
Article: Electronic $US5.00

The study of fractions is foundational in mathematics, yet it is among the most difficult of topics for school students. Learners must grasp concepts or procedures more complex than those they already have available for application. Bereiter suggests that when educators overlook this learning paradox, they simplify learning and miss important factors in the promotion of learning. The implication is that understanding how the learning paradox can be resolved will improve the educator’s capability to support more effective concept formation in their students.
This research aims to implement Piaget’s theory of biological intelligence combined with Bereiter’s notion of cognitive bootstrapping using a novel approach to reinforcement learning (as a form of machine learning) and attempt to develop concept formation: fraction sense from whole number sense.
Taking a cognitively constrained view of fractions (Egyptian fractions) the simulation implements a model of assimilation-accommodation using a number-line world which expresses conservation (measurement) as the precursor to the development of fraction sense, in a computer-based multi-agent system.
This modelling problem presents many technical challenges, but a solution may provide two many significant benefits. Firstly, by providing a simulation of the evolutionary process in the development of fraction number sense from whole number sense, we may gain more specific insight into the developmental barriers that exist and how these might be overcome. Secondly, and more generally, we gain insight into the emergence of higher-level cognitive structures that are essential in the transition to formal reasoning.

Keywords: Concept Development, Constructivism, Emergence, Evolutionary Bootstrapping, Fraction Sense, Learning Paradox, Machine Learning, Modelling Cognitive Processes, Piaget, Knowing Circle, Simulation

The International Journal of Learning, Volume 15, Issue 2, pp.233-244. Article: Print (Spiral Bound). Article: Electronic (PDF File; 960.816KB).

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.


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