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The Nature of the Object as an Integral Component of Numerical Processes 1

This paper presents an outline of research studies indicating the existence of qualitatively different thinking in elementary number development, drawing on empirical evidence obtained over the last ten years. Evidence first signalled qualitative differences in numerical processing (Gray, 1991) which was seminal in the development of the notion of procept (Gray & Tall, 1994). More recent studies of the role of imagery in elementary number processing (Pitta and Gray, 1997) indicate that qualitatively different outcomes may arise in the abstraction of numerical concepts from numerical processes because children concentrate on different objects or different aspects of objects which are integral components of numerical processes. The focus of our work has been to consider this difference in thinking and its consequences. What is it that children are doing differently and why? Using rich empirical evidence we develop a cognitive theory which may account for these differences. Our research paradigm has focused on extremes of mathematical achievement and examined children's interpretations of arithmetical symbolism and the associated imagery. Our conclusions highlight the influence of different cognitive styles influencing the nature the object which is an integral component of children's numerical processing. We suggest that such differences effect the cognitive shift from concrete to abstract thought and will have consequences on children's numerical development. 1 Our grateful thanks are extended to the four anonymous reviewers of the PME review panel whose detailed and perceptive comments have contributed to the form of the current paper.