Abductive processes in conjecturing and proving
Abstract
The purpose of the present study was to build a cognitive model to identify and account for possible cognitive processes students implement when they prove assertions in Calculus, specifically a cognitive model that would help to recognize creative processes of an abductive nature. To this end, Peirce's Theory of Abduction and Harel's Theory of Transformational Proof Scheme have been used. The result has been the construction of the Abductive System whose elements are {facts, conjectures, statements, actions}; briefly, conjectures and facts are ‘act of reasoning’ generated by phenomenic or abductive actions, and expressed by ‘act of speech’ which are the statements. At the base of the construction of the Abductive System there is also the intention to show that the creative processes own some components, and to separate this process from the belief that it is not possible to talk about it because it is something indefinable and only comparable to a “flash of genius”. The common denominator with Peirce's work is the philosophic spirit on which both works are based. Peirce wanted to legitimate the fact that abduction is a kind of reasoning along with deduction and induction, in contrast with many philosophers who regard the discovery of new ideas as mere guesswork, chance, insight, hunch or some mental jump of the scientist that is only open to historical, psychological, or sociological investigation. The definition of Abductive System allows the researcher to analyze a broader spectrum of creative processes, and it gives the opportunity to name and recognize the abductive creative components present in the protocols. From the didactical point of view, it allows to recognize the variety of the components of the creative processes, in order to respect them (usually it is not done this way at school) and to improve them. Therefore, this framework could help teachers to be more conscious of what has to be (1) recognized, (2) respected, and (3) improved, with respect to a didactic culture of “certainty”, which follows preestablished schemes.
Degree
Ph.D.
Advisors
Milner, Purdue University.
Subject Area
Mathematics education
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