Building and testing a cognitive approach to the calculus using interactive computer graphics
Doctoral thesis
English
OPEN
Tall, David Orme
This thesis consists of a theoretical building of a cognitive approach to the calculus and an empirical testing of the theory in the classroom. A cognitive approach to the teaching of a knowledge domain is defined to be one that aims to make the material potentially meaningful at every stage (in the sense of Ausubel). As a resource in such an approach, the notion of a generic organiser is introduced (after Dienes), which is an environment enabling the learner to explore examples of mathematical processes and concepts, providing cognitive experience to assist in the abstraction of higher order concepts embodied by the organiser. This allows the learner to build and test concepts in a mode 1 environment (in the sense of Skemp) rather than the more abstract modes of thinking typical in higher mathematics.\ud The major hypothesis of the thesis is that appropriately designed generic organisers, supported by an appropriate learning environment, are able to provide students with global gestalts for mathematical processes and concepts at an earlier stage than\ud occurs with current teaching methods.\ud The building of the theory involves an indepth study of cognitive development, of the cultural growth and theoretical content of the mathematics, followed by the design and programming of appropriate organisers for the teaching of the calculus. Generic organisers were designed for differentiation (gradient of a graph), integration (area), and differential equations, to be coherent ends in themselves as well as laying foundations for the formal theories of both standard and nonstandard analysis.\ud The testing is concerned with the program GRADIENT, which is\ud designed to give a global gestalt of the dynamic concept of the gradient of a graph. Three experimental classes (one taught by the researcher in conjunction with the regular class teacher) used the software as an adjunct to the normal study of the calculus and five other classes acted as controls. Matched pairs were selected on a pretest for the purpose of statistical comparison of performance on the posttest. Data was also collected from a third school where the organisers functioned less well, and from university mathematics students who had not used a computer.\ud The generic organiser GRADIENT, supported by appropriate teaching, enabled the experimental students to gain a global gestalt of the gradient concept. They were able to sketch derivatives. for given graphs significantly better than the controls on the posttest, at a level comparable with more able students reading mathematics at university. Their conceptualizations of gradient and tangent transferred to a new situation involving functions given by different formulae on either side of the point in question, performing significantly better than the control students and at least as well, or better, than those at university.

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