@MISC{Mills_fromthe, author = {John Mills and David Tall}, title = {From the visual to the logical in mathematics}, year = {} }

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Abstract

Mathematical concepts are more than just the logical sequence of definitions and deductions which make up the formal framework of the subject. In mathematical research it is first necessary to develop a framework of linkages between ideas before they are sorted out into a precise deductive sequence. Yet, in teaching, the subject is so often presented in its final organized form in a way that the majority of undergraduates seem to find notoriously difficult. We believe that the a major source of the studentsâ€™ difficulty lies in the lack of an appropriate intuitive background to provide an environment within which the formal ideas can be interpreted and hence understood. The arrival of more powerful computers offers the opportunity of manipulating concepts visually in real time, and this facility can be put to good use in gaining insight into the logical relationships involved. The recent introduction of the RISC (reduced instruction set) chip in the Archimedes computer gives such a gain in speed that it is now possible to draw highly irregular curves in an interactive manner to give new insight into subtle theorems of mathematical analysis. As an example, let us cite the theorem