Visualization and reification of concepts in advanced mathematical thinking

Graduation date: 1995A concept image is that collection of all images, pictures, symbols, definitions\ud and properties associated with any given mathematical concept. One of the\ud most important components in the mental representation of concepts in the concept\ud image of advanced mathematical thinkers is visualization. This component, in\ud turn, is indispensable in the intuition process which is essential to advanced mathematical\ud thought. The visual aspect of intuitive reasoning in mathematics falls into\ud three main categories - diagrammatic reasoning, which is predominantly though not\ud exclusively graphical, analogic reasoning, relying heavily on non-mathematical experiences\ud as models for abstract mathematical concepts, and the use of prototypes,\ud the selection of one typical example as a representative of the concept.\ud This study was designed to examine and describe the nature of visual images\ud used by advanced mathematical thinkers, as prototypical, analogic or diagrammatic\ud images. We also sought to identify hooks, which provide initial access to the concept\ud image, we looked for links among them, and for image schemas, which provide\ud the mental 'scaffolding' for the concept image. We sought evidence of progress in the construction process of concept images by looking for the interiorization and\ud condensation stages, during which time the concept is internalized and all related\ud information is condensed into a gestalt, and in particular, the reification stage, an\ud event which produces a radical restructuring of the concept image.\ud Nine case studies are presented and analyzed, in which advanced mathematics\ud undergraduates, mathematics graduate students, and mathematics faculty were\ud extensively interviewed, and their responses audiotaped and transcribed. The interviews\ud were reflective in nature, comprised of a series of questions, which were asked\ud regarding twenty-one different mathematical concepts. A detailed analysis of each\ud individual, in light of the above questions, is presented, summarizing the individual\ud nature of the concept image in advanced mathematical thought