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