USING DIAGRAMS TO UNDERSTAND GEOMETRY

This paper describes ARCHIMEDES-STUDENT. a computer program that const ructs and modifies its own representations of diagrams from instructio ns supplied by a human who is demonstrating a theorem of geometry. The program's representation permits it to make inferences from its const ructions and to find a justification for the conclusion of the theorem . It is argued that the sort of perceptual reasoning displayed by this program represents one important aspect of understanding because it r elates the abstract mathematical theorem to knowledge of spatial relat ions. For humans this approach grounds abstraction in experience and t hus provides a more compelling demonstration than a formal proof. Beca use ARCHIMEDES-STUDENT is a well-defined computer program, it provides a precise suggestion of how this aspect of understanding can be achie ved.