Images and Definitions for the Concept of Function

Images held by 271 college students and 36 junior high school teachers for the concept of a mathematical function were compared to the definitions they gave for the concept. A questionaire was designed to exhibit the cognitive schemes for the function concept that become active in identification and construction problems and to make possible the comparison of these schemes with the definition. Many of the definitions and even more of the images were primitive among all but the mathematics majors and the teachers. Discrepancies between image and definition were frequent for all subjects who gave the Dirichlet-Bourbaki definition.