Concept Image and Concept Definition

Concept image and concept definition are two ways of understanding a mathematical concept distinguished by the mathematics educators David Tall and Shlomo Vinner. A concept definition is the formal, agreed form of words used to specify a mathematical concept precisely, while a concept image is the whole cognitive structure a person actually associates with the concept in their own mind, built up gradually through experience with examples, non-examples, pictures, procedures and impressions encountered over time. Tall and Vinner observed that a person's concept image can differ substantially from the formal concept definition, and different parts of an individual's concept image can even be evoked in different contexts without the person noticing that those parts conflict with one another or with the formal definition, a state they called a potential conflict factor; such a conflict becomes an actual source of confusion or error only when the incompatible parts are evoked together and the person notices the clash. The distinction has been widely used in research on mathematics learning to explain why students who can correctly state a mathematical definition may still reason incorrectly about the concept it defines, because their day-to-day mental image of the concept has not caught up with, or has drifted away from, the formal definition itself.