Intuitive Generalizations

Abstract

When learners are able to abstract the properties of a concept and apply them onto a different context, or raise them to a higher dimension, then this is an indication of a firm understanding of that concept. This process is natural to Mathematicians and to Mathematics instructors who assume that it is obvious to students at any stage of learning. The transfer of mathematical concepts, ideas and procedures to a new and unanticipated situation or domain involves high cognitive skills: when mathematicians suspect a similarity between two domains, they are conjecturing or selecting a set of logical relationships, extracting them from a domain, and mapping them onto another seemingly remote area: all this is achieved in what looks like an instinctive natural self-evident manner. In this article, I shed the light on selected cases of intuitive generalizations in Calculus, Geometry and Discrete Mathematical Structures and suggest that students intuitive scope can be widened if they are constantly guided to focus on the generalization itself and what made it happen, more so than on the result being generalized.