The geometric, in which objects are given visually and their properties are described. By middle-level university, however, teachers no longer ask students for answers about, or even have them think about, familiar mathematical objects. One needs to understanding students' current knowledge if one wants them to amend or extend what they know. There is evidence that knowledge is more enduring when it is learned in a meaningful context, through reasoning from relatively primitive concepts, by explaining to others, and by reflecting on one's own knowledge growth. Different logical and experiential paths can lead to the same mathematical ideas.
Intuitive knowledge is both an asset and a liability.
Tom Cooney noted that teachers' beliefs greatly influence what they do in the classroom. While some beliefs are malleable, others are strongly held onto. It is often difficult for teachers to overcome the idea that the way they learned mathematics, which is often procedurally, is the best. He suggested that one approach might be to put preservice teachers in mathematical contexts they understand, but which cause perturbations. For scientists and philosophers, it is one that applies to what is defined and only to that.