Mathematical understanding – Common themes in philosophy and mathematics education

We present different characterizations of mathematical understanding given by mathematicians, philosophers of mathematics, and mathematics educators. One purpose is to illustrate the diversity of these characterizations. Although the descriptions of understanding may seem incompatible, the paper ends by pointing to some shared themes. They include an emphasis on qualities such as relations and unification. Additionally, we note that re-presentation, including visual representations, is thought to play a role in understanding.