Varieties of
Mathematical Experience

  1. Creation and Discovery
  2. Understanding
  3. Problem Solving


Three types of mathematical experience can be distinguished:
Creation and Discovery,
Understanding,
and Problem Solving.
The creation of new mathematical concepts and the discovery of brand new mathematical results is usually done by professional mathematicians. It is not an experience common among students.
There are several levels of understanding a piece of mathematics, but an important distinction can be made between understanding the validity of each step and understanding the reasoning sufficient to be able to do similar mathematics oneself. Thus a good test of understanding is the ability to solve problems.