Tacit Knowledge

In maths there is a lot of tacit knowledge. A student at the beginning of learning a subject will often be able to say the exact same words about the subject as at the end of learning it. But at the beginning their ability to solve problems in this subject will be limited. At the end it will be increased. They will not be able to express with words what changed in their understanding. But in some sense the words they were speaking at the beginning were vacuous. At the same time that tacit knowledge could really be tested: by asking them to calculate or prove related things. If they can’t, their words are just repeated theorems etc. And they may even be convinced themselves that they do understand those words; yet they don’t.

As Von Neumann said: “In mathematics you don't understand things. You just get used to them.”