I somewhat wish to contest Tom Kern’s answer:
The goal of most education is an internalised understanding, which in turn reduces cognitive load when building upon what you learnt (which in turn may be a deeper understanding of the same subject). Memorisation of some things almost inevitably happens on the way there, in particular when solving exercises. It may also be an important tool on the way (example below). Finally, seeking for efficient memorisation may structure one’s knowledge and expose patterns and similar.
However, memorisation alone rarely causes understanding or a sufficient reduction of cognitive load for application. In particular, students who only rote-memorise can (and often do) completely miss the goal of a course and can be completely lost when they need to build upon their understanding on the next level. And even if they can solve a task, they may need much more time to apply their memorised knowledge.
For example, when teaching quadratic equations, the goal is not that students can regurgitate a quadratic formula, but that they know when to apply it, transform a problem so they can apply it, exclude implausible solutions, get a better understanding of equations and logic in general, etc. Now, when learning all of these things through exercises and similar, having a quadratic formula readily available is extremely useful – so useful that you indeed might want to instruct students to memorise it as soon as it comes up (and motivate why). However, even if you don’t, students who seriously work on exercises and similar will inevitably memorise it over the course.
Thus, at the end of the teaching unit, not having memorised a quadratic formula is a strong indicator that a student also missed the goals of the course (or is severely underchallenged). However, the reverse does not hold: Students can have memorised that formula without reaching the goals of the course. An exam needs to assess whether students are able to properly apply the quadratic formula, not whether they can regurgitate it. It should not make much of a difference whether you write a quadratic formula on the blackboard or not – with the main exception probably being students who suffer from anxiety and for whom not having to memorise the equation will provide security and a ward against memorisation flukes.
I would therefore only instruct students to memorise something (or even assess their memorisation) if all of the following apply:
- it helps achieving the actual learning goal (which is never the memorisation itself),
- you expect that for some students memorisation as a side effect (of doing exercises and similar) will be not optimal,
- you do not want to leave figuring out the above to the students, either because the usefulness of memorisation is not apparent a priori or they are not sufficiently independent learners yet. (Here, you need to compromise between your students learning math or learning how to learn.)
Otherwise, I do not see a need to make a distinction.