What's the difference between practice, drilling, and memorization?
In the psychological literature Practice has a formal definition, which I know through Anders Ericsson; I think it originates with him but am not certain. Practice includes feedback on performance, and it's executed for the purpose of improvement. The distinction is important because it differentiates practice from performance (which is done for the pleasure of others) or play (which is done for one's own pleasure) or the routine execution of a task (which is done to achieve a goal).
Thus, if I practice guitar I'm trying to improve, and I'm monitoring my performance for the sake of noting errors and thinking of new ways to do it. Performance and play of the guitar differ in obvious ways. Routine execution might apply to a task like handwriting. My handwriting is pretty bad, despite thousands of hours of execution, because during all of that time I wasn't practicing. I was just writing to get something on paper.
Drilling and memorization don't, so far as I know, have definitions that have been carefully thought through to draw important distinctions.
To me, "drilling" connotes repetition for the purpose of automaticity, using the technique of thoughtless repetition.
"Memorization" connotes the goal of something ending up in long-term memory with ready access. . . but the word does not imply anything about the technique one uses to achieve that goal.
Thus, students are unlikely to practice the multiplication table, but they would practice the violin, or writing an evocative description of a scene.
Students might drill in an effort to learn the multiplication table, but I hope they would not. As I've defined it, it's hard to think many school-related tasks that would be well-served by drill. Perhaps a very basic motor skill, like a particular run when playing the guitar? Again, this would be repetition without thought.
Much more common would be memorization: activities undertaken in the desire to commit something to memory so that it is readily accessed. This would include deep processing (i.e., thinking about meaning), generating cues for oneself, etc. A student who wants to memorize a poem, for example, could try to do so by drill, but it's a terrible way to learn something. Much better to think about the meaning of whatever it is you're trying to remember, and tie it to things you already know.
Again, let me repeat that practice does have an accepted definition, but I've made up my own for drill and memorization.