How do most people practice? Why is it so inefficient? Let me try to explain by describing the average person practicing any given piece on the piano. The advice you always hear is “oh, Billy, just play with both hands really slowly with a metronome. Slowly, you can bring the piece up to speed with the metronome.” While some of this advice might be good, like using a metronome for example, this is very bad advice in general.
First of all, the movements you need to play a piece at speed are quite often different from the movements you needs to play the piece really slowly. Also, it is very inefficient to practice hands together and try to bring a piece slowly up to speed. You will develop bad habits that way and it will take you a very long time (if ever) to be able to play the piece at a good performance level.
In order to prevent these bad habits from developing, you should do a little thinking on what movements you will need to play the piece at performance speed. If you can figure those movements out, you can then slow down those movements (keeping the same general movements, however) and then practice them that way. This is a lot better than just starting off practicing a piece really slowly. It makes a lot of sense to have a teacher show you how this should be done for the pieces you are playing.
I also strongly advocate practicing pieces hands separately at first. This is the best way to work on bringing a piece up to speed in the most efficient manner. If you can’t play a piece at performance speed hands separately, then you will certainly not be able to play the piece hands together at anything close to performance speed. For that reason alone, it makes no sense to start practicing by learning a piece slowly with the metronome. Don’t fall into the trap that the majority of piano students fall into. Find the most efficient ways to practice and then use the methods to your advantage to improve at a much faster pace than everyone else.