In just about anything you read on practicing the piano -- any book, any blog, any DVD -- and also with most teachers, you will find that the idea of playing very slowly is the way to learn a piece and to master its difficulties.
The general idea is to play extremely slowly and gradually increase the tempo until you are playing at the desired tempo of the piece. This may sound logical at first, but does it really work? More to the point, would it produce the desired musical effect, a performance with passion and dazzling brilliance?
If you wanted to run very fast, would you start by walking very slowly and gradually increasing the speed until you were running? Not really: walking and running are quite different from one another and require different coordination. Walking does not necessarily prepare you for running. Slow playing does not necessarily prepare you for fast playing.
Here are a few of the inherent pitfalls of very slow playing:
1. It can be very un-musical. Notice I say can be. Very slow playing encourages very note-wise playing. What I mean by note-wise is that familiar sound of a total beginner playing, for example, Twinkle Twinkle Little Star. Each note is emphasized equally, giving a very stiff, stodgy, boring effect. There is no feeling of movement or destination. In fact, the destination is always the very next note, which is not really a destination at all. There is no sense of a phrase, or what musicians call a "long line." If you speak a sentence this way, or a line of poetry, you will hear the effect -- it's very robotic. Yet this is often how people play. In fact, playing this way is encouraged by many teachers for the playing of "finger exercises." When you play a piece of music slowly, note by note, you are often playing it as if it were some kind of exercise. This will simply not translate to gorgeous, exciting playing at some later time.
2. It encourages inefficiency. If you have a piece with very challenging technical aspects, whether it be rapid scale passages, arpeggios, octaves, or whatever, you will only achieve speed and power by being physically efficient, that is, having no wasted motions. Playing each finger with a lot of articulation will not give you a dazzlingly fast arpeggio. Instead, there must be the feeling of a grand sweep with the arm with the fingers precisely in the right place at the right time, but not with each finger making a separate and distinct action. Yet when you play very slowly, you are likely to do just that -- make a separate movement with each finger, for it is difficult to get a "sweep" with the arm at a slow tempo. If you want to master the technique of playing fast, you have to play fast. You will have some stumbles and plenty of "wrong notes" at first. But you will gradually train your body (hands, arms and fingers) to be efficient. Playing the piano is very athletic, and learning to play it is very similar to athletic training. You must learn to increase your reaction time, the ability of your body to respond very quickly. You can't do that by always playing slowly. I've found that most people who try playing slowly and gradually increasing their speed never succeed at playing fast and demanding pieces. They haven't trained their bodies to respond at those speeds.
3. The sound is distorted. Imagine listening to your favorite song on a recording played at a very slow speed. It sounds warped and distorted. When you play very slowly, your ear loses the connection between tones, and can't hear the whole phrase as it should sound. You are essentially giving a false or distorted auditory image to the ear. Since I believe the ear is of the utmost importance, and is really "running the show," I don't believe it is desirable to give it a distorted image so much of the time.
How, then, do we learn a new piece, if not by playing it slowly? The first answer is outlining. For a full description, see my previous post on this wonderful, life-altering method of learning. You play a "sketch" of the piece, hands together, seeking to play the most important, structural elements, and omitting the small details. You play at the the desired tempo, or as close to it as you can, but since you are omitting notes it does not feel terribly fast or demanding at the early stages. You gradually fill in more details. Some will come easily -- more easily than you might have expected. Others may not and you will need to do some "spot work" on those passages. Spot work may involve some very slow playing, so you can observe exactly what your hands are doing and make corrections to the problems you find. But the spot work would be in small doses, only as much as is really necessary.
It is impossible to describe here how I work on every type of technical issue. But to summarize, I am making sure that the arm, hand and fingers are all working together for maximum power and speed, as well as for a focused, precise touch for slow and delicate passages. I give specifically-designed "set-ups" for the student to practice the specific technical challenge. But if the passage being worked on is a fast one, the practice will necessitate playing fast -- even faster than you may ultimately play it -- to increase efficiency and reaction times.
Ear work, through transposing, also helps you to learn the piece. When you are new to transposing you will probably find you need to go quite slowly. But here it is "worth the price." The benefits you are getting by transposing outweigh the problems with slow playing. And you need to give your ear time to really hear the relationships.
It's pretty simple, really: you become what you practice. If you want to play hands together, don't practice hands separately. If you want to play musically, don't practice mechanically. If you want to play boldly, don't practice timidly. If you want to play fast, don't practice everything slowly!