A lot of people would assume the answer to this is yes, of course you'd have to look at your hands when you play; how else could you find the right notes? However, as with many aspects of playing the piano, the best way is often not the way you might assume.
If you taught yourself to play the piano, and/or play mostly by ear, you probably look at your hands. Why not? Why would you just stare off into space? You could do that, or your could close your eyes. In fact, many people might find it enjoyable to play with eyes closed, as it usually enhances the emotional response to the music.
But if you are reading music, then the question arises: should you look at your hands? My answer is an unequivocal NO. If you learn to read music by constantly looking down at your hands, you will never be a good sight reader. Of course, it seems easier at first, to just look down and "find the notes." However, you will always be dependent on doing just that, and will probably find it nearly impossible to change that habit. "Easier" is often not the best indicator of long term success. The problem with looking at your hands is that you never learn what I call the "geography" of the keyboard.
We are all born with a level of kinesthetic awareness; some people have a hightened awareness and may be what we would call kinesthetic learners. Others perhaps don't have the sense very well developed. But regardless, you can learn to improve your kinesthetic awareness through the right kind of training. A high level of kinesthetic awareness is critical to play an instrument such as the piano. If you depend on your eyes (visual skill) you will short-circuit the development of your kinesthetic skills.
When you are a beginning student, the books that introduce reading have your hands placed in such a way that you are playing five notes in each hand. This way you can easily learn to feel the notes without looking, since they are right under your fingers, so to speak. Gradually the book will move to music spanning six, seven, eight notes and more. You learn to feel these distances without looking as well, and also to move your hand to a new position as necessary. It's a gradual step-by-step approach, and if you adhere to it (and have a good teacher to help you), you will be able to do all of this without looking at your hands at all. Naturally, beginning music is not going to have you doing giant leaps all over the keyboard. (That is a skill which you will also be able to do without looking, in time. After all, blind pianists do it all the time, so we know it can be learned.) Soon it will be second nature -- IF you don't "cheat" and start peeking at your hands.
The main reason you can learn to "feel your way" is because you learn to read by interval. That is, you see the notes on the page as distances from each other, not as absolute values. The problem with learning to read by thinking "that's an A, that's a C, that's an F" and so on, is that it will encourage you to look at your hands to locate that particular note. Reading by interval teaches you to navigate your way around the keyboard. (See my earlier post on Sight Reading.)
What about "mistakes?" Don't you have to look at your hands when you have a wrong note, so you can correct it? Absolutely not. As a beginning or intermediate student, you will have many wrong notes. That's just part of learning. But if you look down at your hands every time, you will become dependent on it and all the work of trying to not look at your hands will be undone. Instead, you must learn to let your ear inform you where the "wrong note" is: are you too low or too high? By how much? Then you learn to adjust. This is the real mastery: to adjust for a wrong note and continue, not just stop, look, "fix" and then start again. (See my earlier posts on wrong notes.)
Looking at your hands is also just too slow for advanced music. When you are a beginner, your music will be moderately slow. But as you play faster music, the eyes would not be able to process at those speeds. The body is much faster than the eyes. And if the hands are far apart on the keys, you can't actually see both at once.
At the lessons with my students I am quite adamant about learning to play without looking at hands and I make sure they understand that this applies to their practice at home as well, of course. Yet somehow, every now and then I'll have a student who I find is not really progressing in their reading as well as I thought they would be; sure enough, they finally "confess" that at home they look at their hands first, to "find all the notes," and then try to transition to not looking. It just doesn't work.
I often say that if I could take two students of equal natural ability and who spend an equal abount of time practicing, the one who learns NOT to look at their hands will make progress twice as fast as the one who looks. It is that important.
What about after you've memorized the piece and are no longer looking at the music? Can you finally look at your hands? The answer is YES, but you'll barely need to. At this point, the eyes will be your final "ace in the hole." You will just be keeping a general eye on things, but you can now use your eyes to make sure you nail that giant leap. You may not even need to, but it will give you additional security. I believe that the eyes (visual) are a resource that we need to learn to use wisely. Your primary tools are the ear (auditory) and the kinesthetic. The thinking mind (intellectual) and the eyes are secondary (although they do have their roles to play). I'm certain this is true in most sports as well, since almost all sports require speed and quick response time.
This subject is related to my earlier post about immediate gratification versus long-term success. Looking at your hands may seem like a good idea because it appears to help you find the correct notes. But you will sacrifice long-term success with this habit, because playing the piano is highly physical, and you have to focus on training the physical aspects of playing.
