How to Play the Piano to Amateur Concert Level Status

This is not just about playing the piano, but playing the piano well, to Amateur Concert-Level Status.
Always place your fingers on the keys before you play them. That way you will always know that you're hitting the correct free piano sheet music key. (You don't have time to look at them all when you're playing fast.) It will also ensure that you don't hit the keys too hard, which you should never do.
Either lift the finger from the key (while keeping contact with the keys with the other ones,) and let it drop on the key, or use *slight* arm pressure, with very relaxed fingers to press the keys.
After the key has been pressed, slide the fingers to the right or left. The fingers should not stick, but glide across the keys. That way you don't lose contact with the ivory (as Josef Hofmann said,) and you don't need to look at the keys again to re-set your hands in the proper place.
In places where you need to go a long ways (i.e. 2-3 octaves,) you may need to look at the keys to see where they're at, but still, you must TOUCH the keys first before playing them. This ensures maximum accuracy (very important at high speeds,) a SOFT touch (very important for expression, and not hammering the keys,) and very little effort or work to do for your fingers, which makes it easy to play long, fast passages without mistakes, and without fatigue.

When you're going for accuracy (i.e. no wrong notes,) touch the keys with just the tips of the fingers. It should still be a soft touch, not bony. That will ensure that your finger fits easily into the key without possibly touching the key to either side of it.
Never hammer on the keys. That is the way most amateurs do it, and it sounds horrible, not to mention cause them to hit tons of wrong notes, causes fatigue, makes it harder to memorize pieces, etc.
When you play with a good touch (fingers touching the keys before playing,) your fingers will almost play by themselves, and you can have a good deal of songs memorized simply by touch (as well as by sound, and by chords also,) because the fingers naturally know where to go next.
The best books on playing piano that I've read are Joseph Hofmann's "Piano Playing, with Piano Questions Answered," and Josef Lhevinne's "Basic Principles in Pianoforte Playing." Those are worth more than all the "method" books combined.
If you start out with the right technique (playing from the fingers, and touching the keys) you can skip 10 years of practice and mistakes, and get to the more interesting pieces much sooner.
Still repetition does help to make you more consistent. Repetition will help you gain a lot of experience very quickly. The above-mentioned tips are the things you would eventually figure out if you did enough repetition on the piano. So it's good to know them up front, so you can save yourself a lot of wasted time.
Learn songs that you like. You will naturally play the piano better / more musically if you learn songs that you like. But the musicality is pounded out of a lot of young would-be pianists by forcing them into songs that they don't like. Passion and sentimentality is the center of art, and if it's not there, you don't have art.
So if your children are learning the piano, help them find some songs that they would actually ENJOY learning, in addition to what they're already learning. Give them something that they enjoy, and they will keep up with it by themselves.
For the most accurate playing (with the least amount of error,) play mostly with the fingers, raising them and dropping them on the keys, without moving the hand or wrist, or the arm. The fingers, even when they're lifted considerably, may only move about 2 inches from the keys. Whereas even a small movement of the arm may go farther than that, thus considerably increasing the chances for error. Moving the arm also re-shifts your 'base of operations,' making it harder for the fingers to come down on the correct keys.
However, playing with only the fingers all the time may fatigue the fingers, so it must be distributed somewhat with weight from the hand, slight movement from the wrist, weight of the arm (for more fortissimo passages,) and sometimes even movement of the forearm.
Even though you play with the fingers for piano man maximum accuracy and speed, the fingers, hands, wrists, arms and shoulders should never be stiff. They should always be relaxed.
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