Perhaps the most formidable technical challenge faced by all piano players is this: How do I keep track of and coordinate what is going on in my left hand with what is going on in my right hand?
Getting the left side of your body-mind and right side of your body-mind to be on the same sheet of music requires two complementary skills: independence and coordination, the yin and yang of integration.
Integration Principle #1. Your brain can only consciously do one or two things at a time. Nobody is smart enough to do a gazillion things at once.
Integration Principle #2. The logical extension of Principle #1 is to reduce the number of things you need to pay attention to.
Integration Principle #3. The logical extension of Principle #2 is to turn four things into three things and three things into two things and two things into one thing.
Integration Principle #4. You want to study the music in a way that enables you to turn two things into one thing. Memorizing a bunch of apparently unrelated notes is certainly not the way to get there.
Integration Principle #5. You need to combine notes into meaningful patterns that make sense to your musical and auditory and kinesthetic brain. In other words, you need to chunk individual notes together into meaningful patterns, just like we chunk individual letters into words and words into sentences.
Integration Principle #6. You need to reconceive of two events that occur at the same time into one single event. For example, I cannot think the following: “OK my right hand is going to play the note D with my right pinkie and my left hand is going to play the note G with my left pinkie.” I need to hear and think and see and play them as if they we the very same integrated event, not two separate events that just happen to occur at the same time.
Integration Principle #7. No flams. A “flam” is when a drummer intentionally lets one hand lead the other hand in order to produce the equivalent of a grace note that occurs just before the main note. This creates a hip “thwap” kind of sound. While grace notes are indeed played on the piano to beautiful effect, there should never be any unintentional flams between hands. One way to eliminate flams is to outline the important notes where the left hand and right are playing together and make sure that you play them as a single integrated idea, not two separate ideas that just happen to occur at the same time. Think of these as “anchor notes” than connect your left and right sides.