When I was a young student, my teacher insisted I first learn each hand separately and only then play both hands together. I found this more than a little frustrating. I always felt I had to start all over again when I put my hands together; all of the proceeding work with hands separate was a waste. Eventually in a pique of rebelliousness I quit playing with my hands separately, except for when I was asked to do so in a lesson.
I have often wondered about where the notion of learning each hand separately originated. I can picture a scenario like this: JS Bach tells CPE Bach he should pay extra attention to the right hand passage he continuously mis-plays. CPE tells his student he should play the right hand a couple times to fix a problem. Beethoven then tells his student Czerny that his right hand is very sloppy and he needs to fix it before the next lesson or he will use the ruler. Finally Czerny tells everybody to practice each hand separately.
So let’s break the cycle. The piano is a two handed instrument requiring a constant partnership between the hands. A good sight reader doesn’t run through each hand before playing, he just plays both hands immediately.
Now before you completely write me off as a crank, I will allow brief hands separate work but only in the interest of clarification of technical details and fingering. After that put them together!