Kids have small hands. Really little ones. Adults, even small adults, have much larger hands. As most teachers work largely with children, they know their hands best. These kids have hands that span only 5 keys. This makes it easy to teach them. 5 keys, 5 fingers.
Unfortunately, adults, even those with small hands, cover 6 keys. If you were to shake out your hand, either of them, and place them without any contraction on the keyboard the distance between your thumb and pinky would be 6 keys. It actually takes some muscle action to only cover 5 keys. Most adults took lessons when they had smaller hands so they are very used to covering only 5 keys.
6 is better however than 5. Works better with popular songs. Go ahead and try it. Put your thumb on C and your pinky on A, rather than G. Check out where the tunes you love are. Few of them have only 5 notes. Many have 6 or more. See also the next few paragraphs.
What do you do with those fingered pieces? Too many of them are based on the idea of "five finger postions" and they will no doubt tie your hand or hands in knots putting in numbers that validate the idea that you simply must ONLY cover 5 keys.
