Art Markman in his HBR Article says you do not have to think differently, but you have to think different things to be innovative, to be able to come up with innovative solutions for the known problems. He argues this on the basis of how our memory works. He says when you are in the kitchen, your memory of every associated with how the cooking happens comes in the front just like all the rules and records of a game come to forefront when you want a game. So when you think about a problem from different perspectives you invoke different parts of your memory and get different solutions with the help of these various slices and dices of your memory.
Interesting take, I think.