The Mental Keyboard
You should be able to run a keyboard in your mind. I try doing it as a mental exercise and see what works and what doesn't. I think, its really important to memorize intervals (initially by visually highlighting, but later by sound too). The most important interval to memorize, IMHO is the fifth-interval. If you can sing, then trying humming the notes as you are doing this exercise. (Again, why complicate life than it already is? If you want a reason why you need to hum or even learn this exercise, my answer would be "It's just an exercise", just do it for that sake.1)