I don't answer emails, browse the web, or instant message while watching a video on my computer, so why would I want to do that on a phone? It's one thing to see a small alert and totally another to try and compose a reply.
You will find by focusing on one thing, or a very small number of things, you accomplish more than trying to do a bunch of things at once. We just aren't wired that way, and while you may think you're doing more, the science behind it shows that you're not.