When we got Blackberrys at a previous employer, they became known as “Crackberry’s” because using them in the middle of meetings became addictive. Head down meant “tuned out”.
Since our meetings are all phone meetings, nobody can see if you’re paying attention or not. You may have even dropped off and dialed in to a different meeting.
When I have two phone meetings at once, I sometimes dial into both and switch back and forth. I have some guy ping me if something comes up for me.