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That's not true. Studies show that many pupils are better able to absorb information while doodling or by other repetitive actions. Many many famous authors have used long walks as a way to help themselves write.

The brain is very good at ignoring some sounds. Personally I'm fond of "white noise" generators - I have my own perfect combination of rain, wind, flowing water and nature noises that cancels everything else.

Maybe compared to silence, but often not net if it's displacing other stimulus.

I'm not entirely sure it's even always true compared to silence. Sometimes I feel like other bits of my own brain can be competing for attention and that occupying them can be a win.