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Attention training doesn't transfer
Studies have mostly used video games, and players get better at those tasks. The same pattern appears in stroke patients who lose attention to one side of space, a condition called neglect: they improve on the trained task and then don't improve on other tasks. Attention training looks hyper domain specific.
What the evidence does back is cutting out distractions. Turning off notifications, closing the email window during focused work, and removing distracting noise reduce the stimuli that capture attention.
A brain that devotes its resources to a few things instead of many pays less for its attention, so fewer choices lower the cost.
Does online brain training transfer to other skills?
In 2010, Adrian Owen and colleagues published a study in Nature with 11,430 participants who trained online for six weeks. The trainees improved on the tasks they practiced. There was no evidence that the gains carried over to untrained tasks, even ones that resembled the trained tasks.
Do action video games improve attention?
In 2003, C. Shawn Green and Daphne Bavelier reported in Nature that regular action video game players outperformed non-players on tests of visual attention. They also showed that non-players improved after training on an action game. Whether those gains reach far beyond the tasks tested has been debated in the years since.
Do heavy media multitaskers filter distractions worse?
Eyal Ophir, Clifford Nass, and Anthony Wagner reported in a 2009 PNAS study that heavy media multitaskers were more affected by irrelevant items than light multitaskers. The heavy group was slower when distractors were added and found it harder to switch between tasks. The study was correlational, so it could not show whether multitasking caused the difference.