Every time your attention is shifted from one task to another, the Rule of Disengagement followed by the Rule of Activation is employed. The Rule of Disengagement requires neurons active for a specific attention task to be disengaged and the Rule of Engagement requires neurons required for the new attention task to be sought and then activated.

This results in a time delay (perhaps as much as a 10th of a second or more) as one rule delivering attention to a task is disengaged before the next rule for giving attention to a different task is activated. The time delay prevents attention multitasking! Further, if you keep switching your attention that can add up to a lot of time. If you’re driving, even a few 10ths of a second can result in a quite considerable distance being travelled. Studies confirm talking on a phone while driving increases road traffic crashes because stopping distances are considerably increased. 
If you have an efficient working memory then task shifting can occur very quickly indeed. Shifting is still not attention, however. The fact remains: attention multitasking is neurologically impossible
You can of course do more than one thing at a time. At this moment in time you are doing more than one thing at the same time, as you read this post, perhaps drinking a coffee while your heart, of course, beats steadily away. You might be monitoring the traffic too if you’re near a busy road like I am. However, doing more than one thing at a time and attending to more than one thing at the same time is not the same thing at all. 

Everyone – even men! – can do more than one thing at the same time. No one – not even women! – can attend to more than one thing at the same time. 
It’s time to put attention multitasking, for either sex, in the False Doctrine Box.

Men can’t multitask – and neither can women – when it comes to paying attention.

(Adapted from the sources: Medina, J. (2009). Brain Rules. Seattle: Pear Press 
Medina, J. (2014). Brain Rules Updated and Expanded. Seattle: Pear Press.)