Many grammarians identify the participles in introductory phrases or clauses as "dangling" when they are followed by an illogical noun or pronoun. This is usually the subject of the sentence. Here's an absurd example, with the dangling participle in purple and the pronoun subject in blue:
Swarming aggressively, I was attacked by bees.
The participle 'swarming' is said to be dangling because in syntax it modifies the first noun or pronoun following the introductory phrase, which is 'I'; but in logic it modifies 'bees'.
I think it's misleading to call these things dangling participles. That implies that the participle is in the wrong place. But as you can see, the problem is with the noun or pronoun rather than the participle. And the best fix is almost always to leave the participle where it is and to re-write the sentence with the logical noun or pronoun as the subject, followed by an active verb (in red) and its object (in green), like this:
Swarming aggressively, the bees attacked me.
This is pretty obvious in such a simple and outrageous example. But here's the same problem in a sentence you could see in any business document or email:
Listening to the market, the price was dropped by 15 percent.
Well, no … the price is an inanimate abstraction; it cannot listen to real life. And there's another problem usually associated with these misbegotten subjects -- passive verbs.
The solution to both problems, as in the swarming bees story, is to change the subject to the logical noun or pronoun, followed by an active verb (red) and its object (green):
Listening to the market, we dropped the price by 15 percent.
So let's stop blaming dangling participles when the real problem is illogical subjects. That term focuses the student's mind on the noun or pronoun, where the real trouble lies.
Watch the video for a more entertaining presentation of the same stuff.