Let's say you're sending someone an email, maybe to thank them for visiting you in the hospital. Would you say "I appreciate you taking the time to stop by" or "I appreciate your taking the time to stop by"?
Believe it or not, some people have pretty strong feelings about which of these sentences is correct. For many of us though, it's the kind of thing that gives us pause.
First, a quick grammar refresher. A gerund is a verb with an "ing" ending that's used as a noun -- for example, "his singing" or "the dog's barking." What often gives us trouble is whether to use the possessive form of the noun or pronoun that proceeds a gerund.
When grammarians addressed this issue in the 18th century, they said the possessive should never be used. Don't say "his singing" or "your stopping by." But other grammarians jumped in later and said the possessive is always right, and it's the other form -- "him singing" or "you stopping by" -- that's always wrong.
The truth is, there doesn't seem to be a hard and fast rule for this one.
Merriam Webster's Dictionary of Concise Usage points out that if you look at writers over the past 100 years, you'll find that many use both forms. For example, Flannery O'Connor wrote, "He approves of this one's being a girl." But she also wrote, "I can't see me letting Harold C. condense it."
There do seem to be some patterns. For example, personal pronouns tend to prefer the possessive, but the possessive doesn't always work. In this example from William Faulkner -- "I would certainly insist on you all coming here" -- changing "you all" to "you all's" makes the sentence sound strange.
Sometimes, it's better just to remove the pronoun entirely. If the subject of the sentence is the same as what's going to be the pronoun, just get rid of the pronoun. "The students were happy about their having completed the difficult course" isn't wrong, but it does sound redundant. It works fine without "their" -- "The students were happy about having completed the difficult course."
In any case, if you're still sitting there, trying to decide whether to write "your stopping by" or "you stopping by," just pick the one that sounds best to you.