This is a tricky one. I almost always got this wrong when I was young and most of the time I still deliberately misspeak because I don't want to sound uptight, but the correct answer is counterintuitive. Most of us want to say, "I appreciate you considering the matter" because we view this sentence just the way we view "I appreciate your consideration," but they are completely different. Why?
Because the latter, "I appreciate your consideration," emphasizes the term consideration. That's what I appreciate. Your consideration. But when I say, "I appreciate you considering the matter," I could be talking about one of two things – I could mean that I appreciate you and then the emphasis is on you, or I could mean that I appreciate you considering, in which case the focus is on the term considering.
So, if I' m focusing on you, it is technically correct to say, "I appreciate you considering the matter." But most of the time, what people mean is that they appreciate the considering, so the right way to say that would be, "I appreciate your considering the matter." It sounds stiff and formal, and because many grammarians consider both to be correct because they mean separate things, I wouldn't worry too much about it if you get it wrong.
Sigrid Macdonald is an author and an editor. You can find her at http://www.bookmagic.ca/