Back to Fowler:
And some words from the conservative Theodore M. Bernstein in The Careful Writer"
Except under the most formal and unusual circumstances, there is nothing whatsoever wrong with ending a sentence with a preposition, if that is what makes the sentence flow clearly and easily to the eye and ear of the reader. Bad sentences are bad sentences, no matter how they are constructed.
I am not against people like commasense, who appears to understand the language and tries to work with it, but those who mindlessly apply rules and strictures. They serve no purpose except to confuse poor innocents like KidCharlemagne, who seems to be brainwashed by their pernicious nonsense.
And textbooks are made to be read. There is no reason to be especially formal in them. Don’t change the sentence at all.
I DO appreciate your help but do you always have to backhand me on the way out?
Sorry. That wasn’t my intent. You asked a perfectly sound question. I was just venting against the world in general.
Sorry I’m late here.
Frankly, in a textbook, we would just plain avoid it by rewriting the surrounding text, if at all possible. “I know that game” would do in a pinch in most contexts.
If we absolutely had to stick to the text as written (for instance, in a late-stage proof, where no one had caught this earlier and it would be too expensive to rewrite a lot of text), we might go with “I know the game to which you are referring.” But we go far, far out of our way to avoid such a thing, both because we know the kids hate it, and because almost all of us agree with Fowler and Bernstein that such tortured, pompous sentences are worse than the alternative. And sometimes we just let the prepositions lie where they fall.
Contrary to what most people I talk to assume, the choices we make are more often based on market realities than on principles. Most of us are painfully aware that “correctness” in grammar is an artificial construct and a historically late development in English, despite what some English teahcers believe, so we don’t put much stock in such “principles”. I’m possibly more aware of this than many in my field because I’m married to a linguist.
I see the whole preposition ‘rule’ to which you refer as more of a guideline. If it eliminates an awkward sentence, good. If it creates an awkward sentence, it’s not as good.
The preposition rule usually falls apart in the presence of phrasal verbs like shut up, put up, whacking off (see Beavis and Butt-Head Do America), et cetera. In a lot of areas, you have to destroy the sentence to make it work. I can’t fathom what happens to “I don’t know what you’re talking about” aside from phrasing the question differently.
For the most part, just follow common sense.
For those who are curious, the original promulgator of this silly rule (and the equally silly rule about splitting infinitives) was Robert Lowth, a British bishop who wrote A Short Introduction to English Grammar in 1762. He was aided and abetted in this foolishness by Lindley Murray who expanded this document in his popular 1794 book, English Grammar.
It’s possible that I’m misunderstanding you, but if not, then here goes:
Words can have more than one part of speech. The “prepositions” in phrasal verbs like this are really particles; they’re connected to the verb and not the prepositional phrase. In fact, there’s no prepositional phrase at all; it’s a direct object, as in the sentence She looked up the number. If you needed it in a subordinate clause like the number which she looked up, it is incorrect, both formally and informally, to reword it as the number up which she looked. Furthermore, for intransitive phrasal verbs (like shut up and whack off) I don’t see how anyone could ever make this mistake.
Hrm. When teaching English grammar to speakers of other languages, which I do everyday ;), I am often reminded of a comment made by an elder grammarian. A young man was a firm proponent of the use and acceptance of Ebonics as a valid dialect of English. My friend simply stood and told him that, yes, one should endevour to speak in the manner of the educated and respected members of one’s own community. He said, “But dis town ain’t got no smart folks.” She sighed, mentally patted him on the head, told him he’d go far, kicked him outta class.
So, should I teach these people how to speak correctly, or appropriately?
Based on your friend’s comment as to what one should “endevour” to do, wouldn’t those be the same thing?
Well thanks a lot for looking at the link I supplied. That’s almost exactly the example that the editors of the OED–which most would say is a rather formal “document”–used.
I used that example in my OP. 
It’s amazing that in AP English at a good high school I used to get lambasted for ending sentences with prepostions. Was no one else even recently taught to avoid this? Fricking Mr. Clark.
I seem to remember some wag who explained that the kid’s bedroom was on the top floor of a lighthouse, so the question to father became: “What did you bring that book that I don’t like to be read to out of from up around for?”
I always thought a better reponse to the Harvard guy would be, “At Yale, we know that in this case the problem is not that the sentence ends in a preposition, but rather that ‘at’ is superfluous; it’s role is already covered by the word ‘where.’”
One of the alltime great wags, Stephen Fry, has it a book about Australia: “What on earth did you bring a book to read out of about Down Under up for?”