This question is more important than you might think. The answer to it basically shapes the way that the United States feels about the language it speaks (or teaches to those who have enough cultural clout to go to college in the first place). And it brings up the politics of using language as well. It touches everyone. Everyone has a stake in the issue.
So, at no extra charge to you, the reader, here's my opinion.
There are six iron-clad rules of English--four grammatical, two mechanical. Only six. If you follow these rules, you have the greater part of speaking grammatically. Everything else you can find by reading a decent dictionary or glossary of usage.
The grammatical rules are:
1) Verbs must agree with their subjects.
2) Pronouns must agree with their antecedents.
3) Dependent clauses must be connected to a sentence.
4) Compound sentences can only be connected by a comma followed by a conjunction, or by a semicolon.
The mechanical rules are:
1) Semicolons connect independent clauses; commas connect dependent ones.
2) Punctuation never goes outside quotation marks.
The opposition to really hard-core grammar instruction is analogous to the opposition to technological progress. Because since the Enlightenment, both grammar and progress have been used as the yardstick to define "civilization" and "savagery." (I.e., if you use excruciatingly proper grammar / come from a culture that's had an industrial revolution, you deserve to be considered "civilized." If you don't, you don't.)
As an English teacher, I can't say this backlash is wrong--historically, this grammatical yardstick has been used to exclude and marginalize a lot of social groups. In the grand old days of the British Empire, American English was cause enough for the U.S. to be seen as a "savage" country--and what was true in the United States was doubly true in any of the other colonies. A bit closer to home, the idea that "only 'savages' don't speak 'properly'" got used to exclude a lot of different American social groups from positions of cultural power (e.g., African-Americans, poor rural whites, and various non/European immigrants).
However. As a postmodernist, I believe that language is a game--not an absolute quantity, but something we've all agreed to play along with (the word "cat" means what it does because that's what we all agree it means). So, if language is a game, then we have to play by the rules of that game if we want to be understood. And that's where grammar comes in.
So, there needs to be improvement in the attitudes of both sides. Grammar is not intended as a tool to marginalize anyone outside the hegemony--it's important to use for people to get what you're saying. At the same time, though, there's a lot more slack in grammar than most people realize.
The training I received as a TA was mostly concerned with indoctrinating us not to be Professor Umbridge in any way, shape, or form. And I can't say that the university powers were wrong in that. As professors, I and my colleagues occupy a significant political position. And (refer back to what I said above) people who teach English occupy an even more significant one. (Remember, civilization vs. savagery--gatekeepers of the Western Tradition--dispersers of the canon--etc. etc. etc.) And I think it's a good idea to be aware of that position--too many professors forget that a casual remark sometimes has all the force of The Word Of God for an intimidated first-year comp student.
However, avoiding oppression of your students is only part of what makes an effective teacher. And from what I've gathered, the major part seems to be effective explanation. Or, to paraphrase what someone said to Robert Graves in Good-Bye To All That, "They'll do and die--but they've got to know the reason why." Or, to put it another way (and to take this out of the trenches and into the cozy classroom), a teacher will get better results from his/her students if the students know exactly why they're doing what the teacher asks. If the students can see a strong, real-world reason for the in-class activities or the homework, they'll give it the old college try. If they don't, the students will be more likely to think "Oh, this teacher's just being a two-bit tin-plated Grammar Nazi because s/he gets a hard-on from making students jump through hoops."
I think this is what's wrong with composition instruction in classes today. Teachers are not explaining the methods to their madness enough.
...but you know what? I'd like that to be the whole truth, but that's too cheap, easy, and simple. Because it's equally true that students are resisting, either because their ideologies conflict with the instructors', or because students are just too fucking stupid to understand what's going on.
Remember what I said before--there are no innocent victims here. Everyone is equally guilty. The teachers are guilty of being Grammar Nazis--and the students are guilty of being dumbasses who just see comp class as something to get through as quick as possible so they can get their degree and start making money at their "real jobs."
I'll say this again, and I'll say it in caps and boldface, because it's an important point.
THERE IS NO INNOCENCE HERE. EVERYONE HAS A STAKE IN THE WAY ENGLISH IS TAUGHT IN COLLEGE. EVERYONE IS EQUALLY GUILTY. THIS MEANS YOU. YES, YOU, READING THIS NOW. YOU, SITTING THERE IN YOUR OFFICE SAYING "HOW COULD THESE KIDS BE SO STUPID?" YOU, SITTING THERE AT YOUR DESK, PRETENDING TO PAY ATTENTION BUT REALLY SAYING "WHEN IS THIS CLASS GOING TO END? I WANT TO GO BACK TO MY ROOM AND RIP A BEERBONG." YOU, WRITING THE SMUG OP-ED COLUMNS ABOUT WHAT'S THE MATTER WITH KIDS TODAY AND WHY THEY AREN'T AS GOOD AS YOU AND YOUR GENERATION WERE. YOU, NURSING OLD GRUDGES BECAUSE YOU THOUGHT THE PROFESSOR LIKED THE LESBIANS AND GOTH GIRLS BETTER THAN S/HE LIKED THE WEEKLY PAGES OF FIRST-YEAR BULLSHIT YOU TURNED IN.
...a bad ending to this piece, I know, but the whole issue just gets to me.