I teach mathematics at the university level. While I do not expect anyone to speak and write perfectly all the time, I do feel that students who emerge from college should not be regularly misspelling words and using poor grammar. So when a student sends me an email using such constructions as "Me and X met" rather than "X and I met," I feel it's almost a duty of mine to correct them, yet I usually refrain, because I don't think the student would understand my pedagogical intent and would instead label me as the grammar police, especially since I am a mathematics instructor. Is there any way I can help students develop better writing skills without becoming despised?
You can post a statement at the start of class saying that you may on occasion provide feedback to grammar or phrasing mistakes even in such mundane exchanges as email simply as a way to improve the clarity or professional tone (versus colloquial tone) that a students should have as they move forward in their career.
You can generate a "learning toolbox" folder, create a document of "Some Common Best Practices in Communication", and make this available to the students.
You can incorporate grammar and phrasing as grading assessments, leaning on the statement that being able to communicate the approach and results from mathematical analysis with clarity and professionalism is an outcome for the entire math program (and therefore becomes a part of your course).
Whatever path you decide to take, you must be clear to the students when you are making any statements about their grammar and phrasing whether your statements are requirements that the students must correspondingly follow because they are part of your assessment (grading) metrics. By example, the default statement that you might need to repeat a few times throughout the class could be "Unless I say otherwise, any comments that I may make back to you about your grammar or phrasing are simply to help you appreciate how you could communicate in a clearer, more professional manner". Alternatively, should you decide to grade grammar and phrasing in some manner, do so fairly against all the outcomes that your course should provide, do so with at least implicit blessings from a quorum of your department colleagues, and do so using standards that students can access before you start the assessments. This avoids the undue perception that you are unfair or capricious, grading too harshly for grammar and phrasing as only a small part of the course outcomes, being the only faculty member who cares about such things, or using standards that no one else uses.
You should continue to refrain from correcting your students' conversational English. For the reasons you identified, correcting their speech or an informal email is unlikely to lead to a good outcome because it's so personal. There's a good chance they'll experience it as insulting and intrusive, they won't change, and it certainly won't help your relationship with the student.
But you are free to red pen your students' written assignments with corrections, meaning that if you want to work on improving your students' formal language skills, even a math teacher is free to give essay assignments, e.g., "Tell me your career goals and why you're taking this class."
English is not like mathematics in that there is typically not a universally accepted prescription of what is right and what is wrong. In particular, what is considered "right" is highly dependend on social and geographical background. Even things that you have been taught as being unequivocally wrong might be considered perfectly fine in different social or geographical circles.
So unless you plan on doing an extreme deep dive in different dialects of English, what you will end up doing by systematically correcting your students' English is imposing your personal view of what is right and wrong. Doing so can significantly impact the confidence of students speaking/writing English that would be perfectly fine in their social/geographical/cultural background.
Since you are a mathematics instructor and (presumably) do not have formal training in the pedagogy of langauge, you should probably refrain from "correcting" your students' English.
Ask them whether they want this feedback.
Is there any way I can help students develop better writing skills without becoming despised?
Yes: change careers and become an English professor.
But if that seems like too drastic a step... consider that people who make English mistakes1 fall into two groups:
The first group is people who aren't particularly interested in improving their English2, and who will not welcome advice on doing so. That may be simply because they don't see the value in getting beyond "can be understood", or it may be for more considered reasons; there is a great deal of politics and history tied up in the question of which varieties of English are considered legitimate "varieties" and which are viewed as "incorrect". Even well-meant "corrections" can easily come across as patronising or worse.3
The second group is people who do want to improve, and who welcome advice on how to do so. Many, but by no means all, are non-native speakers.
Both these groups are large. There are enough in the first group that if you make a habit of giving unsolicited advice, a great deal of it will be wasted effort. There are enough in the second group that you could devote your entire life to helping them without ever running out.
That being the case, if you want to volunteer English help to people, it makes sense to focus on the latter group where your efforts will be productive and appreciated.
So ask them whether they want to be helped. At the beginning of the term, you could let your students know that you're willing to provide feedback on written English to those who want it, explaining your pedagogical intent and why you feel this might be worthwhile, and then see who asks for it.
If you don't get enough interest from your own class, well, there are academic editing services who will pay for somebody willing to provide that kind of advice in technical writing. (Some don't pay very well, but since you're looking at doing this for free anyway...)
I sympathise with your position. I used to be the guy who offered unsolicited English correction to everybody, whether they wanted it or not. I eventually realised that it was largely unpleasant for my associates to be getting advice they hadn't asked for, and I also discovered that a side job in academic editing was a great way to get paid and thanked for correcting people. So now I expend my energies in that direction, and everybody is happier for it.
For which, read "make English mistakes" as "do not follow the version of English that I consider to be authoritative".
For which, read "improving their English" as "changing it to more closely resemble my preferred version of English".
For instance, there are far more people who speak Indian English than Australian English. But many who acknowledge Australian English as its own variety would view the quirks of Indian English as "errors".
There are two ways to approach language: prescriptively and descriptively. You are leaning very strongly towards prescriptivism, while most linguists are much more on the side of descriptivism these days. There's always a balance to be found between the two in real life, though.
Prescriptivism is the belief that there is a right way to use a language. Descriptivism is the belief that the way people use a language is the right way, and that all you can do is describe that way. In a way, prescriptivism is like treating the way that language is used as people doing math (they can definitely do things very wrong) and descriptivism is treating them as the math itself, where if you see a conclusion that collides with your world view (assuming the math was applied correctly) you adjust your world view.
There is definitely some room for prescriptivism: learning a language is definitely based on it. And when the meaning of the statement is becoming unclear, there might be a place for it too. Some languages also have things that are very hard for language learners, but native speakers are very consistent about it (e.g. when to use "de" or "het" in Dutch) and in some environments, some prescriptivism might justified.
I don't think prescriptivism has much of a place in a Math class, except in places where the meaning is becoming unclear.
So yeah, especially, for something like "Me and X met", you should continue refraining commenting on it. Or perhaps better: don't just hold back commenting but see it as learning about how language is evolving.
While I respect the opinions of those of you expressing a disdain for prescriptivism, I personally believe that education and discourse would benefit from a return to it.
Excuse me while I expand on what this actually means:
While I respect the opinions of those of you expressing a disdain for teaching made-up rules by rote and rigidly enforcing them, I personally believe that education and discourse would benefit from a return to teaching made-up rules by rote and rigidly enforcing them.
That's all prescriptivist grammar is: made-up rules, which must be learned by rote because there is no logical reason to prefer them over any other rule. Each prescriptivist rule was made up arbitrarily, typically by one person who lived hundreds of years ago, typically a man with high enough social status that enough other people accepted his personal preferences as authoritative wisdom.
So the reason you might think it's wrong to begin a sentence with a conjunction, split an infinitive, or end a sentence with a preposition, is because some long-dead guy preferred it that way so he made up a rule saying you had to write and speak like that. And he wouldn't have had to make up the rule if people were already following it, so all of these rules represented active campaigns to change how other people speak and write, so that it conformed to the rule-inventors' preferences.
Try this: invent your own grammatical rule. How about "never use a semicolon after an adjective" or "each noun in a list must have its own article"? Now imagine yourself teaching your new rule to your students and enforcing your rule when your students email you. Would it be reasonable for you to do this? Surely not, because it is not a teacher's role to impose their own personal preferences on their students. But then what makes it any more reasonable for you to impose the personal preferences of some guy who died 100 years ago? His preferences are no more rational or less arbitrary than yours.
Now suppose all of that hasn't convinced you that prescriptivism isn't beneficial to your students' learning. But how about this: your students are smart enough to know that your grammatical rules don't make any sense, and perhaps they'll think the rest of what you're teaching is just as dubious.
I once read the thesis written by a coworker, in the degree program for Computer Science or something similar to that. It was full of typos, spelling errors, and grammatical errors, and I found that to be detrimental to the reading of the paper. Her professor had the policy of "this isn't an English class." But, that doesn't mean you shouldn't do basic proofreading! How do I know you don't have similar errors in the math or code expressions? It gives a rather poor impression, tainting any assessment based on the work being reported on.
I think you need to explain that language varies with the context, and writing up results constitutes a more formal context than spoken conversation. Even then there is a range of contexts: spelling mistakes on a hand-written exam can be overlooked, but a typewritten term paper should have been checked before handing it in.
The issue is: does the student consider a quick email to be an informal context that can use colloquial language similar to speaking in person? Then would a correction be useful because the writer actually doesn't know, or would it be annoying because the writer does know how to use a more formal mode, but was simply not using it here?
Whether it would be fruitful to point out such mistakes in email depends on how well you know the specific student. Noting it on their work, on the other hand, is appropriate, and they should be prepped with an introduction that their results should be documented in a more formal tone.
Just do it occasionally, not every time. If someone has terrible grammar then just correct one mistake, not all of them.
Tell them when you first meet them that you think writing skills are important, so you might occasionally correct grammatical mistakes, and you hope they find it useful. You could tell them the second sentence of your question, which puts it very clearly.
Some of the other answers look more thoughtful than mine, so don't take my advice without reading them. Overall, some advice might be suitable in one country but not in another. For example, in some countries, a mathematics instructor may not be free to give assignments that are not about mathematics.
I am French and obsessional about the correctness of my beloved language. I make lots of efforts to write and speak properly and suffer when reading emails and listening to some people speaking (including on the national radio).
I am also a father of two children. One of them is like me: grandiose language skills :). The other one is the destroyer of the future of French: not only he communicates as if he was taxed on the nr of lttrs, but in a sentence of eight words he manages to fit nine errors (slightly exaggerating).
Both had the same education, same schools, same everything. Both are very good at maths and sports.
The older one will have an easier life, the younger will suffer because of how he will be perceived - despite all my efforts to improve his French.
So what should you do? Nothing. It would especially be a pity if you marked based on their language abilities, some people just do not get it.
In a properly functioning university, every academic in every field is a member of the grammar police
The pedagogical intent of correcting grammar mistakes is self-evident --- it is to teach and assess proper grammar. It is perfectly appropriate (and generally desirable) to correct grammar in any field of tertiary instruction, including mathematics and other STEM subjects. As to whether you choose to do this in activites outside the scope of your own courses (e.g., in emails from students, etc.) that is a matter for your own discretion. I tend to agree that it is a duty of academics to instruct students outside the scope of coursework, including giving instruction in soft-skills, spelling, grammar, etc. In any case, in a properly functioning university, every academic in every field should be a member of the grammar police.
As a secondary matter, I think you need to reconsider unreasonable fears about how you will be perceived by your students. You are there to be their teacher, not their buddy. Properly functioning universities do not "let the inmates run the asylum". Students receive instruction from experts who know more than them, and if they "despise" receiving feedback on the quality of their work from said experts, then they are in the wrong place, and are free to refrain from pursuing tertiary education.
With regard to the specific issue of correcting grammar in student emails, I don't have a strong view either way. It is legitimate to give students instruction on their grammar in this context if you wish, and it may assist them in learning to craft more professional emails. That is a useful skill for the student to take into the professional world after graduation. On the other hand, you might decide that this is going beyond the scope of the intruction you want to give, or you might decide that you want email communication to be a forum where students needn't worry about these issues. Either approach is legitimate and it is within your academic discretion.
When students write material of whatever nature for class, it is a good idea to ensure they are using an appropriate academic register and to point out egregious or repeated spelling and punctuation mistakes and so forth. If a student had written "Me and X went ..." in an assignment, whatever the subject, it would be perfectly apposite to add a note saying "wording too informal for an academic essay." Who could possibly disagree?
However, whereas it is beneficial to correct someone's writing when feedback might be expected, the same is not true when it comes to correcting everyday communications. It is extremely rude, if not offensive to correct people's speech or writing while they are in the course of communicating directly with you.
Imagine, for example, a situation where you had submitted an article for publication in an academic journal. You would most definitely expect, if not welcome, corrections to slips and errors in your writing within that article. At an outside chance, you might even welcome some feedback on the English (or whatever language you are using) in your cover letter. However, suppose having had your article accepted subject to revisions, you emailed a query to the editor regarding who you should contact with regard to a serious typo and and in their reply they wrote:
By the way, the correct grammar here is not "Who should I contact?", but "Whom should I contact?"
Your might well think that in addition to being pompous, this behaviour was both rude and unhelpful. You would be correct. [You might also think that the editor was spectacularly ignorant about the facts of the language, although this would be a side issue, as in the Original Poster's case.]
Of course, the Original Poster tacitly knows all of this. The reason they are asking the question here, is that they want to know whether their position as teacher should allow them to trample on the normal sensibilities of their students, and correct them when it might be otherwise be considered rude or intrusive (i.e. correcting a student's email). Or rather, they want to do it anyway but aren't sure how to.
I suggest to the Original Poster that they instead model the behaviour that they would like to see in their students. In other words, I suggest that they treat all their correspondents, whether or not they are students, with the same courtesy and respect that they would expect from their own colleagues. Hopefully, this will rub off.