Parts of this piece are written for instructors/mentors and students, with the first part more directly for instructors. If you are just here for "what to do" advice, skip down to "advice I'm going to give students".

Context
I've written before about how your grad school inquiry email better relate directly to the person you’re emailing and that a cover letter should center your expertise relevant to the position. But, I have now had numerous interactions that have compelled me to make a change in my scicomm course syllabus [1] this semester: adding in a module on professional communications.
My syllabus is packed (even though it's down to about half of what it was the first round). So, why am I adding in another layer? As far as I can tell, undergraduate students are not getting deliberate (or adequate) training regarding the importance and practicalities of writing professional emails. (Or most other dimensions of writing, which is why my co-author Stephen Heard and I wrote our upcoming book: Helping Students Write in the Sciences.)
Today, I'm also writing to say that I think you (as in we all, us faculty and instructors) should be teaching about professional emails.
Three reasons why teaching professional emails is our responsibility
As a first-generation college student myself, I had no idea what professional emails were supposed to be. (Granted, email hadn't been around long, especially for high schoolers when I was coming up.) I got lucky, though -- I was raised a letter-writer, and I probably just went at email the way I did letters. I was also taught to type by following formulaic templates for office memos and the like. So, I got some aspects of professional emailing right, because emails got ported in from those analog genres.
For various reasons, the undergraduates we see today are in a similar boat regarding limited exposure to/models of professional email correspondence. I know this is true because nearly every email exchange I have with an undergraduate student makes this gap in training very clear. This has been the case at least as long as I've been teaching in higher ed (since 2015).
Someone has to to help these early career learners amass the professional skills they need to "cold-call" email people for internships, request new-to-them professors become their honors project supervisors, make effective inquiries about working in a lab or on a field team, and more. If I don't build it into my syllabus, these students probably won't ever receive this email advice. (I say that with confidence because (a) my course is a spring, senior-level course -- so, it's one of their last chances. And (b), I direct assessment for our department, so I can tell you: professional emailing is not part of the formal curriculum.) Of course, that means if you don't build it into your courses in some way, the students around you are just as likely adrift.
And a bonus fourth reason: the online advice is junk. I did a web search for advice about how to write a professional email, and let me tell you: if that's what our students have to resort to, it's no surprise we're not satisfied with the emails we receive. The materials I found were either were not comprehensive enough, not at all calibrated to an academic setting, and/or didn't offer a sufficient/accurate template.
So yes - I think it's our responsibility to share with students what we know about how people like us prefer to be contacted via emails. If you don't have capacity for that (and most of us don't, for all the reasons under the sun!), feel free to just link them over here.
I'm going to combine the advice below with an initial assignment, and then build email professionalism into the other major assignments. My intent is to provide students with multiple opportunities to practice and receive feedback during the semester.
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It's going to make a bit more work for them, and a fair bit more work for me (to give all that feedback). But, I'm channeling the most valuable piece of teaching advice I ever received (make only one major change per semester), and this round, email professionalism is it.
Advice I'm going to give students, starting next week
1. Don't assume your email is being ignored.
It might be your email was overlooked or totally buried. The people you are contacting get dozens or even hundreds of emails every day, and they have a host of professional and personal responsibilities besides emails. At a minimum, waiting 24-48 hours to respond is perfectly reasonable [2], so don't get upset or concerned about a brief delay.
If you haven't heard back from someone in a week or so [3], you can respond to the original email you sent. Just change the "to" address back to the person you're emailing, and write something like: "In case this email didn't reach you last week, I'm re-sending it. Thank you, signed your name, etc." Keep your reminder brief and very polite.
If you've sent a couple such reminders over the course of 2-4 weeks, then you might need to assume that person is not going to respond. (If this is the case with an instructor in a course you're in or an advisor you're working with, then you may need to reach out to your department chair to ensure you're getting the appropriate feedback and communication.)
2. Don't tell people things they don't want to hear.
This could also be framed as "what you say can and might be held against you."
The ultimate "unforced error" I see in emails is people saying things that are likely to irritate, insult, or otherwise upset the recipient. If you're asking for a favor (which is what most career-related emails do), you do not want to aggravate the recipient. They are much less likely to be sympathetic to your request, make time for you/your ask, or offer the kind of opportunity or support you are looking for.
Here are some examples of what not to do and how to do it better:
Don't imply that you aren't aware of or don't respect the time and intellectual investment that someone has made. In other words, don't email an instructor or TA and tell them you will/did miss class because you're too busy, aren't interested, prioritized something else (especially another course), etc. Even if that's true, just don't say it that way! Also do not ask the instructor/TA to do extra work to make up for the work you missed/skipped.
Don't email people you don't know and assume they have the time or inclination to make their professional network immediately available to you. Personally made/maintained professional relationships are one of the most valuable parts of someone's social capital. People rarely can/will risk having a stranger (in this case you) reflect badly on them with their professional contacts.
Don't tell people more than they need to know. If you're applying for a job, for example, share very little personal information. Any personal information you share (including when you can start a job, whether you're willing to work/live in a certain place or work on certain topics, etc.) can be negatively interpreted by the potential employer. For example, you might inadvertently imply that you (a) aren't qualified, (b) are immature/unprofessional, (c) don't respect the nature of the work, the employer, etc., and/or (d) see the job only as a stepping stone/don't plan to stay in the position very long.
3. Keep your email as brief as possible, while still being professional.
Tell the recipient why you're emailing in 1-2 sentences.
Then, make clear what kind of action or feedback you need.
Most of the time, that's all you'll need in an email.
But, be sure to always use complete sentences, and avoid abbreviations and acronyms unless you can be 100% certain the recipient understands them.
Don't use emojis in professional emails.
4. It's not just a good idea to format emails as if they were professional letters; it's necessary to do so.
Most of the people you and other students email are old enough they have actually stamped and mailed personal and professional letters. This means that the people who receive an email expect the email to be formatted like a letter, not like a text or chat message. That's gradually changing as generational turnover happens, but for now, letters are still the standard emails are judged by.
What does a professional email/letter look like?
Starts with a formal greeting - If the person has already told you what to call them, do what they said. Your first clue is how an instructor refers to themselves in their own emails to you -- look for the name they use when they sign off their emails. If you're not sure, "Dear Professor LastName" is a good guess, because it's respectful while still being pretty general. Or, you can google the person and use their title (e.g., Dr., Professor, Dean, etc.). Whatever you do, don't use just their first name unless they've already told you to. And, use a formal greeting, such as Dear, Good morning, etc.; no "Hey Professor" even if you're on good terms with the instructor.
Briefly and immediately gets to the point - Getting right to the point helps busy people skim for important information. The people you email probably get hundreds of emails daily. You need to put the point of the email at the beginning. Why? In case they don't even make it to the end of your email. Making your point succinctly and immediately also demonstrates that you are able to communicate what you need to in a professional, efficient manner.
Optional: you can include one more brief paragraph that includes more information about the situation you're writing about.
If you're emailing about an internship, student research, or job opportunity, be sure to include information about your skills/experience. The most efficient way to do this is to attach a current resume or CV.
It can also help to "name drop" a little - if someone else told you to contact the person you're emailing, mention that and briefly describe the context that led to the recommendation.
Including this information about yourself helps the person you're "cold calling" to understand if your interests and skillset match what they are able to offer.
End with an actual closing - You can simply say, thank you, sincerely, I look forward to hearing from you, or best wishes.
Sign your email
Your "signature" will likely have three lines at least:
On the line beneath your closing phrase, include the name you prefer to be called. That is valuable information to the person you're writing. There is a "code" or short-hand in emails that means whatever name you use to sign your email is the name that people will use when they reply to you. So, if you use your middle name, a professional nickname, or the like, use that.
On the line below your preferred name, provide your full name.
Next, include your student ID # (if relevant, e.g., if you're emailing a professor or TA about grades, etc.), or any relevant affiliations (e.g., PhD candidate, SciComm Lab, University of Wyoming).
5. Can you ever send a casual email again?
Yes!
Emails can be much more casual (and even function/be formatted more like texts) once you know someone well and personally.
But, if you're a student and you're taking multiple courses from an instructor or doing research in a professor's lab, you should always maintain professional email habits, even if you've known them for years. That includes calling them Doctor (Dr. in email) or Professor, unless they direct you to call them something else.
Why does all this formality matter?
These slight modifications to your query emails could help you get more replies (quicker and potentially more positively, too).
Folks who don't know you are very busy, and if an email doesn't use professional formatting, it may just seem like spam to ignore/trash.
Furthermore, if you're emailing someone you don't know, your initial email is your first "interview" with a stranger who will hopefully give you professional opportunities or discuss your grade in more detail, etc. You'll be more likely to get their attention in a positive way if your email demonstrates the professionalism that they'll likely expect from you in an internship or other job training situation.
Lastly, if someone does take you on for an internship, research experience, or the like, you'll be representing their organization or research group. They need you to do that professionally. That means, they need to see from you, from the beginning, that you know how to professionally format/write any emails you send out on behalf of their organization.
NOTES
[1] I share all my syllabi here, with each posted at the end of the semester.
[2] Of course, there are a couple of exceptions here. (1) If you're dealing with an emergency, you may need a faster response. Just recognize that you still may not get one. (2) If you're emailing someone (particularly an instructor or TA in a class you're currently taking), they may have provided different time frames for you.
[3] Your own short-notice situations are not usually urgent enough for the recipient of your email for them to get back to you immediately. What does this mean? Above all, don't wait until the last minute to request reference letters, extensions of deadlines, or feedback on materials that you need to submit somewhere else in less than two weeks. Similarly, don't expect an instructor/TA to respond to you immediately if you notify them overnight or day-of that you will be missing class, an exam, etc. (While I abhor the attitude that students should be penalized for their own life circumstances -- deaths, illnesses, caregiving, job responsibilities -- there are still too many people in academia who don't have sympathy for these issues. So, if you see one coming, do what you can to be proactive about notifying people like instructors and research supervisors.)
[4] Of course, there's a flip side. You should absolutely use complete sentences, full punctuation, etc. And generally speaking, no emojis. (Basically, professional emails aren't texts, even though we pretty much all use emails, sometimes, as texts. Let's say it this way: how we (probably) email our friends is not how we do professional emails.)
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