I have a student who's facing suspension. The student got a note from the dean's office saying s/he is suspended but can write an appeal letter if s/he wants the appropriate deanling to reconsider the case. When the student emailed me, I realized that the student really had no sense of how to write the appeal. I'm guessing some students would know, but a lot wouldn't, especially first generation college students. So I thought I might be able to help a tad.
I'm only considering grade problem suspensions, here, though some of what I have to say may apply in other sorts of cases.
If you get one of these letters, you should start by sitting down with yourself and thinking about what happened. You might lie to a prof, and that's irritating, but if you lie to yourself, that's plain stupid. So be totally honest and think about why you had grade problems.
Next, consider if you really want to be in college, or if being suspended is a good chance for you to do something else for a year (or forever). If you don't want to be in college, do something else. There are lots of meaningful ways to spend your career or earn a living. Find one and try to do it well. Focus on making your live worth living. If that means getting a college education, fine. Maybe taking a year and going to a community college would work well for you. If so, do that.
If you decide that you want to appeal your suspension, read the suspension letter carefully, and see what the letter says in terms of an appeal process or reinstatement. If you want to go to that school, take those instructions seriously.
If you have an opportunity to write a letter appealing your suspension, view it as an opportunity. There's some person on the other end who's going to take time to read your letter, take time to review your record, maybe make a couple phone calls. Don't BS that person around.
Further, in all likelihood, that person wants you to succeed and wants an excuse to let you come back. That's how people in student services are. But, if you don't succeed, they may be held partly responsible, so they aren't just going to let you return for the fun of it.
When you're ready to write your letter, write down what you know about why you failed. Did you party too much? Did you get sick? Whatever it is, write it all down.
Now look at the list and think about how you can change the things that contributed to your failure. Can you change your living situation? Can you get yourself into a study group? Can you go to a math center? Write every idea you have down. If you have time, visit with your advisor or a tutoring center or whatever on campus, and see if they have ideas.
Now, again, be honest with yourself: are you willing to do what it takes to change the things that contributed to your failure? Can you make that commitment? Really? If not, then quit writing and do something else.
If so, then write down how you're going to change what you can change.
Then start drafting your letter. Have someone else give you feedback on your draft, preferably someone with a good deal of college experience.
Here's what I think the dean wants to see in the letter:
1) Yes, you take responsibility for the failure.
2) You have thought about why you failed; explain in short what you did (or what happened) that contributed to the failure.
3) You have thought about how to change what you can change, and have already taken some steps to make those changes. Explain in short what you've done so far.
4) You have thought about what further changes you will make once classes start, and can outline your plans.
5) A thank you for the opportunity to appeal the suspension, and an appreciation for someone taking the time to reconsider the suspension.
That's it, I think. That's what a dean wants to see. If the dean is convinced you can succeed, s/he may allow you to enroll again. S/he may suggest that you take specific steps to help yourself succeed. If the dean thinks you're not going to succeed, s/he won't allow you to enroll, but may give you suggestions to help you prepare to request reconsideration in another semester.
In either case, if you want to go to that school, take those suggestions seriously and follow through on them. Take your commitments to change seriously and follow through with them. You can't control a lot of things, obviously, but you can control some things.