I've moved twice, and served on more SCs than I can remember at all three schools.
Looking for a new job as an assistant prof
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I've moved twice, and I never use inside letters. Past institutions and noted advisers only. I used committee members until I got tenure. I will never use inside letters, and it has never hurt me. My colleagues were not bad people. They were just fully, even overly, invested in me staying.
#1 - why you want to move is info you can share with any one of your letter writers. Some of that info is productive (e.g., more resources). A lot of it is not - particularly at higher ranking places (e.g., location, family).
#2 - it is a wasted letter. Letters should be spent extolling your virtues. They should scream to SC, get this person before their market value skyrockets. They should not be talking about anything negative. It literally depresses your value. Even if it is a very valid reason. Still works against you in the grand scheme of things.
#3 - If SC members really want to know why you're wanting to leave, they will reach out to you or to your networks to find out, signaling a deeper investment in you up front. A lot of times this happens before the fly out. A whole paragraph on why you want to leave is just entirely too much and reaks of desperation. A well-placed clause is enough to signal good faith and prompt the insider or weak tie call. If a school really wants you but does not think you are moveable, they will reach out, hoping to get lucky. If you seem desperate to move from the cover letter and reference letters, they will bury your file.
In the end, I did not rely on inside letters because I did not have to, and that itself is a good sign. Cultivate relationships across the discipline, and then you won't have to rely on people who have a vested interest in you staying put. Rather, you'll easily rattle off 3-4 names who want to see you upwardly mobile.
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As an associate prof who’s moved twice and served on several SCs I would concur with the suggestion of getting one letter writer who is neither your advisor nor at your current institution (ie, an ideal tenure letter writer). As an AP, yo should be from my view in a positionto get someone like this to write a letter for you. Then make the rest from your committee, no one from current job
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A letter from a colleague can help put to rest concerns that you're leaving because no one likes you, or that you're getting pushed out for malfeasance of some sort. As a SC member I would be concerned if a candidate did not have at least one friend in her current dept.
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I think what this thread proves is that there is no generalizable advice. People have very different opinions on what they're looking for, perhaps even within the same SC. Add to that unknown variation that may be present at the departmental or institutional level and who gives af. Do whatever you think is best and hope that whoever reviews your file is on the same page.
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For those that argue for one letter from a non-committee member not affiliated with your current department:
Is it better to have, say an Associate or Full that is in your area and initially met you through work-related networking, or another Assistant, maybe close to coming up for tenure, who is a superstar in the discipline but not necessarily in your area, and who you first met through social rather than work networking? Or is it a wash?
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I am going on the market in the Fall as an advanced AP, and was happy to get a letter writer in my dept in additional to a couple of outside letters. It works if your colleague is going to write you a strong letter, and doesn't have a conflict of interest, i.e. is invested in you staying. For the record, I got a campus visit for a job where I wrote in the cover letter that I was interested in their location for purposes of my research. Depending on the context, I think it is okay to mention that you are interested in a locale for research, or even family. Unless it is a super hot shot program that can get anyone, SCs are invested in finding candidates that are not going to stay a few years, and then move up. If you signal that you are interested in the area, that appeals.