Could faculty members earn a degree while being employed?
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Normally someone who holds the rank of a professor already has an earned doctorate, so there really isn't any need for additional qualifications. It's not really clear to me why a faculty member would then want to go on to pursue an additional degree. To my mind, it would be one of those warning signs that they're not entirely serious about working in their given field, and might choose to move on to "greener pastures" in a few years.

So, in short, I don't think there's any real advantage in a professor obtaining an additional degree.

However, if one is a lecturer or adjunct, then returning for a further degree in the field in which one is working makes sense from a career perspective.