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Bo102010 comments on Why do people commit mathematical mistakes? What are the mechanisms behind them? - Less Wrong
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The single most common mistake I have made, and that I have seen when tutoring others, is sloppiness. Especially losing track of a negative sign, either not noticing one in the original problem or forgetting it when writing out intermediate steps. That is why I have started a few years ago trying to be very careful when working things out. As I wrote in a comment a few years ago (I don't remember where, here or OB or HN, it was about math for engineering), when studying math or anything else that may be safety critical, don't settle for an A, do your best to get a perfect score, make it a regular habit. Even if someone else is checking your work when it is critical, don't trust them, they can make mistakes too.
One mistake I noticed when tutoring a high school student was what I might call "failure to take seriously the rules."
We were studying Geometry, and many times the student would make a big assumption (e.g. the angle is 90 degrees) without noting or questioning whether it was true.
When I'd ask him about it, he would say "Look at it, it must be 90 degrees!" or "If it's 90 degrees, then I can solve this other part over here and be finished." When I'd explain "You can't assume it's 90 degrees," or "You're assuming what you're trying to prove," he would grudgingly go along.
So, I think there is a class of math mistakes that come from "a failure to realize that rules in math are not like 'rules' in your everyday life - they are ironclad and irrevocable."