One of my best friends is working on her Master's in Mathematics Education to try to become a high school teacher. However, some of the professors in her advanced undergraduate math courses seemed to have little patience with her and made her feel dumb, so she doesn't feel very comfortable with mathematics proofs right now. In fact, she has never taken a math course on mathematical reasoning and proofs, so I think that's another big reason she doesn't feel comfortable with proofs right now. I recently offered to help her understand proofs better, but I don't know when we will start. I hope we will start soon, but it's hard to say since her summer coursework has taken her a lot of time this summer. When she finishes soon, she will probably not want to do this before the fall semester (if so, I wouldn't blame her).
I had worked with her before on some proofs, and I can tell from those experiences that I really believe she is capable of learning this. I really believe she has more talent than she realizes.
I tried looking for a free online math book on mathematical reasoning and proofs so that she wouldn't feel forced to buy a book. The only such book I found was number 72 on this list of free online books (the list is not exhaustive):
The book is called "Proofs and Concepts: The Fundamentals of Abstract Mathematics" by Dave Morris and Joy Morris. It seems to me to be a pretty good book, but I would like a second opinion. If someone knows a better free online book for such a study of proofs, please let me know. I can't find any others right now. The closest I can find is number 48 on that same list, but it's not directly about the methods of mathematical proofs.
I plan to borrow some material from the books "Proof, Logic, and Conjecture: The Mathematician's Toolbox" by Robert S. Wolf and "How to Prove It: A Structured Approach" by Daniel Velleman that I have. I believe these are very good books on the subject. Daniel Solow's book "How to Read and Do Proofs" is also very good, but I do not have a copy (I had the chance a few years ago to look at it).
I had originally posted this on sci.math (except for differences in the last two sentences of the first paragraph) back in early June before she started her coursework this summer, but no one had replied. So I will try here since this list is primarily about post-calculus mathematics education (and this is a post-calculus course).