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I am a qualified mathematics teacher but I have left teaching because I could not tolerate the behaviour of students. Now I am a mathematics tutor and I love that I get to teach students who are eager to learn. I had fantastic results with tutoring. Some of my students had failed repeatedly and after working with me they got top grades. These were master's students. I also have a couple of school students who are like little scientists and love to work with them. I know that success comes down to enabling them to understand the intuition behind the concepts.

However, I do not want to help my students to only improve their grades but I want them to learn to think mathematically. I would like to create tutoring groups that focus exactly on teaching university-level students how to think mathematically. However, I am finding it tricky to find an appropriate book to use as a foundation for tutoring. I do not want to assume too much knowledge. A first-semester university student should be able to follow it. So far I have thought of using a book about abstract algebra for this purpose. This is because that was the subject that changed the way I saw mathematics and it got me comfortable with understanding and writing proofs. I am not thinking of just going over the abstract algebra but concentrating on understanding the reasoning and motivation behind it all. Enabling them to see how theorems are born etc. Now I am thinking that maybe abstract algebra might be too much. I am looking for a book that helps university students learn how to think mathematically and most importantly how to write proofs. I want to emphasize that it mustn't assume too much knowledge. Thank you in advance.

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    This is really a question for matheducators.stackexchange.com. Jan 25 at 12:40
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    If you search Amazon for "The Nature of Mathematics" you will turn up some related books. The top choice isn't cheap, though. I actually wrote such a book, though won't promote it here, nor is it listed early in the amazon recommendations.
    – Buffy
    Jan 25 at 12:53

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