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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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Here are some really friendly proof books (with PDFs available online):

These books do a good job of not just asking students to prove things, but also gently walking students through the whole idea of what it means to prove something and how one actually goes about proving things in a variety of different contexts.

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    $\begingroup$ “How to Read and Do Proofs: An Introduction to Mathematical Thought Processes” by Daniel Solow should also be added to this list (and IMHO at the top). $\endgroup$
    – Aeryk
    Jan 26 at 18:28
  • $\begingroup$ @Aeryk thanks for the suggestion. Hadn't seen that one before, but I like it. Added. $\endgroup$ Jan 26 at 20:08
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You might take a look at Mathematics and Plausible Reasoning by Polya. First volume is probably sufficient for a starter.

https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096/

It's not 100% what you are looking for, but I think a vector in similar direction. IOW, the emphasis not on proofiness but on how to make good guesses. The subject he mostly uses is number theory and of a easily grasped slant (not abstractions, but observations on primes and the like).

I had it as a college freshman in a special course. I didn't put anything like full effort into the course...but I still remembered it being fun and motivational and special and different (from calc 3 I was in for regular math).

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    $\begingroup$ Thanks for the suggestion. Polya was on my mind too but not this particular book. $\endgroup$ Jan 25 at 20:03
  • $\begingroup$ hals und beinbruch $\endgroup$ Jan 25 at 21:27
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    $\begingroup$ @KristinaDedndreaj There's also the other book by Polya, "How to solve it". It's more geared towards pre-university, but may be very useful if your students have trouble solving (math) problems. $\endgroup$
    – Pablo H
    Jan 26 at 16:27
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    $\begingroup$ @PabloH I know about "How to solve it". Polya is a true master. $\endgroup$ Jan 26 at 20:13
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Many of us have had good results doing this with a class run in Inquiry-Based style. There are many, many variations but basically the students are responsible for working their way through the material. The teacher is more of a guide than a lecturer.

I'll describe the classes I have run, to give a feel. On the first day I give them notes that are pretty much just definitions and statements of results. I ask them as homework to prove the first four results working entirely on their own, without books or the internet, and when they come back I assign four people at random to present at the board. Other class members gently discuss, criticize, give examples and counter examples, and gradually come to a consensus about how to do it correctly, usually by modifying the initial argument. I say as little as possible, including not nodding or head-shaking, so that they do it instead of looking to me. But if the group agrees on something that is wrong then I do not let it pass. I might say "Giving three examples that the square of an odd is odd is not a proof, we need to show it holds in all cases," and it is up to the group to go at it again.

You need a right-sized group (maybe ten to twenty) of students who have seen at least some reasonably hard mathematics (maybe Calc II?). However, under the right circumstances it is beautiful-- you can see them becoming mathematically-minded people in front of your eyes.

For a taste, I have an Intro to Proofs book and there are many other suitable ones at the Journal of Inquiry-Based Learning in Mathematics. For more see the web site of the national organization, COMMIT.

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    $\begingroup$ I like this. Do you think I can do something similar in online classes? $\endgroup$ Jan 26 at 20:27
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    $\begingroup$ I've never done it online. I'd try the national organization and look for your regional branch. I know that the folks in my region are great and your region may have an emailing list that you could ask on. But my opinion would be that I'd try it. You may need some rules to enhance group dynamics (students have to keep the camera on and the mic open, for instance), but it seems worth a try to me. $\endgroup$ Jan 27 at 17:11
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I think the following books can help.

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Byers, William. How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton University Press, 2010. PUP link.

Emphasizes the creativity in mathematics.

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My favorite book so far is Jay Cummings "Proofs: A Long-Form Mathematics Textbook." It not only has great information, it is well-written and an enjoyable read.

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