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I'm on a semester break and thankfully I was recently provided with an opportunity to hire any private teacher I like to learn even more math while on break. However, I don't really know how to approach it. What are we going to do? They'll start to explain material that I can read in the book? Solve homework together? I think it's very counter-productive because I can learn more from my mistakes. Or should I come up with questions like "Could you explain why we use open sets in differential calculus?". But I can just google them and get the answer I need.

At the same time I really want to use this possibility since I was offered it. I don't want the opportunity go to waste and at the same time I feel like mathematics is a one man road. Maybe someone can provide me with an option on how to use the opportunity without being bored or counter-productive.

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    $\begingroup$ I would suggest you look at this question matheducators.stackexchange.com/questions/17989/… which the OP suggests that people can self-study and don't need a teacher. Some of the answers (including mine) discuss the benefits of having a teacher in math and these might point you in the direction of an answer for your question. $\endgroup$
    – Amy B
    Aug 12, 2020 at 8:19
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    $\begingroup$ For what purpose is this study for? What sort degree are you doing? Are you doing remedial study, or hoping to work on stuff ahead of schedule? The sort of math you'd need for a physics degree would be very different to the sort of math you'd need for a data science degree. $\endgroup$
    – nick012000
    Aug 12, 2020 at 11:28
  • $\begingroup$ @nick012000 I just finished my first semester, analysis I, II and linear algebra I. I plan to start analysis on manifolds in fall semester at an online university and analysis III plus a few other courses at my local uni. $\endgroup$
    – average math enjoyer
    Aug 13, 2020 at 2:21

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I agree that you need to do a lot of the heavy lifting. Furthermore, I would try to use texts that are suitable for self-study, basically always. Feel like the teacher-salary-justifiers try to push books that need an instructor more. I.e. even with a paid teacher, I think you should pedagogically optimized materials.

All that said, I do think a tutor can benefit you by challenging your knowledge with drill and interrogation. Something like how tutorials work in Oxford or Cambridge or whatever.

You can also, keep a notebook for your self studying and just write questions in the back to be discussed with the tutor. It will probably end up being more time efficient and accurate and interactive than SE questioning. You can plow through several in a session at once. (Also pretty much how you should use office hours.)

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  • $\begingroup$ I agree. At least at first, this should be an oral exam style interaction. Looking for problems and weaknesses all the way to elementary school material. This would take an experienced teacher though. $\endgroup$
    – Peter Saveliev
    Aug 12, 2020 at 11:54
  • $\begingroup$ Agreed. Select a topic/teacher where there is a strong match. For example, if you have someone strong in topic A, but "OK" on topic B, pick him/A to work on. Not B. Not even if you prefer B. (Assuming you plan to do both A and B eventually, of course.) $\endgroup$
    – guest
    Aug 12, 2020 at 12:19
  • $\begingroup$ Thank you, I think this is a very good idea. A good teacher will be able to fill in the gaps in my knowledge I'm unaware of. I'll follow through with your advice. $\endgroup$
    – average math enjoyer
    Aug 13, 2020 at 2:19
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My suggestion would be that you pick a book with some difficult exercises, attempt to work through the exercises yourself, and then your tutor can help you with the exercises you got stuck on. This works best if they give you hints which lead you to the solution, rather than just solving the exercise for you. The tutor can also check exercises you believe you have solved, as you might have overlooked certain cases or omitted important details.

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They'll start to explain material that I can read in the book?

(1) Depending on where you are in your mathematical development (which you don't make clear in your posting), you might take on a challenging text, for example, Michael Spivak's Calculus (4th ed), and work it through with the guidance of a private teacher.

(2) If not Calculus, then Gilbert Strang's Linear Algebra (also available in MIT Courseware).

(3) If not Calculus or Linear Algebra, perhaps Real Analysis by Karl Stromberg.

The point is to take a rigorous, challenging text and proceed through it methodically under the guidance of an experienced teacher.

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    $\begingroup$ I'm currently using exercises from Baby Rudin. I think they are very difficult for me, at least from what I tried to do. I was only able to solve a few of them. At the same time this book is so popular that when I Google solutions to it I found multiple different books, with complete answers to all exercises. $\endgroup$
    – average math enjoyer
    Aug 13, 2020 at 2:16

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