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I am doing fine in my classes and enjoy math. That is not the issue.

During my spare time, I am doing more advanced stuff than my class is currently doing. My question is though, would I benefit more from a tutor / teacher in learning stuff outside the current classes I am taking or is learning on my own the superior?

Especially since I am content with my current school marks and more interested in stuff outside the curriculum, I do not see a reason to pay substantial amounts of money for a tutor as it is more pleasure than work to me. But, would a tutor be of much benefit in this case? Would I have a hard time finding a teacher / tutor that would help me learn more advanced stuff without breaking the bank?

I should note I am still in high school. As such, most tutors in my area cater towards just showing people the stuff they need to pass final exams. I wish to know more than just cookie cutter explanations, where should I be looking for tutors if I choose to get one?

I guess one reason I ask is in Robert Greene's book "Mastery", he puts a heavy emphasis on the apprenticeship phase and how many "Masters" had someone teaching and devoting significant time to them along the way.

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I don't have time to write a whole answer, so I'm just leaving a comment: In general, a tutor isn't really necessary for a lot of math. It really does help to have someone who will look over proofs when you start writing those, but a lot of people on this site will do so... (so long as you don't swamp us with a lot of "check my proof" requests. :)) – apnorton Jun 13 '13 at 14:02
It sounds to me like you don't have any particular area that you wish to explore. I'd encourage you to work on some of the problems on, and see if there are any particular areas in which you are interested in further study. That will also help guide you in finding a suitable mentor. – Calvin Lin Jun 13 '13 at 15:19
I have never heard of that website before, thanks for that! – nitrous2 Jun 13 '13 at 16:30

Most people find some stuff pretty hard to study in their own, and having a tutor with knowledge and experience can't be bad. That's one of the main didactic reasons universities exist, otherwise most people would simply "learn on their own".

Anyway, as you advance in your studies, you'll find it easier to study and learn new stuff on your own.

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I should of noted I am still in high school. – nitrous2 Jun 13 '13 at 14:00
I think this only reinforces my point. @nitrous2...though I think it's always a good idea to try to learn new stuff by your own. – DonAntonio Jun 13 '13 at 14:04

The answer to your question depends very much on you. Some people (including me) learn well in isolation. Even if someone explains something to me, I often have to go away later by myself to review it and make sure of the details. Other people (and I think they're the majority) learn better by interacting with others --- possibly teachers or tutors, possibly fellow students learning the same material. Obviously, the answer to your question depends on which of these learning styles is yours. If I had to guess, I'd guess the second, simply on statistical grounds; as I said, there seem to be more people of the second sort than of the first. But you have lots of knowledge about your own learning style and can surely make a more informed guess than I can.

On another matter, raised in earlier comments, in my experience, university professors are generally willing to help sincere, intelligent students even if they are not enrolled in classes. Of course, there are limits; I can't spend all day answering everybody's sincere, intelligent questions (though it might be more enjoyable than some of my real work), but I'm certainly willing to do it in moderation, and I think the same goes for most of my colleagues.

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I think that the teacher problem has many sides but now you are interested only in some of them. :-)

For me the learning strategy (mainly) depends on seeker’s or student’s skill. I think that sooner or later you should find a good teacher free of charge and I wish you good luck at your way. :-D

I try to illustrate my ideas by my own mathematical way. By the way, :-) you can overview it in my profile.

Personally I never had a tutor, never paid to teachers and I had the opposite problem: to free myself from the classes. :-) But I was powerful – I was one of the first students in my country in school math competitions.

But even already in my high school my school teacher, Volodymyr Prokip said to us that if anybody wants to know a subject deeper, then catch me, grasp my button and ask. :-) I remember as we were walking together along some kilometers from the school and were talking about matrices. :-) Now we are working in one institute and his room is placed in a couple of steps from mine. :-)

Personally I think that a bad teacher tries to find followers, but a good teacher shares with you his methods and strategies which you can use for solving you own problems. :-) I was lucky to find good teachers in my university (but, despite I think that the teachers from my university are good mathematicians, I still avoided the classes, :-) because their level was too low for me and the curriculum was bounded). I am grateful to prof. Taras Banakh (who is now my scientific consultant for the “Doktor of Science” thesis), under the influence of which I was fathomed into the idea to understand instead of selecting through methods, as jimmies, searching which of them can solve a problem. Banakh is still followed by apprentices observing his work and able to learn from the method «do as I». :-) Also I am grateful to my scientific advisor for my “Kandidat of Science” (PhD) thesis, dr. Igor Y. Guran. When I was working for the thesis, we tried to solve his mathematical problems, using my powers and his experience. I.Y. recommended to me directions of development, helps me with advices and as a reference book. :-) And he still does these. For instance, recently I sent to him my «The referee confession». :-)

Further, there should be scientific mathematical schools consisting of teachers, masters and their disciples and apprentices. For instance, I am a mathematician from Lviv topological school, which has the genealogy going through Karl Weierstraß and Carl Gauss to 17th century. :-) And the line of the school is to be fathomed by a seeker.

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I think when you start studying a topic, a teacher will benefit you but once you get into it, self-study will become much easier as you would have the basic concepts covered.

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I think it very much depends on the style of learning that suits you. I was very impatient when I studied high school and university and I couldn't listen to teachers. That's why for me, books were the best. But some people had problems reading and understanding books, and they indeed appreciated tutors.

I'd say try books first. There is a plenty of resources online nowadays, like this site. If you find that you don't understand something, ask your teacher. If that is not enough, maybe try a friendly university professor. If you are really into math, they won't send you away.

I don't know how it works in your country/school, but my university lectures are open to everyone. My math teacher at my elementary school (I studied in math class) was very much open to any extra lessons in breaks.

Also math competitions are a great place to meet friendly teachers who are willing to spend time with talented students. There are plenty of them in Cadana as well.

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Could someone else confirm that? "try a friendly university professor. If you are really into math, they won't send you away." Especially since I am still in High school I fear most places wouldn't take me seriously unless I attended. – nitrous2 Jun 13 '13 at 14:08
My country's education system is free for every citizen. Where do you study? – sm4 Jun 13 '13 at 14:09
Canada, unfortunately not a free system here. – nitrous2 Jun 13 '13 at 14:10
Do you have math competitions? That was a great place to meet friendly teachers who would spend extra time with talented students (no need to answer, this is just a comment :)) – sm4 Jun 13 '13 at 14:11
@nitrous2: In a reasonably large mathematics department you do stand a good chance of finding someone who’d be willing to work with an interested high school student, at least to the extent of being available to help out when you get stuck. – Brian M. Scott Jun 13 '13 at 23:26

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