I mean how someone goes from a student to become a teacher or a professor ? What is that state feel like ? Are you confident of the knowledge you have conquered over the years ?or there is part of doubt in there ?

I am current in the 3rd year out of 4 to get my degree, and I am struggling to learn the subjects to the fullest, always questions about things that confuse me and I can't answer , I can't solve problems with confidence (a lot of times I can't solve them at all), I don't have much confidence about my solution etc. I have a feeling that even if I get my degree those doubts and confusion won't go away like that, if there is someone who felt that way their opinion and point of view are more that welcome . You can imagine you are giving advice to your old self , an advise you never got but you needed.

For example because you did good on your finals doesn't mean you know abstract algebra , it's like the piano, you can't call your self a pianist because you know how to play well 5-10 songs. So when someone can be called a mathematician ? Or even that at least he knows abstract algebra. Also, I can observe that my knowledge on statistics and probabilities are a bit weaker over the pure math like real analysis number theory etc(because i study them more and give them more attention). Does every professor know deeply all the subjects ?

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    $\begingroup$ Most mathematicians nowadays do not know all subjects deeply $\endgroup$ – J. W. Tanner Nov 21 at 23:35
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    $\begingroup$ It's probably a better idea to talk to someone who knows you in the math faculty at your university, than to solicit advice from random strangers on the internet. $\endgroup$ – Gerry Myerson Nov 21 at 23:54
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    $\begingroup$ @GerryMyerson You can’t really call the people on this place “random” when it comes to advice about careers in mathematics. We do have tags for this sort of questions after all. $\endgroup$ – k.stm Nov 22 at 0:00
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    $\begingroup$ This video of Feynman might encourage you. $\endgroup$ – Shaun Nov 22 at 0:08
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    $\begingroup$ @GerryMyerson That people here don’t personally know the questioner is already said with the word strangers, no need for topping that with random. Anyway, I’m done argueing words here. As I said, I only wanted to make a point about not selling this site short. That’s all. $\endgroup$ – k.stm Nov 22 at 16:05

Given that I don't know you, nor does anyone else on this forum probably, I cannot say whether your doubts are justified or not. I can only share my limited experience.

Almost my entire (short) mathematical career was centered around algebraic number theory. I've taught a few courses in abstract algebra over the years, from group theory to algebraic geometry. My knowledge of statistics and real analysis is extremely limited, somewhere around second-year level. Certainly nowhere near research level, I have no deep understanding of anything related to those subjects.

I started to gain confidence in my ability as a mathematician once I became a teaching assistant, when I got to see the work of dozens and dozens of other students. Seeing their perspectives, their approaches, and hearing their questions greatly broadened my own mathematical toolbox, and it also made clear to me that my mathematical abilities were at least above average.

At first, teaching (tutorials, as a TA) was quite frightening and I did not think I was fit for the job. After a few months of weekly tutorials I was able to answer most questions quite comfortably, and deliver coherent and somewhat interactive lectures (of 20-30 minutes). After about a year I was comfortable discussing problems instead of simply answering them, and comfortable admitting that I didn't know an answer, and maybe even investigating the question together with the students. Around this time I started considering myself to be an actual teacher.

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  • $\begingroup$ your answer was very reassuring to me, i dont have someone to talk about this things and this was very helpful to me , thanks for the time you dedicated to write this . $\endgroup$ – Petros K Nov 21 at 23:59
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    $\begingroup$ I second this answer. I got my PhD a couple years ago, and I am only an "expert" in one or two very specialized areas. I assume you are an undergaduate? An undergraduate education only gives you a broad overview of some basic concepts in various areas of math. You will not get a good idea of what it's like to be a mathematician until you are in grad school for a few years. At that point, you will get some experience doing research and teaching. $\endgroup$ – Nick Nov 22 at 0:17

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