Advice about studying high level math

[Ozma]

Sorry if this is the wrong section to post this, but I didn't find anything more appropriate. I'm asking for advice because I seriously need help. I'm self-studying mathematics (at a university level; unfortunately, I can't attend for various reasons) and in the future, I would like to become a researcher in pure mathematics. However, I'm afraid I've developed a bad habit and a bad mindset. I believe that to become a researcher, it's important for me to be able to solve as many problems as possible on my own, and to do this, I always try not to look at the problem solutions (even though I have access to them). The problem is that by doing this, I often get stuck for a long time on a significant percentage of the problems I tackle, and if I eventually give in and look at the solutions, I find myself demotivated because I would have never come up with that idea on my own. The fact that I make mistakes or fail to reach solutions within reasonable timeframes is seriously making me doubt my abilities. I wanted to ask for your input to understand first of all if this is normal, if any of you work as a mathematics researcher and have had the same problems (but still managed to secure a position in academia), and most importantly, if I shouldn't feel inadequate if, at some point, I look at the problem solutions. My biggest fear is sabotaging myself and risking not fulfilling my potential because I rely on problem solutions to move forward. Thank you for the time you'll dedicate to me.

 

[blamocur]

I am not a math researcher, but in my youth I did spend too much time trying to learn various chunks of relatively advanced math on my own. The main thing I learned is that even the best of books is no match to a good teacher. I would strongly recommend finding online classes or lectures. Personally, I've benefited a lot from some of the best Coursera classes several years ago, but they were mostly in engineering.

The second best thing would be a friend, or, even better, several friends, for studying together. But do try to find online classes and help from someone who already understands the subject.

I think it is a very good idea to do as many exercises as possible, but getting stuck is completely natural, especially when studying on your own. Getting hints is better than getting answers. If your field of study turns out too advanced for this forum I'd suggest asking questions on Stack Exchange.

 

[Steven G]

What level math are you studying?
As long as you improve in solving problems that is good. If however you don't improve then that is another story and maybe math is not for you.
In solving problems you need to understand how to solve it, not just memorize some boring steps. Then again, since you say that you want to study pure math than I suspect that you already know that.

 

[Steven G]

Can't go to college?
Why is that? Is it a financial reason, physical limitation, lack of time, no local college near you, etc?
If you can't go to college just for financial reasons, then imo that is not an excuse. Just walk into any college you want to on the first day of class and ask the professor if you can sit in their class. Be real nice to the professor and give the impression that you will be an excellent student in their class. etc etc. I personally never had turn down a student. After all, the reason I became a college instructor was to educate students so why would I turn down a request from a student to sit in my class? Take a full load of classes if you like in this fashion.

If in the future you can afford to go to that college you can ask the professor to let you enroll in their class with the understanding that you will not attend any classes and that they will submit the grade that you earned while you audited their class. Of course, while you are auditing their class they will have to allow you to take exams and spent their free personal time grading your exams. They will more likely agree with that especially if they see that you are a serious student.
Good luck!

 

[blamocur]

Just walk into any college you want to on the first day of class and ask the professor if you can sit in their class.

Many years ago I've done something similar. I wasn't as civilized and didn't even ask for permission, just sat in the back of the classroom through the whole semester without ever being questioned. And this was a class by one of the best researchers and teachers in the field in one of the best universities in the world. I have no regrets about attending but still feel somewhat guilty about not asking. Could my volunteering at this forum be a subconscious attempt to make amends ?

P.S. This days lots of colleges post videos of classes and texts of lecture notes online, so there is no need to sneak into classrooms.

 

[umbertonap]

Sorry if this is the wrong section to post this, but I didn't find anything more appropriate. I'm asking for advice because I seriously need help. I'm self-studying mathematics (at a university level; unfortunately, I can't attend for various reasons) and in the future, I would like to become a researcher in pure mathematics. However, I'm afraid I've developed a bad habit and a bad mindset. I believe that to become a researcher, it's important for me to be able to solve as many problems as possible on my own, and to do this, I always try not to look at the problem solutions (even though I have access to them). The problem is that by doing this, I often get stuck for a long time on a significant percentage of the problems I tackle, and if I eventually give in and look at the solutions, I find myself demotivated because I would have never come up with that idea on my own. The fact that I make mistakes or fail to reach solutions within reasonable timeframes is seriously making me doubt my abilities. I wanted to ask for your input to understand first of all if this is normal, if any of you work as a mathematics researcher and have had the same problems (but still managed to secure a position in academia), and most importantly, if I shouldn't feel inadequate if, at some point, I look at the problem solutions. My biggest fear is sabotaging myself and risking not fulfilling my potential because I rely on problem solutions to move forward. Thank you for the time you'll dedicate to me.

It's great that you're studying mathematics on your own and want to become a researcher. It's normal to get stuck on problems and have doubts. Even professional mathematicians face challenges.
Looking at problem solutions can be helpful when you're stuck. It's not a sign of failure or being inadequate. It's a chance to learn new approaches and improve your skills.
Math research involves collaboration and learning from others. Asking for help or looking at solutions is okay. It shows you're willing to learn.
To improve your self-study I used to try and find a balance between solving problems on your own and seeking help.Join math communities or study groups to discuss problems with others.Practice problems from different areas of math.Reflect on your problem-solving process and think about different approaches.Celebrate your progress, even small steps.Remember, success in math isn't just about solving problems alone. It's about resilience, curiosity, and learning from others. Enjoy the learning process and keep pursuing your passion for math!

 

[khansaheb]

solve as many problems as possible on my own, and to do this, I always try not to look at the problem solutions (even though I have access to them)

Read the problem statement carefully and then try to solve the problem - first time before looking at the solution.

If fail to solve it time - look at the solution all the way to the given answer.

Now close the solution and your first attempt - and work all the way to the answer. If you cannot get the answer - open the solution manual to analyze and understand the reason for your missed attempt. After that close the solution manual - start the solution anew.

If you go through this process - you will become an expert in this topic. It will take some time to progress - but you will be in total command of those topics.

 

[Steven G]

Many years ago I've done something similar. I wasn't as civilized and didn't even ask for permission, just sat in the back of the classroom through the whole semester without ever being questioned. And this was a class by one of the best researchers and teachers in the field in one of the best universities in the world. I have no regrets about attending but still feel somewhat guilty about not asking. Could my volunteering at this forum be a subconscious attempt to make amends ?

P.S. This days lots of colleges post videos of classes and texts of lecture notes online, so there is no need to sneak into classrooms.

Did you really do that? That is a great story. The colleges I went to and the colleges I taught at never had more than 25 students in a class (as a undergraduate student my advanced math classes only had between 5-10 students) so it was hard to sneak in for the whole semester. Please tell me that these were lecture size classes, otherwise I will have to change my opinion of you!

 

[blamocur]

Did you really do that? That is a great story. The colleges I went to and the colleges I taught at never had more than 25 students in a class (as a undergraduate student my advanced math classes only had between 5-10 students) so it was hard to sneak in for the whole semester. Please tell me that these were lecture size classes, otherwise I will have to change my opinion of you!

Not sure how this will effect your opinion, but I'd guess 30-40 students in the class.

 

[Steven G]

Not sure how this will effect your opinion, but I'd guess 30-40 students in the class.

...and the professor never realized that you were not formally in the class! That is a amazing!
I already forgave you!