Before I move on to the main idea of this post, I need to tell you some background information about myself. Hopefully, it proves useful for you in giving me advice. I'm a 16 year old high school student who just recently got interested in mathematics a couple of months ago. I was never interested in mathematics in elementary school and middle school. However, all of a suden in the middle of my high school journey I got interested in it.
Now back to the point.
Currently my mind is in a state of conflict! I do not know how I should self-study mathematics!
My original approach to self-studying math was to merely solve problems that caught my interest. For instance, I would search online (like this http://maaminutemath.blogspot.com/ ) or in a book for a problem that caught my attention. Then I would do my best to solve it. I would play with it like a toy. I would even try to create my own problems similar to the problems I solved. However, I realized this approach leads me to a lack of foundation. In other words, I have gaps in my knowledge!
Because of this worry, I set myself a goal to focus on filling my gaps and building foundation. I bought an Art of Problem Solving book called Introduction to Algebra because of this. After awhile on working on the problems in the book, I got bored. The problems weren't challenging or interesting. It was the usual find x and applied to some word problems. In fact ,I already did algebra before in middle school, yet I felt like I should have continued to work on the problems no matter how dull to fill my gaps. UGH! It's so fustrating.
Afterward, my mind diverted to something else. It was the Putnam problems like these:http://www.math.niu.edu/~rusin/problems-math/MP1975.html .I wanted to know how to solve them! They were very mysterious to me (and still are) because I do not know how you solve them, yet I want to solve them. So, I bought a book on how to form proofs since the Putnam problems involve prooving stuff. I wanted to know how to solve these problems, but I lacked foundation(and still do). I did not know where to start(and still do). I did work with the book I bought but eventually I reached that same experience as the Introduction to Algebra book.
For some time, I think I burnt myself out, and I quited solving problems. Currently my head is in chaos with all this overwleming information in front of me(the internet and masses of books at library), yet not knowing where to start or where to end or am I researching too much on this problem that's way above my head like the Putnam.
How do you guys self-study mathematics, especially you very experienced mathematician out there? Do you force yourself to solve dull or uninteresting or easy or unchallenging problems in order to have a firm understanding of things?
Please mathematicians with great experience I need some guidance, tips, or advice. I did ask my math teacher for advice but she said not to worry, which is not the kind of advice I was looking for.
Addendum
I'm sorry it took me time to respond to this thread. I was sorting thoughts in my head. Now I decided to post my thoughts as it might be helpful for others.This might not be the finest addendum for this thread as it seems to me to be fragmented, but I did my best shot in writing it. Another thing, I may sound egotistic or cocky in this. But remember these are my thoughts, and my thoughts could change in the future.
First I'd like to thank you guys for responding to this question. Your guys replies are informative and useful. Nevertheless, I think there is no best answer in this thread, although I already chose one based on how agreeable it is to what I'm saying. I think it's fair to say that advice given in mathematics(or in other fields) is complicated. Why? It's because everyone is different. Some piece of advice might work for one person but at the same time might not work for the other. In the grande scheme of things, the world is too complicated.
Currently I'm ignoring all advice because I think the best advice comes from oneself. In addition, I have been keeping myself an intellectual journal. Whenever a negative emotion appears in my mind, I describe it in my journal. Then I do my best to replace that emotion with a positive emotion. In other words, I redefine how I see things.
For instance, if I approach a math problem that causes me anxiety, I would find a way to change that emotion to a positive one. Suppose I find that other emotion that I want to use to replace the negative emotion , and it's the emotion of assurance. Specifically,change my negative emotion so I can think that something good will happen. I would say to myself, "although the problem may be tough or frightening, if I do my best to solve it even though I might fail, I will gain better insight into the problem than if I didn't attempt to solve it." I know this is cheesy. But this method has worked for me for the past two days. It's practical for me.
In conclusion, this how I'm approaching mathematics in a general sense:
First of all when I do mathematics, I should just do it and approach it. Secondly, I should treat mathematics like a game. It shouldn't be extremely serious. Curiosity should be my driving force in solving problems . Plus, I should be more playful to gain that "likening/love" as a motivation with the problems even if they're boring, dull or easy. However, if the problem takes too much energy and time, I'll think of this nursery rhyme:
"For every problem under the sun, there's a solution or there's none. If there be one, think till you find it. If there be none, then never mind it." That last bit of the quote I do not take 100% serious. If I put the problem off in the side, I won't put it off forever. Instead when the time comes when I acquire the necessary knowledge(either be it a month or year), I will get back to it.
Lastly, I should ignore all given advice on how I should go about doing math(I know that sounds cocky or arrogant). But I am I and they are they. What works for me may not work for them and vice verse. Another thing, is that I found that I was too focused on getting advice in mathematics instead of actually doing mathematics. Here's a quote that can perhaps add to what I mean:
"It's one of the best theories that when people give you advice , they're really just talking to themselves in the past."- Austin Kleon
You probably got the whole point of my strategy; I'm redefining the way I approach things and getting rid of the negative emotions in my head. This strategy is what I have been doing for the past two days. As of now, this strategy is still working for me.
Again, I want to emphasize these are my thoughts, and my thoughts are pretty messy. Do not take it all too serious what I wrote as an overall character of me. Feel free to critique what I wrote as well. I just wanted to share it as it might help other people.
Best Answer
Do the things you like. If something you're doing isn't fun, stop, and do something else.
Mathematics is hard enough that if you are doing it it should be because you enjoy it. There is no point in doing it to become a mathematician if you don't enjoy it; you'd be in the position of someone who is trying to become a rabbi, but isn't Jewish, or maybe someone who is wants to become a professional violinist but doesn't like playing the violin. It's not just silly, it's insane.
If you find some topic uninteresting, skip it. Perhaps someday it will become interesting to you, perhaps because you need it to solve some problem you want to solve, and you can learn it then. Or perhaps it never grabs you; so what? You aren't going to run out of things to learn. (Or if you do run out of subjects that seem interesting, perhaps you should take that as a sign that you don't really want to be studying mathematics.)
If you find you're burned out and you don't want to do any more problems, stop and do other things for a while. I would not be surprised if you found that that after a few weeks off you discovered you were no longer burned out, and you wanted to start again. At sixteen, a few weeks or months seems like a long time. But when you look back on it, it seems a lot less important. When I was a teenager I loved programming the computer and I did it all the time. But I burned out on computer programming when I was sixteen and stopped doing it for a year and a half. Then one day it was interesting again and I went back to programming. My 18-month vacation did not prevent me from having a productive career as a programmer, or from becoming a well-known expert on computer programming. Do I look back on my life and wish I had that extra 18 months, do I wonder if I wouldn't have been more successful with 39 years of practice instead of only 37½? Of course I don't. Instead I look back and it seems that taking a year-and-a-half vacation from programming was most likely the right thing to do at the time.
But suppose you did burn out and didn't want to solve math problems ever again? What then? All I can think is "So what?" The world would have to do without your mathematical contributions. That's okay; you can find something else to contribute instead.