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How do you deal with the feeling of not being smart enough for math?

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This is your first year as a bachelor's student in math. The way you go about mathematics changes as you move from computational classes to proof based classes. It takes time to adjust, and don't worry if it takes you some time.

Any time you get a grade that you aren't happy with, don't take it as not being smart enough. Look at it as a sign that you need to adjust how you are studying. It's not just enough to study "a lot," how you study matters too.

Try strategies like spaced repetition. Implement study schedules like the Pomodoro method. Look at each proof, and don't just study it line by line, but divide it into "chunks," which can become techniques for new problems.

It's not about being smart, it's about learning from your mistakes and getting better.

The smartest guys I knew in undergrad, who I thought were going to be stellar mathematicians, all decided to do other things when they graduated. One went into teaching in Alaska, and another got into drugs and grass roots campaigning. I think of my undergraduate cohort, I might be one of two or three people that went on to be professors.

I feel that I'm a smart person, but I wasn't as smart as some of my friends. Being smart isn't everything.

I'm curious, though. Where are you studying?
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Don't worry, I felt almost the same in my first year now doing phd anyway ;)
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I had this feeling back then when I was a freshman. In fact, I failed my Real analysis paper. Now, I'm starting from scratch. What I am doing now is to focus more on understanding math and developing curiosity and interests or "passion" one may say, and try not to get set back by exam marks. Currently, working well for me. Maybe you could give it a try! All the best, stranger.
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I did the exact same when I first touched Real Analysis. I felt I had a great grasp on it all and I understood the content from a "broad" point of view but when it came to writing proofs for specific prompts I would often lose little marks everywhere for not being "rigorous" enough.

In the end it was my attitude towards improvement in the unit that let me score a Distinction in the end - a grade some would be disappointed by but I was proud of.

Keep your head up. Study with others. If you dont give up youll do well :)
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I'm a working mathematician (\~postdoc level), and I've published a few math articles. This feeling never went away for me, but it *did* take on a different flavor as I obtained more "mathematical maturity" somewhere during my Ph.D.   


Particularly, I don't believe I'll ever be one of those "peerless, far-seeing mathematical geniuses" that push the boundary of their discipline further than anyone else, let alone someone who opens up entire new rich fields of research on their own. Lots of us "mathematically grow-up" on the stories of people like Grothendieck, Poincaré, Von Neumann, Erdös, etc. (there are many such stories, and they're glorified in popular books like Bell's "Men of Mathematics"), so naturally I saw these people as idols and something to strive toward. Eventually, reality hit and I really beat myself up for not being (basically a prophet) like them.  


I eventually took solace in noticing that, more often than not (although there are a rare few exceptions), these peerless geniuses kinda generally *sucked* at expository writing and communicating their ideas to other people (even to other specialists within their own field, to say nothing of fledging grad students looking to learn their techniques, and you can definitely forget about explaining to laymen).   


So, my compromise with crushing imposter syndrome became: "if I can't be someone who can commune directly with math gods like these people, I can at least work on being someone who can bring the tablets down to mere mortals like me". I'm not super smart and I'm not exceptionally clever or quick, but I'm stubborn and I love this shit and I \*want to understand\*. So that keeps me going.
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This feeling get's me every so often. First thing i can recommend is staying commited and keep doing what you do. Things will become clear eventually.

Once you have a little time under your belt, about a year will work i guess, think about where you are now in your understanding compared to where started. Is there a difference? A quite big one i would guess.  That is all you can ask for but you can influence it yourself.

A lot of feeling "not smart enough" comes from the feeling of being the one that knows the least. But that is to be expected since you are there to learn in the first place.

I feel intimitated myself when i talk to professors, phd students, postdocs etc because they come up with ideas and solutions to things i struggle with seemingly effortless and on the spot.

But they have been where i was and have a lot of time on me, so it is just natural. Once i had the same time the difference wont be as big anymore.

To be clear: I do not claim to get where they are or talk someone's effort, work, dedication down.

Give yourself some slack, you know more about your struggles than about anyone else's. Talk to other students and you wont feel so left behind.

There will be tests where you are glad about the mark you got and those where you are not. This will happen no matter how you prepare.

Had oral exam in prob.theory. Lectures where great, i  was able to talk about proofs in detail while there and anticipated where many of the proofs would go etc.

My mark was/is a 2,7 (C+), 1,0 is the best 5,0 the worst. That still annoys me, but i accept it as the impression of the professor of the 30 minute "talk/quiz" we had that afternoon, not of me being a student overall.

Could go on but will leave it here, wrote already more than intended.
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I've got a math PhD, and I still feel this way more often than not. It's just a matter of time before the rest of you figure out I don't belong!

While I've phrased that as a joke, it's true. I do feel this way pretty much any time I sit down to do research and don't make significant progress. Don't let the Imposter Syndrome get to you. It's a frightful monster that comes hunting for pretty much everyone at some point or another.
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I have recently finished my first year Math PhD coursework/TA duties. I would like to explain my background a bit and then explain why you should not be worried about your experience(s).

I was a nerdy kid and did many math competitions, etc. until high school. I nearly failed out of high school due to excessive partying and laziness (you get the gist) and proceeded to work for my family’s business for 3 years, entering college at the age of 21. I started my first year as an English major, and quickly rediscovered Math and my love for it. As an undergraduate, I found the courses doable and not so challenging until I encountered Real Analysis. Why was it so hard? Here is my answer:

Your first encounter with conceptual or proof-based mathematics will always be challenging. Most curriculums are “banking methods” - they deposit information, you return that information on an exam, you almost certainly forget the previous deposit and the cycle continues, and this happens for each course (my experience). Basically, you solve well-defined problems given an explicit procedure that will do the trick... then, Real Analysis. Multiple solution methods, having to identify the question being asked, understanding intuitively what from your toolbox will be needed to solve the problem - the average student, even in Mathematics, will have a hard time making this leap. The one simple fact is that time and discipline are a mathematician’s best friend. One can cram for a computational exam, but success in higher level math (in my experience) is directly correlated to the consistency with which you study the material, i.e. 30 minutes per day beats 3.5 hours once per week.

The truth is, a Math PhD is not reserved for the gifted (that helps). Rather, it is reserved for the individual who can consistently struggle through challenging material and self-doubt. Believe in yourself, maintain balance between your studies and hobbies (important), and you will develop into a promising mathematician. It is helpful (I think) to equate the PhD pursuit to professional sports... THINK ABOUT THE TIME IT TAKES TO REACH THAT LEVEL. Do not be upset that your progress does not fit into your timeline, progress in every endeavor is nonlinear. Do something good for your brain and body every day, dull the negative self-noise, and I think you will be proud of yourself a year from now. Feel free to reach out if you wish to speak further.

P.S. You’re a young person on r/math, that’s saying something.
by
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To answer the question directly... in my opinion the best way to deal with the feeling of not being smart enough for math is to simply study mathematics and display to yourself that you are perfectly capable of understanding the concepts. What does this look like? I will give you an example, but keep in mind this can appear in many different forms (and how it will appear for you may be different from me).

I took introduction to advanced mathematics last summer and went through the entire course feeling like I did not understand anything and I would never be able to. It was a crazy semester for me (40 hours of research, 3 summer classes) and I felt a lot of shame that I overloaded myself and didn't give myself the time to digest the concepts and enjoy studying mathematics. Fast forward to this summer - I started reading my abstract algebra textbook for the fall semester and as advised by the preface, reviewed appendix A to ensure my foundational understanding was sufficient. So far I've spent the summer writing up over 100 pages of notes reviewing what I previously learned last summer, and everything is clicking... like I've had some sort of awakening. I'm reading through the text, thinking about the concepts, working through the suggested exercises, and even coming up with my own problems. Previously I was very scared that I was not "good enough" for math and physics... after this experience I now feel that anyone who is willing to read, think, take notes, and get stuck on problems is "good enough" for anything. Given sufficient prerequisites understanding and a high quality textbook, I feel I can learn anything - math, physics, whatever.

Now just to comment on your situation a bit:

There is this notion of "mathematical maturity" that is hard for me to explain in a simple manner, but is essentially something you acquire over taking many different math classes. In my opinion you are doing very well for someone who lacks a math background and therefore "mathematical maturity" acquired from experience. Regardless of experience, real analysis is one of the hardest math courses undergrads take, so the fact that you did "average" is very acceptable. Over time I feel that you will find ways to study more efficiently, perhaps by focusing more on understanding definitions, doing more proofs for practice, or whatever your point of improvement may be.

The fact that you care about understanding the material will be your saving grace - take care of yourself, and enjoy the journey - welcome to the club!
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There are a lot of topics in math and you’ll probably find one or two in particular that come more naturally or that you find more fascinating than the others. Or it could be an applied field. Keep going, with an open mind.

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