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To introduce myself: I'm an undergraduate mathematics student in Germany. Currently I'm studying in the second semester and until now I'm doing well, but I still got the feeling that my ability to develop proofs (or just solving complex issues in general) needs to be improved.

Let me explain myself: Sometimes, (especially) when it comes to proof tasks I'm being given, I feel like I dont know how to find my way through the proof, it rarely happens that I not even know where to start. Somehow I'm missing the feeling that my thoughts guide me to the correct answer, which I always used to have in school. This makes me feel a lack of "creativity" or in other words the ability to find my own path to reaching the solution.

If there is any advice you could give me, I would appreciate that.

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Skill at formulating proofs, just like an art or a craft, depends on the number of techniques you have at your disposal. Practice practice practice and you'll learn new techniques and equivalencies the more you do. –  Ataraxia May 16 '13 at 1:04
This is certainly an important question, but I think it would be more feasible to answer if it were made more specific in some way. Is there some difficulty you are having that is atypical in some way? –  Trevor Wilson May 16 '13 at 1:05
That being said, one piece of advice would be to make sure you understand the definitions thoroughly before trying to prove complicated things about them. Sometimes this means you have to make up simple questions about a definition on your own and then answer them before moving on. –  Trevor Wilson May 16 '13 at 1:08
It may not be a lack of creativity you are suffering from, but probably a lack of experience. Not everyone can generate ideas like Terry Tao can on the fly, but I think (and many, including probably Terry Tao will agree here) that to get better at generating them the best way is to DO LOTS OF MATH. Read lots of books, do lots of proofs, read the masters à la Abel. If you want to be creative you'd better know what's already been done forwards and backwards. Let hardened experience make up for any perceived lack of "creativity." –  kigen May 16 '13 at 1:48
Every time you see a proof that you don't think you could have come up with on your own, read it over and think carefully of a one-sentence hint that you could have given yourself that would have helped (the simpler, the better). Do this hundreds of times and you will soon have plenty of brainstorming material for any occasion that you're stuck. –  Erick Wong May 16 '13 at 2:04

1 Answer 1

up vote 5 down vote accepted

I remember having a similar feeling in my first undergraduate year. I could comprehend the proofs I was taught, and could mimic them afterwards on very similar problems, but I felt I lacked the creativity to actually think of proofs myself. Two realizations I had helped me get through this stage:

  1. Practice actually helps. The more you prove, the more tools are added to your inventory, even if this is not immediately apparent. In two years from now, you'll look back and may not even understand what you found difficult.

  2. Each of these proof "tools" was developed by someone very smart, generally over long periods of time thinking about the problems at hand. If you managed to come up with all the tricks and techniques by yourself, without first seeing some similar examples online or in books, you would indeed be a genius.

In short, you shouldn't be feeling too bad about not "getting" proofs immediately - this does not mean you aren't creative, just that you have more to learn and that things that are presented as trivial actually took quite a while to get to. One could almost say that this is what university is for - to save you the time it would take to reinvent everything yourself.

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Helpful answer, I especially like what you mentioned in the last sentence: learning at university is indeed a fast process, especially when it comes to mathematics, where nobody is able to "know everything" (considering that even professors are highly specifiedin certain fields). –  Josh May 16 '13 at 7:00

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