Attitude towards exercises in mathematics I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too difficult to solve. I spend quite some time to think about them. When I fail, I google to find the solutions. Most of the time, I get the solutions.

However, there are some downsides to this attitude. First, I would spend too much time on a single problem. So I feel that my progress is a little bit too slow. Other than that, when I read a solution, I don't get the real understanding of the problem. When I read a proof, my brain is working mechanically to check every statement, so I don't know what exactly is going on.

I start to wonder whether I'm doing this correctly. What I want to ask here is, what should I do when I encounter difficult exercises? Should I think about them myself until I get the answers? Or should I skip the difficult ones and move on to the next chapters, then go back after I gain more understanding?
It depends on the textbook. However, most authors like to include a few exercises that they feel will challenge even very good students (and there is at least one respected author who deliberately includes open research problems among his exercises), so it's not necessarily grounds for panic if you fail to solve some of the exercises.

It's always a good idea to at least try all of the exercises. Some authors use the exercises to present useful facts (typically "prove that such-and-such", and sometimes entire well-known concepts that are peripheral to the main development are defined only in an exercise); trying to solve it will help you remember the result later even if you didn't succeed in proving it.

If you get stuck, however, just leave it be and carry on -- unless it's a majority of the exercises for a chapter or section that fail you. Return to them later, and if they still seem interesting and hard even then, you can consider asking for help here.
Mathematical concepts or theorems is like a tool, it will be awkward using it first but with time &amp; practice, the better you become! 

I was a 4th year math &amp; stats when NCSA Mosiac, Gopher and Kermit was still around (read: 2400baud modem). No googling for answers and we relied on Teaching Assistant or the smartest guy in class for help (sorry, very few gals). My own recipe for approaching a question is:-


Take a theorem and apply it to the problem, does it fit? 
If not,take another theorem and try fitting it 
What is the result of applying this theorem?  
Repeat from 1) until you are satisfied with the answer


Exams test how well verse you are in applying these concepts, and you only have 3 hours in the finals to show-off. It's like Lego, only that the blocks comes in different shapes and sizes and each term, the throw in new blocks just to confused you.