What is the best way to develop Mathematical intuition? [closed]

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the intuition to do so.

I can say that a big part of mathematical intuition comes from experience. That being said there is no "best way". I like to think about experience like this:

You're stuck in thick, dense jungle like the amazon rain forest. You have a machete, and you're chopping down plants struggling to find a way out. You walk around headless, knowing not even of a path to the nearest village.

However now say you are flying in a helicopter with a reasonable view around. Now not only can you see where you would be in the jungle, but can also see where the nearest village is. You can see the terrain, map out a path, avoid crossing big streams, etc.

This is the analogy: In the first situation you are stuck with how to move, in the next one with a lot of experience you can see many connections and the way out.

Now we come to your next problem. You say that when going through exercises they ask for proofs. What I would recommend if you have never ever seen a mathematical proof is to go through an article like this one.

Of course it is not possible to grasp these techniques at once. You should try out a few suggested exercises, look at how things are done first, look at how other people do things. Why do they think like this? Slowly after a while, your mind will get used to it.

I am willing to edit my answer and improve on it if you provide more detail on the specifics of your difficulties.

Here is a true story. One year a student questionnaire on my first year analysis course wrote: "Professor Brown gives too many proofs." I thereupon decided that next year the course would have no theorems, and no proofs. It would however, have "Facts" and "Explanations". (I got this idea from an engineer!) Part of the deal of course would be that an Explanation should explain something. Another point is that unlike the good old days when Euclidean Geometry was studied at school students have no previous experience of the words "Theorem" and "Proof". Also the study of Grammar has gone as well, so they are unfamiliar with the structure of language.

What is called a "proof" is really an explanation but written very carefully.

In other second year lectures which needed an explanation of a particular case of $A \subseteq B$ I asked the class "What is the first line of the proof?" and then, to be written further down on the board, "What is the last line?". By the end of the course, they got the idea!

Also students need training in writing carefully. See a course called Ideas in Mathematics I gave.

I also found that the best way to develop intuition was to write things out very carefully, and explain them to others. When you have written something out 5 times, you may see how to improve it a bit. Then another bit. And so on.

The idea that a proof is found one step after another is just not so. For 9 years I had an idea of a proof in search of a theorem. It took that long, and lots talking and writing, to piece together the gadgets needed to make the idea of the proof actually work and prove a theorem.

The composer Ravel said that you should copy. If you have some originality, this may appear as you copy. If not never mind! Your new ideas also may only appear at the 5th copy, as the wheels of the brain start to unclog. In fact it has been said that Newton was an inveterate copier!