How to do proofs in your head

Question

I think I have a problem. If I am asked to do a proof, no matter how simple, I need to use a pen paper. I think I'm fairly quick but if someone asks me to do the same proof in my head it takes me significantly longer. I am an undergraduate student and often in class I see people who are able to do proofs and answer questions incredibly quickly without writing anything. It is well known that there are tricks that allow you to perform computations with large numbers incredibly quickly in your head. Are there similar tricks for doing proofs?

Answer 1

As in so many things, my recommendation is to practice.

Take a problem, try to visualize it, and see how far you can get doing it mentally. When you are stuck, write down what you have done so far, and either continue on paper or continue mentally.

The more you get accustomed to keeping ideas and chains of reasoning in your head, the better you will be.

Historical note:

Charles L. Dodgson, better know as Lewis Carroll, who was a quite good mathematician, did this often. He collected the problems, he solved which he called "pillow problems", in a book which is still available:

http://www.maa.org/press/maa-reviews/the-mathematical-recreations-of-lewis-carroll-pillow-problems-and-a-tangled-tale

It is an enjoyable read, and I highly recommend it.