How to explain proofs to audience smoothly? I have given 5 to 6 presentations till now (on abstract algebra topics ). Many times to me it appears that most of the people in the audience have not understand the most of the part. Some people ask me to give intuition of the theorem and proof, which is difficult to me. Some people from the audience told me to be smooth while proving the theorem. I think smoothy means next step of the proof should appears to linked from previous one. My audience comprises of masters and Ph.D student of computer science and mathematics. > Question : How to explain proofs smoothly?

I totaly agree with eepperly16 comment. One other way to make the proof more concrete is to find one or two examples and work on them simultaneously with the proof.

For example , if you want to prove any theorems on Group theory, you can always illustrate what you actually do working on $\mathbb{Z_7}$ or something less trivial like the dihedral group.

Anyway my point is, use as many examples as possible, so the audience can work the proof on their head with given examples.