How to explain proofs to audience smoothly?

Question

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?

Answer 1

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.

Answer 2

You should be able to at least approximately know how much an "average" person that comes to listen to your presentations know about the topic(s) that you are going to present. Then you should try to prove (if you really must prove results during the presentation(s)) what you are aiming to prove by the methods and intermediate results that are suitable for an "average" person attending your presentation.

If they ask for "what is the intuition behind some result" then you should prepare yourself in such a way that you master enough knowing of concrete examples that are special cases of something you are going to prove. If giving of a concrete example does not explain in enough detail "how did someone came to the result that you are proving" then you should be able to know some history of how ideas that you use originated in the first place and to explain in enough detail a path that led to them.

When they told you to be "smooth while proving the theorem" , I think that what they´ve meant is that your presentation of some result should go by some path that is as natural as possible and as precise as possible and as non-chaotic as possible. When I write non-chaotic I mean that, when presenting proofs, it is perhaps the best not to jump from one to another concept or intermediate result in some messy way but to explain everything in a way that makes it crystally clear.

Answer 3

First of all there are different parameters to give a good talk. If some people are sleeping, then, it shows that you are fairly poorly presenting your topic.

There are two points that I would like to mention. The first is that you should aim at an average person attending your talk. When you rehearse your talk, you should be able to be sure that this person will definitely understand what you are talking about.

The second point is actully what I want to add to the other answers. It depends on the talk. There are two cases. First, you prove something and the main thing is not how you exactly proved it; rather the main ideas you used in your proof(s). The second version is really about your proof and how you really proved it, for example: "a short proof of Fermat's little theorem". Even in this case, a presenter should not talk about the details. Just imagine someone is giving a talk and he/she is defining alot of notations and in every page some formula is derived from another one (for some reason?).. Even if the presenter is great, this is very difficult to follow for the audience. If the audiences want to learn exactly how it is done, they will read the paper or the handouts. Therefore, the presentations should focus on the main ideas and make it interesting for the others.

In conclusion, presentations should be perfectly clear. The proofs should talk about high level objects and ideas. The number of equations and notations should be as little as possible. Every slide should be linked to the previous one. If possible graphical explanations should be preferred. It is also a good practice to rehearse it and get some feedback from your colleagues, before you actually give the talk.