## Monday, October 21, 2013

### Teaching without a net

As a grad student I was teaching the linear-time Median finding algorithm and I FORGOT
that I needed to solve the more general problem of selection. After less than a minute
of trying to see what was wrong I told them
I am sure that Median IS in linear time. I will consult sources  and redo this tomorrow.
I then did the rest of the lecture (which didn't require knowing the Algorithm for Median) and the next day I did the linear Median Finding Algorithm correctly.

Note that I was teaching well known material. So I KNEW that what I was saying was true even if I couldn't  prove it. I also KNOW that I could look it up. I was TEACHING WITH A NET.

When I taught Grad Algs a few years ago I sometimes didn't quite know how the PROOF went  BUT I knew that the STATEMENTS I made were correct, and the algorithms and proofs were out there. In one case I emailed the original author with a subtle point I was stuck on. (It really was subtle- the author himself had to think about it). TEACHING WITH A NET

Last semester some of my Ramsey Theory course was taught WITHOUT A NET. Not in termsof the statements of theorems, but in my attempt to find easier proofs of theorems--- sometimes my alleged proof DID NOT WORK. And there was no book I could consult, nor person I could ask, to help me out on these new proofs''. One of my attempted simplifications (of the Canonical Ramsey Theory) DID NOT pan out in the end.

This semester  I am teaching an honors interdisplinary course on Fair Division (nicknamed 'Cake cutting'). I've pulled material from a  variety of different subfields (math, CS, AI. Yes AI!). So I have put some things together that are new''(not worth-publishing-new but new in some sense). Some of them have been wrong, or to be more fair, not quite right. But WHO CAN I ASK? Nobody! This is truely teaching WITHOUT A NET. I have made about 2 incorrect statements (both of which were prefaced with this might not be quite right') but the bigger effect is that every day I wonder if what I am saying is correct.
The effect on the actual course is mininal-- but my mentality going in `will I make a mistake today that I cannot recover from'' is... interesting.

What to do if you are wrong? Own up to it ASAP. Every minute you fumble around you lose the classes interest.

Is the course working? I think so-- they are learning and having fun. It helps that they are honors students who chose to take this course.

1. I don't understand what you are worried about.

If I am not mistaken one of the main techniques of teaching, is by setting an example.
By teaching without a safety net you are setting a very good example especially if you make a mistake once in a while (unless you are like me and when you try to prove something you get it right on the first atttempt. Sadely this is not a good example for mear mortals, since it scares them away).

I think the fact a teacher makes a mistake once in a while actually makes the lecture more exciting. It seems that people remember things better when something exciting and related happens, maybe it is some primordial trait.

2. Actually I'm not worried, the course is going well and I don't want to fumble TOO much.

I have seen profs use a CARICTURE of what you are saying as an excuse to not prepare.

I fully agree that its good for students to see mistakes--- when math is FIRST discovered the proofs are informal, not quite right. When it gets polished up it loses some of the excitement of discovery. I am, albeit not quite intentionally, restoring that.

3. I'm not sure I understand: you are working out the proofs in advance of class (I assume?). So if you have the proof worked out in advance then your working assumption should be that it is correct. (Of course, it is always possible that you find a bug while teaching, but this should not be happening every time!) If you *don't* have the proof worked out in advance, then I think it's insufficient preparation, not something to be emulated.

4. I do work things out in advance BUT since these are NOW proofs they are sometimes
incorrect. This is rare- but the fact that it COULD gives every lecture the POTENTIAL to have something not work, and not be fixable.

5. One of the more interesting courses I had was a 1st year Calculus. Our instructor was a distinguished and horribly overworked astronomer, who jumped in to substitute the assigned instructor. He obviously did not have time to prepare: he would come to class, ask us what was the last thing we did, and start proving the next theorem. He wrote out a hypothesis (like, "assume both second order partial derivatives of a 2-variable function exist") then proceeded to prove the theorem (in this case, that the two are the same function). He would jump back between proof and hypothesis (like, "well, assume for the moment also that they are continuous") and continue with the proof, with occasional stumbles. He would go back and erase some of the hypothesis when it seemed they were necessary--sometimes reintroducing them later, when it seemed like they were needed. At the end he would go back and try to reduce the number of hypotheses, and offer alternative proof strategies that would require fewer assumptions.
While this could be confusing, and a bit of an adventure (the "no net" feeling, as viewed by the students). It really gave a feeling for what hypothesis was used where, and more importantly, a feeling of how a mathematician thought. I thought it was great--some students HATED it....

6. What was the subtlety and who was the original author or is it a secret?