Originally Posted 09052012
Like any teacher, I find myself repeating a few items to students. Sometimes in class, sometimes out of class. Here is a list:

Half a proof is no proof, but progress is progress! You just need to reframe your results.

Mathematics is divided into two pieces for each person. There is the stuff you understand, which all seems easy. Then there is the stuff you don’t understand, which is all really hard. There seems to be a hard barrier between the two parts. Our job is to move the line a little bit every chance we get.

It is considered rude to use the word “obviously” when writing or speaking mathematics. It might seem obvious to you because you have been thinking about it for three days. It might look really hard to someone who hasn’t thought about it.

Mathematics is just as much a social construct as anything else. It somehow consists of the way in which we have learned to work effectively and to communicate to each other. We have our own standards for making arguments, and those change depending on your audience. When you are speaking or writing, you have to know who is in your audience.

Sometimes your brain does mathematics unconsciously when you are not paying attention. Your job is to seed your brain with as much useful data as you can when you are actively working. When you reach a frustration or exhaustion, let the problem go and see what your mind can come up with when you are doing something else.

That is awesome. What else can you do? What is next?

I intentionally put you in the position to get stuck on a hard problem. How can I help you find a way to get unstuck?

It only looks easy when a professor does it because they put in years of work that you didn’t see. Mathematics takes time.

Wait. This is [soandso]’s proof. We’ll talk about your idea next. First let’s evaluate this work.
Those are the ones I use the most. Teaching in an variety of IBL styles, I have to do a lot of managing expectations and psychology.