Teaching

I was thinking about teaching math, and I think I know part of why it is hard to do well. You essentially have to operate on a variety of levels of abstraction simultaneously.

At the most abstract level you want to keep in mind the formal reasons why what you are teaching is true. You then need to decide which parts of this, if any, will help the students.

Next you have the generalizations and applications of the concept. Again you have to be careful what parts of this you share.

Then you have the intuitive picture or pictures of what you are currently teaching. The hand wavey big picture bits that make them feel like they are doing something real.

At the same time you have to keep in mind where this topic sits in relation to the other topics. What is the flow of material? Along these same lines, how do you use this topic to build problem solving skills?

At the bottom you have the explicit process/algorithm you are trying to get them to memorize. How do you chunk the steps together?

Below this, in the basement where the students live, you have pure syntax/arithmetic issues. How do you write the solution down? Which basic calculations do you show?

Outside of all of this, you want to somehow convince them to actually pay attention to the material and try to understand it. Do you use humor? Do you use outside videos?