Captain Miller's thought process in Private Ryan "School Teacher" scene?

It has to do with the separation of officer and regular soldiers. He does not get involved in this type of situation. That is what his sergeant is for (hence Mike glances at Tom hanks and then intervenes).

However, Hanks is a sensible, mature officer. He realises that he has to do something to resolve the situation, and decides to connect with them and explain why he is doing what he is doing. Normally officers have to keep their distance, and do not become personal friends with their men as it might compromise the command hierarchy and influence the officer's judgment.

This was a big step to take, and showed the men that Tom Hanks was not immune to the frustration and stresses of the mission. He was sharing it with them, and they all needed to work together.

I think I am going with the process that Captain Miller was just thinking a lot, mainly about getting home to his wife and back to his monotonous (but interesting to him) job as a school teacher. Because of this he was not quick to respond to his sub-ordinate actions.

The reasoning I am using to support this is the ending scene where he faces the tank. Logically he knew he stood no chance he stepped in harm's way. This was a decision to finish the mission of keeping Private Ryan alive with the hopes he can either one day go home or fail knowing he doesn't have to carry burden of home sickness anymore.

If you recall, there was beginning to be a breakdown in the unit. Some were upset at Miller's decision to let the hostage go, as they wanted to exact revenge for the death of Wade (played by Giovanni Ribisi). Horvath (Tom Sizemore) and Reiben (Edward Burns) had just had a standoff with pistols, Upham (Jeremy Davies) was being called a sympathizer and basically everyone else was yelling at each other. The unity of the troop was breaking down quickly.

In that minute you describe, I think Miller was trying to determine how he was going to survive this mission, and get everyone back on track so he could get back home to his family.

Miller recalled that there was a bet going on about what his pre-war job was. In an effort to diffuse the situation and divert their attention, he offers up his career back home; a school teacher. He begins reminiscing about it and eventually calms the troop down and gets them focused on the task at hand; finding Ryan.