Having struggled to find cost-effective ways to project and record the iOS (iPad, iPhone and iPod touch) screen, I was delighted  to discover Reflection.  

This is a MacOS X app that receives the mirrored image of an iPad screen via AirPlay.  That, of course, is wireless so the peripatetic teacher can move freely throughout the classroom while mirroring their iOS screen to a big screen in that classroom via a MacOS X computer. 

As well, one can use screencasting software to capture the iOS screen that presents  itself on a MacOS X screen.  I did a quick demo of the work one can do with Reflection here.  I used ScreenFlow from Telestream but QuickTime Player will capture the screen at zero cost.

The challenge that remains is to be able to represent touch gestures.  This is not possible at present but Apple could enable the transmission of touch events as part of the video stream.  I'm imagining a toggle-able AirPlay preference to send touch event data to the receiving application.