I just finally had time to read all the responses today.
Thank you for your input everyone.
I'm pleased to see how many of you have helped children learn our favortie pastime.

As I stated in the original post, the legal minimum for IL deer is 20 gauge, so that throws out anything smaller. My little girl will already have the shooting basics down pat by the time she hunts. If she's interested, she'll start with an airgun, then a 22 carbine, then a 22 pistol, and only then will she be moving on to bigger things, but I'm concerned about the 20's recoil.

She will also accompany me on my huntsin the ways several of you have suggested. She needs to learn the ways of the woods, how to read sign, how to move quietly, how to use the weather and wind, and needs to observe deer under a variety of conditions so that she can learn their ways too. There is no substitute for personal experience, and I'd like her to learn much before actually pulling a trigger.

I feel that all beginnerhunters would benefit from having a mentor show them the way, but I definitely want my daughter to learn things in the proper order, and demonstrate safety and responsibility on the rangewell before she walks with a loaded gun in search of game.

I am jumping the gun here a bit since she is still so young, but I want to be ahead of the game in my thinking, and to be a good resource for anyone who asks me what might work best.

In spite of the pump's lower cost, simpler manual of arms, and forced time to think before shooting follow up shots, I'm still leaning toward the 20 semi for the reduced recoil, and avoiding having her not able to cycle the action of a pump because her arms aren't long enough yet. I can limit her additional shots by simply loading only 1 shell, and keeping extras handy. Not ideal, but a good start.

It's good to know that there have been problems getting kids to pass the test. I would have thought that with proper study, practice, and well written clear questions, that it would not be incredibly difficult for kids to pass. I truly hope that the main cause for this is that the kids taking the test are not properly prepared, and not that there is a flaw in the testing.

I wish I could set her up with a nice 243 bolt adn not have to consider this so much, but the powers that be have deemed us IL residents too unfit and unsafe to hunt with any weapon that shoots out past 200 yards. [:@]

Thanks again.