My nine year old started lessons a couple of months ago too. Think we are up to eight lessons now. She can trot her school horse around the arena, around obstacles and over trot poles. She's learnt hunter seat and sitting trot too. Hasn't cantered yet but is keen to give it a go!
What I like though is that her horsemanship has come along in leaps and bounds. From knowing nothing, she can now saddle up, adjust her stirrups, rug the pony and care for her gear. She has also been taught the names for parts of the horse, about sand and colic, and worms etc. She's also been taught how to do some ground work and a few random things like how to re-gain a lost stirrup at walk and trot...
I'm a little surprised she hasn't been taught to canter yet, but I'm sure it will come in time :)
Muriel, did your daughter have any horse knowledge before her lessons? What else will she be getting taught?