As the only airbender aang had he would have to responsible for learnig everything aang new about airbending and even what he did not know. It would eventually be up to Tenzin to train the next avatar and evey future airbender afterwards. That would require a lot of training and time.
It would have made sense to me for him to incorporate Air Nomad culture in his parenting to all three of them. Tenzin, being an airbender would obviously make him a little more central to reviving the Air Nation, but that is part of all of their heritege. It was somewhat surprising to see this in his parenting, seeing as while he was always kind of flakey, he was also thoughtful, and it would seem like it would occur to him that this might cause some resentment and all around, just be unfair and even detrimental to the cause.
Remember when he was teaching Meelo to train Momos? Meelo said: "It's hard to be the alpha Momo."(sorry, i forgot the name of this animal) and Tenzin says: "I know."
Being the only airbender among his kids Aang must have put a lot of pressure in young Tenzin. Their vacations were probably only training and studying all the time. That's the reason i believe Tenzin is so severe in his way of teaching.
Not saying Aang was a bad father, but he eventually became something he didn't like.
Yeah, I mean all three of them could be remembering incorrectly for all we know. Going somewhat off topic, was anyone but me surprised that Tenzin actually enjoyed riding elephant koi and everything? I mean, Katara said he was serious and stern even as a kid. It didn't sound like something he'd like.