Skiing with kids in Innsbruck?

We are planning a skiing trip to Innsbruck in February, and considering
whether we should take our 7 year old with us. He is not a bad skier,
but rather slow (it won't take long before he says the same thing about
me , and I don't mind to ski with him for a day or two, but not the
whole week.

This is our first trip to Alps and I have no idea what services are
available for kids in European ski resorts and particularly in
Innsbruck. Are there full day/half day ski schools, how expensive they
are, do i need to make reservations long before? Any other information
you could share?

Thank you,
simon...