Visited La Norma end March/early April 2010, just a day visit as we were staying at Valfrejus just 20mins up the road and it's on the same 3 resorts pass.
The village is typical french 1960's built, with a few ugly blocks but the skiing is varied and interesting, although a few of the lifts (older chairs) were very slow. It covers a wide section of the main mountainside and feels like you are travelling a bit rather than covering the same runs as you sometimes feel in other smaller resorts. Lots of stuff for all grades up the mountain including good snow-sure nursery slopes near the top of the gondola. Overall, well worth a visit if you are in the area, especially on the 3 resort lift pass arrangement with Valfrejus and Aussios.
La Norma is a very friendly resort with plenty of English spoken to supplment our poor French. The ski school was great for the children: 9 and 7 and the ESF piou piou club for the three year old was very patient.
Given the resort's compact size there was very little walking to be done, which is important with three children. The coverage is only 65 kms but it is all good skiing with plenty of nice tree lined runs. Watch out for the black that's not rolled; it's a bit of a challange, according to my wife.
There are two short blue runs served by a drag lift that we could let the children ski on their own. You are able to see the top of the run from the village so we could watch them as we played with the youngest in the little park.
You will not spend an afternoon shopping here at La Norma, but if you have a hire car, which we picked up in Chambery Airport(1 hour away and a really straight forward drive), you can drive to Val Thorens, Vald'isere or Sestiere which are about an hours drive away. The snow at La Norma was excellent, but most resorts were this year. It is a very family friendly resort with some decent skiing.
We were recently disappointed to find that via Orelle we could not get into Val Thorens so instead we ventured a few more minutes up the road to the delightful village of La Norma. We had an excellent afternoons skiing here - tree lined mostly as well. Some excellent snow conditions too. Great for families.