My dc started violin in September, we did open strings until maybe about February, and then started with first finger until now. She is just about to start with a second finger. Tbh, she gets a much better sound out of open strings than with fingers, which gives her more enjoyment and confidence. 6 lessons isn't very long at all and I'm sure the teacher knows what they are doing, but will no doubt explain the teaching methods if you ask.
It depends on the method really but I wouldn't expect her to have started fingers yet. Ask the teacher - they won't mind! Violin moves very slowly at the start and you have to be very patient, it doesn't have as instant results as other instruments. The teacher will explain it all if you ask. Have fun!
dd has been learning for around 5 or 6 lessons now and is bowing open strings. I think she's doing well (but I would) and I'm wondering about how long dc stay on open strings before adding the LH fingers? Don't want to ask the teacher for fear of sounding pushy!! Just curious.