Children traditionally learn through imitation. They track adult movement, and try to duplicate it. As such, children are able to learn without a lot of formal instruction or clear descriptions of how a process should proceed. It's actually very convenient for parents, because outside of trying to explain complex situations, children pretty much learn their social interactions on their own.
But that isn't true with autistic children. Instead of tracking and imitating a process, they are more interested with audio-visual stimulation, such as a sound that accompanies a movement. By circumventing the traditional or typical learning method for social interaction, they become less social and less verbal. Think of it as looking at an animation. Autistic children do not find excitement with a movement that is not accompanied by sound, and so they need both the sound and the action to have their attention captured.
So what is a parent to do with an autistic child? They need to teach the child how to interact in the world socially, and that means working at skills that most children learn through imitation.
Communication is a big step, requiring the parent and care-giver to use alternative methods of communication, such as picture exchanges or sign language. By adding a tactile action to a verbal communication, autistic children can better pick up the communication skill.
Another is social interaction, like eye contact. Here you need to add an action to a verbal request. I've found that tickling my son helps by drawing his attention, and then I will put my face up close to his, touch his nose, and look him in the eye. He then eventually started to keep eye contact with me, expecting the same action to happen again. It worked well for my son, and something similar may help your child.
And finally, writing. While some children can just start to duplicate the writing method by watching a parent, in order to reach your autistic child you need to explain the process of writing to them, or say the letter while helping them write. My son does well with the letter names, though his Preschool teacher suggested that we try using the same description method that they use, for continuity. Both ways my son is learning to write his name, and is already starting to learn how to spell.
The best part of the article I read regarding this study was the last phrase:
"There is nothing in our research that in any way conveys a sense that children [with autism] are any less human, any less deserving of our love and respect, or any less of anything at all. It is that the way they seem to learn about this world is rather different than the strategies used by their peers. By better understanding how they do this, the better we will be able to reach them, and like in any personal relationship, the better they will be able to reach us. ... Their different perspective might give us solutions that others, with the typical mind and brain, might never see."
This is too true! Autistic children are not stupid because they don't learn the same way other children do. They are not creatures, subhuman beings that should be approached with fear. They are people, just like everyone else, but people that learn differently. It's more work to teach an autistic child, but it's also a lot more fun!