If you want to kill the urge to write, simply go through all your child's written material and "correct"--of course that means "mark as wrong"--every misspelling, every punctuation error, every grammatical mistake
Do that, and you will probably find that "encouraging" your child to write becomes like pulling teeth. Writing is very personal, and everyone--even professional writers--take such corrections personally. Never is this more true than when the writer is young, and lacks confidence.
You can protest all you want that "children have to learn to do things they don't like," and that's true enough. But that doesn't mean that academics in general, and specifically writing, are the proper place to press this point. In all academic areas, pressure is likely to be counterproductive. That is, the harder you push, the more the student comes to dread and dislike the subject. And that means you'll have to push even harder the next time, which increases the resistance, and so on and on. This is a battle you don't want to fight, because the results never look anything like "winning."
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First of all, ask who the written document is intended for. . . This can produce several positive results.
If the document is just a story, if it was written solely from the student's own desire for self-expression, then don't make any changes that are not specifically requested. Don't risk stifling the student's joy of self-expression.
If the document is for Grandma, you might say, "Grandma will have an easier time understanding it if you . . ." and suggest just the minimum changes for that purpose. Some misspellings, and some punctuation/capitalization errors are a part of the charm.
If it's a request for a catalog, or other information, more changes will be necessary. If it's for the mayor, the governor, etc., still more changes are needed.
I'll have more to say about Unschool Writing, but this is a good start. Make writing a cooperative effort to help the student communicate, rather than an exercise in syntactical perfection, and you and your student are more likely to continue enjoying the fun of writing.