Should you "correct" the poem or story your child writes at home? My answer has to be a vociferous NO! "Correcting" the creative writing a child brings you may very well destroy the child's incentive to write.
When you ask a child to write a poem and then criticize it, the child may feel that you have betrayed him. You asked for a poem, but you didn't appreciate its beauties; instead you found fault with its commas and spelling.
It's bad enough that parents keep harping about the loudness of the stereo and the condition of bathtubs, kitchen counters, and fingernails, but when a parent also begins to plump the innermost thoughts of the child -- the best ideas which he or she can commit to paper -- the child may very well feel that the effort wasn't worth the hassle.
Because writing has to come out of one's "secret" places, your child does you a great favor if or she writes a poem or story and is willing to show it to you. It is a real gift, a part of the self. If you start tearing this gift apart, you may be doing a great disservice to your child as well as causing him to dislike writing.
If you are fortunate enough to get your child to do some writing, remember that the first reaction must be appreciation. Always find something in the work to praise. Say how much you enjoy reading it, that you appreciate the time and effort that he or she spent on it.
Commend the good ideas; comment on the interesting use of sounds; compliment the interesting use of language.
Even if misspelled words give you the shivers, and disagreement of verbs induces palpitations, my advice is: Don't "correct" thier work.
Remember that ideas are far more important than correct comma placement.
Innovative use of language is far more important than correct spelling.
Although correct spelling and grammar have their virtues, the child will never know where the true emphasis lies in good writing if you concentrate on technicalities.
Eventually you will discover that you do not need to correct. By praising the good qualities of the writing, you are accenting these. The child will give you more of what you praise and will not be inclined to try again the parts which are not praised. As the child continues to write, the good will crowd out the not-so-good. At least, this has been my experience. For example:
If you praise metaphors (you may want to call them comparisons), your budding author will write more and better metaphors.
If you praise humor, the writing will get funnier and funnier.
If you praise imaginative language, he or she will stretch the imagination.
If you commend alliteration, the child will use it again, and most likely will use it better as he as more practice.
The parent who is seriously interested in getting children to write should try some creative writing himself. If adults take writing seriously, if it is a viable adult avocation, then it is a viable kid avocation, as well.
Besides, writing, all good writers read. No writer writes out of a vacuum. So be sure that you have books of both prose and poetry around.
Be sure that you read poetry yourself.
When you find a poem that you think the family will enjoy, read it aloud at the dinner table and invite comments. If someone says, "I think that stinks," don't dispute this critical judgment. Kids are entitled to their opinions, too. Encourage them to share poems that they have read, and don't be too critical of their choices.
Above all, don't expect perfection. Remember that the young writer is a growing individual and a growing intellect. Some things that seem obvious to you will not be obvious to the young writer. Nor should you expect that every writing effort will be sequentially better than the last.
No, don't "correct" their writing; love and appreciate it instead.