In order to obtain a detailed, personalized answer to a question, a survey or essay may request a narrative response. How long or detailed that response should be is usually left up to the responder.
A narrative, whether it's an answer or an essay, is a story written from a distinct point of view. This point of view is usually the author's.
A response is usually an answer to a specific question or series of questions. A response paper or response essay, however, can be written to counter another paper or essay.
If you combine these two terms (narrative and response), it can easily be understood that a narrative response is an answer to a question or a series of questions that is written in the form of a story from a specific point of view.
A narrative response is meant to give the author the opportunity to relate their own experience to the question being asked. A narrative response also gives more information about the person writing it than just a fill-in-the-blank-response might.
Narrative responses can be written in either the first or third person. First person is easier, and the third person "this author" can seem dated.
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