One aspect of story structure is the beginning, which sets up a situation before a conflict occurs. Teach this to kindergarten students by having them come up with their own "Once upon a time" beginnings as a group game. Sit your students in a circle. Have one student say the first sentence of a story or fairy tale, such as, "Once upon a time there was a kid named Joe." Go around the circle with each student saying the next sentence of the story. Continue until you have a complete and funny story.
All stories contain a climax, the point of highest tension or conflict. In children's stories, the climax tends to be a scary point, where the protagonist is most in jeopardy. Get students thinking about climax by having them identify the scariest part of a story and act it out in groups. Help groups as they try to find the scariest point in a story you have read aloud. Allow them to present their dramatization of the climax in front of the rest of the class.
Much like the beginnings of children's stories, endings tend to be conventional and happy. Get your kindergartners to think about the way stories end by having them imagine what happens after "happily ever after." Read a story to your students and ask them to draw a picture of what the main character does after the story's ending. Doing this will help your students realize that endings are arbitrarily constructed, and that there are any number of ways that a story can end.
One of the simplest ways to get young students thinking about story structure is to ask some basic questions. Make this into a group game for your kindergarten students. Read the first paragraph or page of a story aloud. Ask your class who the characters are, where they are, when the story is set and what the characters are doing. Eventually, you should be able to call out "who," "what," "where" or "when" and have students raise their hands. Give a point to the student who raises his hand first and answers correctly.