I don't know if Gus Van Sant has admitted it but it seems clear that this movie has been inspired by the shooting in Columbine High School.
Is there a link between this event and an elephant?
Why is the movie Elephant (2003) called that?
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From the IMDb link in the question:
Gus Van Sant borrowed the title from Alan Clarke's film of the same name, and thought that it referred to the Chinese proverb about five blind men who were each led to a different part of an elephant. Each man thinks that it is a different thing. What Clarke's title actually referred to was the idea of the "elephant in the room", where something is so obvious that to miss it would be the equivalent of not seeing a huge elephant in an ordinary room, yet is still not recognized out of either stupidity or willful ignorance. In this film, the "elephant in the room" is the homicidal rage of Alex and Eric, which leaves them free to precipitate the last-scene massacre at their school.
I always thought it was because of the treatment of elephants in the circus. Time and time again these elephants are abused, beaten, and whipped, which creates an inevitable breaking point and they end up stampeding over everyone. Much in the same way, these pariahs who have been bullied far too long finally explode...
It's an homage to this film: http://www.screenonline.org.uk/tv/id/439410/ Elephant is without question Alan Clarke's bleakest film. Essentially a compilation of eighteen murders on the streets of Belfast, without explanatory narrative or characterisation and shot in a cold, dispassionate documentary style, the film succinctly captures the horror of sectarian killing.
The lack of narrative removes any scope for justification of the killings on religious, political or any other grounds and the matter-of-factness of Clarke's approach debases the often-heroic portrayal - by all sides - of the individuals involved in sectarian murder. Moreover, Clarke's use of a Steadicam to follow the killers before and during the murders casts the viewer as at best a willing voyeur, at worst an accomplice. After each killing, the camera dwells on the bodies slumped on floors or draped over desks for longer than is comfortable, forcing the viewer to confront the brutality of their deaths.