For one perspective, see the book The Open-Ended Approach: A New Proposal for Teaching Mathematics, edited by Jerry P. Becker and Shigeru Shimada (National Council of Teachers of Mathematics, 1997).
Links (not all interpreting "open-ended" the way Becker and Shimada do):
Akihiko Takahashi web site
University of Michigan
A searchable collection of open-ended questions