In this paper we consider prior definitions of the terms "ill-defined domain" and "ill-defined problem". We then present alternate definitions that better support research at the intersection of Artificial Intelligence and Education. In our view both problems and domains are ill-defined when essential concepts, relations, or criteria are un- or underspecified, open-textured, or intractable requiring a solver to "recharacterize" them. This definition focuses on the core structural and pedagogical features that make problems and domains ill-defined while providing a consistent and functional frame of reference for this special issue and for future work in this area. The concept of ill-definedness is an "open-textured concept" where no single static definition exists. We present the most suitable definition for the present goals of facilitating research in AI and Education, and addressing the pedagogical need to focus learners on addressing this ambiguity. (Contains 1 footnote.)