We often understand something only after we've had to teach or explain it to someone else. Learning-by-teaching (LBT) systems exploit this phenomenon by playing the role of "tutee." BELLA, our sixth-grade mathematics LBT systems, departs from other LTB systems in several ways: (1) It was built not from scratch but by very slightly extending the ontology and knowledge base of an existing large AI system, Cyc. (2) The "teachable agent"--Elle--begins not with a "tabula rasa" but rather with an understanding of the domain content which is close to the human student's. (3) Most importantly, Elle never actually learns anything directly from the human tutor! Instead, there is a super-agent (Cyc) which already knows the domain content extremely well. BELLA builds up a mental model of the human student by observing them interact with Elle. It uses that Socratically to decide what Elle's current mental model should be (what concepts and skills Elle should already know, and what sorts of mistakes it should make) so as to best help the user to overcome their current confusions. All changes to the Elle model are made by BELLA, not by the user--the only learning going on is BELLA learning more about the user--but from the user's point of view it often appears as though Elle were attending to them and learning from them. Our main hypothesis is that this may prove to be a particularly powerful and effective illusion to maintain.