Through a series of six guided classroom discoveries, students create, via targeted questions, a definition for deciding when two sets have the same cardinality. The program begins by developing basic facts about cardinalities of finite sets. Extending two of these facts to infinite sets yields two statements on comparing infinite cardinalities that contradict each other. The experiment "More circles or more squares?" resolves this dilemma in favor of the definition of "same cardinality" that Georg Cantor adopted over a century ago. (Contains 2 figures.)