In the research described in this document, the goal was to develop definite theoretical characterizations of understanding in a specific domain of problems. Some performance that provides persuasive evidence of understanding was chosen, and hypotheses about cognitive structures and processes that cause the performance to occur were developed. Performance was analyzed on two tasks in high school geometry, developing proofs and checking proofs. For developing proofs, thinking-aloud protocols were obtained from six students who were interviewed approximately once a week during the year they studied geometry. Proof-checking problems were given to 15 different students during interviews in November and May, and a group of 15 college students also were given instruction on proof checking to ascertain whether it could be acquired. Two characterizations of knowledge that can be adopted as objectives of instruction resulted, one on structural understanding and one in understanding the principle of deductive consequence. (MNS)