The current document is directed to methods and systems for monitoring a dental patient's progress during a course of treatment. A three-dimensional model of the expected positions of the patient's teeth can be projected, in time, from a three-dimensional model of the patient's teeth prepared prior to beginning the treatment. A digital camera is used to take one or more two-dimensional photographs of the patient's teeth, which are input to a monitoring system. The monitoring system determines virtual-camera parameters for each two-dimensional input image with respect to the time-projected three-dimensional model, uses the determined virtual-camera parameters to generate two-dimensional images from the three-dimensional model, and then compares each input photograph to the corresponding generated two-dimensional image in order to determine how closely the three-dimensional arrangement of the patient's teeth corresponds to the time-projected three-dimensional arrangement.