A three-dimensional classification system of mathematical problems is given. A background philosophy is constructed through an analysis of different approaches to problem-solving research, leading to the statement of 5 theses and the objectives for the study. An exploratory study of mathematical problem-solving processes is conducted within this framework. "Think aloud" sessions of students were videotaped while solving problems from incidence geometry. A category system is developed to form data from these observations. Correlation-methods and sequence-analysis techniques are used to analyze the data. Hypotheses are generated concerning systematic behavior (strategies), problem-solving success, modes of representation, mistakes and problem-solving types. (MP)