This paper presents the qualitative analysis of data from a 20-year project analyzing the knowledge and attitudes toward science of undergraduate students enrolled in introductory astronomy courses. The data were collected from nearly 10,000 students between 1989 and 2009 via a written survey that included four open-ended questions, inquiring into students' knowledge of scientific inquiry, DNA, computer software, and radiation. Trends in students' responses were arranged into concept maps that depict patterns in student thinking. Students' responses were also compared with criteria established by a sample of scientists. Students were familiar with empiricism in science and understood that science tries to explain the world but were not as attuned to the need to support arguments with evidence as scientists would expect. Students had a narrower conception of DNA, yet often related a blend of accurate and inaccurate ideas. The accuracy of students' descriptions of software increased over time, though they were more likely to approach software from a consumer rather than computer science perspective. Students attended overly much to the dangers of radiation, and the accuracy of responses decreased over time. This research demonstrates that students' ideas about science are less focused than scientists would like. (Contains 7 figures.)