Proof schemes and learning strategies of above-average mathematics students

What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove a conjecture.