Proof schemes and proof writing

Proof is a notoriously difficult concept for mathematics students, even for those with good mathematical backgrounds. One difficulty with proof is that students often have inadequate proof schemes, a term describing the types of argument students use to justify proof. This dissertation examined the proof schemes used by students in introductory proof writing courses at six elite liberal arts colleges. The study compares the proof schemes used at the beginning of the semester and at the end, and investigates how students' proof schemes have developed over the semester. The effect of proof schemes on proof writing abilities is also examined. Data is collected using questionnaires, supplemented by interviews. The questionnaires and interviews also ask about students' study habits and beliefs about proof, in order to examine their effect on proof schemes and proof writing abilities. Quantitative and qualitative analysis techniques are used to examine the relationship between the proof schemes students find convincing, the proofs they write, and their beliefs, attitudes, and practices of proof.; This study reveals some challenges that students face when learning to write proofs, including conflicts between the proof schemes that students find convincing and the proof schemes that students use when writing a proof, a lack of understanding of the need for mathematical proofs to be based on axioms and definitions, and evidence that students may "ritualize" proof writing techniques that they are taught, hindering their ability to write proofs. Despite these challenges, most students were able to successfully write a simple proof when responding to the questionnaires. Some evidence shows that students can improve their ability to write proofs somewhat by presenting their proofs to their peers for critique.