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This paper reports on an exploratory study of mathematicians'views on the teaching and learning that occurs in a course designed to introducestudents to mathematical reasoning and proof. Based on a sequence ofinterviews with five mathematicians experienced in teaching the course, I identify four modes of thinking that these professors indicate are used by successfulprovers. I term these instantiation, structural thinking, creative thinking andcritical thinking. Through the mathematicians' comments, I explain thesemodes and highlight ways in which students sometimes fail to use them effectively. I then discuss teaching strategies described by the participants, relatingthese to the four modes of thinking. I argue that teaching aimed at improvingstructural thinking tends to dominate, and that courses that introduceproof, regardless of classroom organization, should address all four modes in abalanced and integrated way.

Mathematicians’ perspectives on the teaching and learning of proof