Proof and Justification in Collegiate Calculus

Manya Janaky Raman

PhD Thesis, Graduate School of Education, University of of California, Berkeley, 2002

This dissertation is a study of mathematical proof from both an empirical and a theoretical standpoint. The empirical component compares the views of proof held by entering university level students with those held by their two types of teachers: graduate student teaching assistants and mathematics faculty. The few studies that have been done at the university level focus almost exclusively on students, and most of those study populations of either preservice teachers or students in proof-based courses. Thus this direct comparison of views held by entering university students and their teachers provides needed data to help understand difficulties students face in making the transition from high school to university level mathematics.
The theoretical component, which is deeply intertwined with the empirical one, is to explicate the notion of proof. To do so, a distinction is made between a private and a public aspect of proof. The private aspect is that which engenders understanding and provides a sense of why a claim ought to be true. The public aspect is a formal argument with sufficient rigor (for the particular mathematical setting in which the argument is given) which gives a sense that the claim is true.
For the teachers in the study, the public and private aspects of proof are connected, through what is called the key idea of the proof. The key idea is the essence of the proof which gives a sense of why a claim is true and which can be rendered into formal rigorous argument. In contrast, for the students, the private and public aspects appear disconnected-in part because the students do not recognize the key idea of the proof, and in part because they do not even realize that the public and private aspects should be connected. It seems, then, that an emphasis on key ideas (including the understanding that they are eminently mathematical) may be an important mechanism for helping students develop a mature and epistemologically correct view of mathematical proof.