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.