Some Beliefs about Proof in Collegiate Calculus
Manya Raman
UC Berkeley
In this study I compare some of the beliefs that
students and their teachers hold about proof in the context of collegiate
calculus. Proof, interpreted broadly,
refers to any sort of justification that an individual finds convincing. The subjects were five first semester calculus
students and five teachers at a large research university. The subjects were asked to prove that the
derivative of an even function is odd. Then, in an experimental approach similar to that used to
study beliefs about proof in secondary school (Hoyles & Healy, 1999), subjects were asked to evaluate five different
responses to this question. These
responses included an empirical approach (looking at functions of the type y=xn
for n from 1 to 6), a graphical
approach (the slopes of the tangent lines at x and –x are opposite),
and a formal proof using the definition of derivative. For each response subjects were asked
questions such as: Is it convincing?
Why or why not? How many points
would this receive on an exam? While
students and teachers had similar evaluations of empirical and formal
approaches, they differed on their view of the graphical approach. This difference, I claim, stems from
students' fragmented understandings of derivative and implicit messages in the
curriculum about what constitutes a good mathematical explanation.
Reference
Hoyles, C., & Healy, L. (1999). Justifying and Proving in School Mathematics (End of Award Report
to ESRC ). London: Institute of Education, University of London.