Methods of
Mathematical Proof,
Part I:
Elementary Methods
CHAPTER 5
As undergraduate students of mathematics pass through the sophomore to
the junior level, a major change in their mathematical career occurs with
the seemingly sudden emphasis on the need to understand and, especially,
to write proofs. Indeed, the most common refrain heard by instructors of
undergraduates in courses such as abstract algebra, advanced calculus,
number theory, and linear algebra, is "I understand the material, but I can't
do the proofs."
From one point of view, it is not surprising that students find proof
writing difficult. After all, mathematicians themselves spend a lifetime ab-
sorbed in the attempt to discover and prove theorems, a pursuit involving
the combination of expertise (sometimes genius), effort, and occasional luck,
which is the basis of all scientific discovery. For these professionals, a
major part of the challenge, and of the beauty of mathematics itself, ema-
nates from what is at the core of the difficulty that most undergraduate
students experience. Namely, there is no formula for writing proofs; writing
even simple proofs involves a certain degree of creativity as well as uncer-
tainty at the outset of where the process will lead. Furthermore, there is
generally no unique correct answer to a problem calling for writing a proof.
For an undergraduate student, then, an assignment that involves writing
a proof is bound to produce anxiety, in contrast with the security most feel
when working, say, to differentiate a function or find the inverse of a matrix.