How Not to Write Proofs

\begin{document} \maketitle \section{Introduction} Learning on how to write proofs is part of the professional training of a mathematician, where (s)he will teach at the shool, community college, college or university level. It is part of the intellectual formation and not necessarily part of for-the-job-training. Like playing the piano, coaching Little League, or brain surgery, this skill is as acquired more by imitation and practice than by wrote learning. So, this page is not a general essay on the subject. It is a collection of some common decificiencies on homework in this course. \subsection{ Example 1} Here is a fragment of a proof which is essentially correct, but the fragment is full of "gaps" (see the previous lesson on proofs.) a+b+c = 1 and a is constant so
C' = aA + bB = aA + (1-a)B
B' = aA + cC = aA + (1-a)C
Therefore, (B'C' is parallel to (CB).
From the last line (which is missing a parenthesis), the reader can tell that the student planned to prove that the line, call it $h$ where $a= const$, is parallel to the line $(BC)$. But there is way too little documentation to warrant this conclusion. \begin{itemize} \item Line 1 is acceptable \\ \item Line 2 does not follow. Indeed the first equality is incorrect, because there is no reason to believe that $a+b=1$. You must qualify with something like this, either in words or symbols \\ \item Line 1.5: The point where the line crosses a side, $C'=h(AB)$ satisfies \\ \item Line 2.5: Similarly \\ \item Line 4 is a \textbf{jump to the conclusion} and should be justified something like this \\ \item Line 3.5: Therefore $(B'C')$ has displacement vetctor $ C' - B' = (1-a)(B-C)$ \\ \end{itemize} The \textit{moral of this story} is as soon as you see the proof, imagine your dumb brother not quite following you. \subsection{Example 2} Another case is missing algebra steps. Sometimes the steps are crucial, so if you skip them the grader can't be sure if you're skipping steps because you think they are obvious or because you don't know how justify them. Reassure the grader that you know by some documentation. After the grader knows you are an A student, you can get away with little leaps. Until you fall into a gap, of course. Then the grader becomes suspicious. A' - A = \frac{bB+cC+(b+c)A}{b+c}
A' - A = \frac{G-A}{b+c}
As all of you will be geometry teachers someday, either professionally, or, as parents helping your kids with homework, you should appreciate the gap between these two line. Missing is, of course, the algebra $ bB+cC+(b+c)A = bB+cC+(1-a)A = aA+bB+cC - A = G-A$. The temptation is huge to be done with the problem and just claim the conclusion after some stabs at rewriting the hypotheses. On a test, this is an easy mark when the alleged justification leads to a different numerical answer than is claimed in the last line. But in geometry, we prove things, not calculate numbers. So it's the logical soundness of the steps that count. \end{document}