Common Errors in Undergraduate Mathematics (2009)
These are primarily for algebra mistakes. Teaching a proof based course, the most frustrating error I saw was people doing a fine proof but in reverse (this is related to the article&#x27;s point about irreversible steps). If I wanted them to prove that the square of any 
 number was nonnegative, they&#x27;d start by assuming that x^2 &gt;= 0 and then driving equations from that until they got something obviously true like 0&gt;=0. This was the proof written &quot;upside down&quot; if every step were reversible. They should have just taken the last step and started from there until they derived their first step.<p>So how was I supposed to grade that? It showed a significant misunderstanding but also often literally had every step necessary for a correct proof. Usually I just wrote overly long notes about how they should write proofs in the future that I&#x27;m sure the students never read.<p>The other fun one was unnecessary proof by contradiction. They&#x27;d start by assuming not P. Then they&#x27;d prove P directly, not using that assumption. But they, that contradicts the assumption of not P. Hence not P is false, so then they conclude P. Nothing was wrong there, they just had a couple extraneous sentences that overcomplicated things.