I remember when I took geometry in high school, like most students it's where I was formally introduced to proofs. However, the way we went about them was strange, it really felt like symbol manipulation. We had a big list of axioms and postulates, and proofs were fairly mechanical. They were done by a table, one side was the result, the other was what you invoked to get it
Result | Method ________________________________________________ 1 Lycanthropy | Given 2 A cute bunny | Given 3 A werewolf rabbit | Flibbergoth's theorem of lycanthropic mammals on (1) and (2) ________________________________________________ ∴ Wolf isomorphism | The harewolf-lupine equivalence axiom on (3)
You can see an actual, non-silly simple example here.
However, I question whether or not this is a good way to introduce proof writing. Certainly the proofs that come out are valid (if done correctly). There's nothing inherently wrong, but even now they feel foreign to me. I don't feel like I really got proofs until I took a discrete math course in college, where proofs were more free, we could choose our own approach, and we tended to write in paragraphs. Certainly we still invoked axioms and theorems and brought up givens, but it was much less structured and, for me, was easier to understand and perform.
(Full disclosure: we did introduce more freeform, paragraph-like proofs by contradiction near the end of the course, but it was introduced as a novelty and never used on any exams or homework).
However, I am not the world. I was wondering if there was evidence to support this method for introducing proofs. Is there any evidence that this improves mathematical reasoning, or is easier for students to learn and execute than more free-form proofs? Certainly it's a lot more similar to the mechanical symbol manipulation most students have been doing up to that point, and I can see an argument that while argumentation and mathematical creativity is still involved, providing the structure to make it more mechanical should make it easier to perform given their likely education up to that point.
Still, I can't shake that I had very little idea what was going on until I got to less structured looking proofs. What does the evidence say? Is introducing proofs in a more structured format helpful? Am I just atypical? I'll grant it's entirely possible I wouldn't have even understood freeform proofs if I hadn't done the structured approach first, but I still have trouble doing the same proofs in table format that I could do easily in a paragraph with sentences interspersed with notation, so I don't know. I've always personally had a more argumentative/wordy approach to doing math (post-arithmetic at least).
Edit: As mweiss indicated, "effectiveness" isn't very well defined. I was trying to get at concepts such as "does the 2-column method help foster an ability to prove mathematical statements, or does it hold back or confuse students more than the more freeform style of proving things?" Overall, I think mweiss attempt to answer the question in terms of whether it provides reasonable structure to guide students or else acts as a straightjacket that stifles their ability to freely think about problems is what I was getting at.