Diffgeometer1 on Nostr: yes, it’s definitely fine to take different roads in proving a mathematical result. ...

yes, it’s definitely fine to take different roads in proving a mathematical result. It only strengthens the belief in that result.

However, it’s also good also to point out to the student that there is no logical flaw using algebra. I’ve seen that algebraic calculation that leads to the conclusion that 2 = 1, a contradiction and students become distrustful of algebra. However, that’s also a good learning experience because the contradiction is an indication that the student made a false assumption at some point during the calculation. Contradictions can’t happen in reality so there has to be an erroneous step somewhere in the calculation and sure enough in the calculation that leads to 2 = 1 the error occurs because the student is dividing by a-b which is only possible if a-b is nonzero. However, since we got a contradiction by assuming a-b is nonzero, we must conclude that the assumption that a-b is nonzero is false.

The student should never believe a result is true simply because the teacher says it. The student should believe that a result is true only if it can be demonstrated to be logically sound. If the student has a distrust of algebra and says “hey, algebra failed. It lead me to 2=1”. The teacher should say “well let’s take a look at your work and let’s see if all your steps are sound.”

After explaining why algebra didn’t fail, it’s fine to consider alternate proofs. If \(0.\bar{9}\) and 1 are different, then there has to be a nonzero number between \(0.\bar{9}\) and 1. At this point, the teacher can ask the student “well, what is that nonzero number?” This argument should be checkmate for the student’s refusal to accept that \(0.\bar{9}=1\).