When is a Proof Not a Proof.

We have all heard of pupils who “got by” in geometry by merely memorizing the proofs given in the book and of teachers who permitted them to “get by.” To such persons those proofs were not proofs. Assuming that such methods are passe if they have not passed all together, what limits are we to approach in the other direction? How long should a student be asked to hold in his memory any completed proof? Is it sufficient to remember the fact and that it has been proved? Are the proofs presented really designed to convince the student of the truth of the fact being proved? Are they the kind of proof that is really convincing to him? Is the logical proof convincing to him? Why prove to him facts that he is sure are true? We ask why he thinks that two triangles must be alike if they have two sides and the included angle the same and he replies that anybody can see that with the implication that somebody around is a “boob.” If he can see no exceptions then is it not a proof to him? Is it not more likely to prejudice him against the geometric proof than to raise any respect for it to drag him through a formal proof under those conditions?